Complex population dynamics: A theoretical/empirical synthesis
Abstract
Why do organisms become extremely abundant one year and then seem to disappear a few years later? Why do population outbreaks in particular species happen more or less regularly in certain locations, but only irregularly (or never at all) in other locations? Complex population dynamics have fascinated biologists for decades. By bringing together mathematical models, statistical analyses, and field experiments, this book offers a comprehensive new synthesis of the theory of population oscillations. Peter Turchin first reviews the conceptual tools that ecologists use to investigate population oscillations, introducing population modeling and the statistical analysis of time series data. He then provides an in-depth discussion of several case studies--including the larch budmoth, southern pine beetle, red grouse, voles and lemmings, snowshoe hare, and ungulates--to develop a new analysis of the mechanisms that drive population oscillations in nature. Through such work, the author argues, ecologists can develop general laws of population dynamics that will help turn ecology into a truly quantitative and predictive science. Complex Population Dynamics integrates theoretical and empirical studies into a major new synthesis of current knowledge about population dynamics. It is also a pioneering work that sets the course for ecology's future as a predictive science.
... All parameters are positive unless otherwise stated. The functional response function G(u, v) is a crucial factor shaping various dynamical behaviors in predator-prey systems [53]. Over the past century, different functional responses have been identified based on various biological applications, including Holling type [16][17][18], Hassell-Varley type [15], Beddington-DeAngelis type [4,10], ratio-dependent type [3] and Crowley-Martin type [9], among others [53]. ...
... The functional response function G(u, v) is a crucial factor shaping various dynamical behaviors in predator-prey systems [53]. Over the past century, different functional responses have been identified based on various biological applications, including Holling type [16][17][18], Hassell-Varley type [15], Beddington-DeAngelis type [4,10], ratio-dependent type [3] and Crowley-Martin type [9], among others [53]. The prey growth function F(v) typically takes the form of a logistic type ...
... (Type III [53]), (1.3) where λ > 0 represents the predation rate on prey and α ≥ 0 characterizes the level of predator cooperation during hunting, h represents the handling time per prey item, e 0 is the encounter rate per predator per prey unit time, and c is a fraction of a prey item killed per predator per encounter. When α = 0, the Type I, II and III hunting cooperation response functions reduce to the well-known Holling I, II and III functional response functions, respectively. ...
This paper is concerned with a predator–prey system with hunting cooperation and prey-taxis under homogeneous Neumann boundary conditions. We establish the existence of globally bounded solutions in two dimensions. In three or higher dimensions, the global boundedness of solutions is obtained for the small prey-tactic coefficient. By using hunting cooperation and prey species diffusion as bifurcation parameters, we conduct linear stability analysis and find that both hunting cooperation and prey species diffusion can drive the instability to induce Hopf, Turing and Turing–Hopf bifurcations in appropriate parameter regimes. It is also found that prey-taxis is a factor stabilizing the positive constant steady state. We use numerical simulations to illustrate various spatiotemporal patterns arising from the abovementioned bifurcations including spatially homogeneous and inhomogeneous time-periodic patterns, stationary spatial patterns and chaotic fluctuations.
... Relaxation oscillations are ubiquitous in real world applications including ecology and biology, e.g., the predator-prey biological model [16,36], ...
... where h c 2 ( √ ϵ) is a smooth curve in (s, √ ϵ) defined in (36). For system (15), when h deviates from h c 2 ( √ ϵ) sufficiently small, the homoclinic orbit bifurcates into two canard cycles, namely, a unstable canard cycle without head and a stable canard cycle with head. ...
... Let Ψ : △ 1 → △ 2 be the transition map following the flow of system (15). Set σ 1 = △ 1 ∩ C a,ϵ 3 and σ 2 = △ 2 ∩ C r,ε 3 , then by Theorem 3.2 in [22], C a,ϵ 3 and C r,ϵ 3 could be connected when h = h c 2 ( √ ε) which is given by (36), i.e. Ψ(σ 1 ) = σ 2 . ...
... El modelo de Leslie [4,5] se caracteriza porque la ecuación de crecimiento del depredador es de tipo logístico. Leslie asumió que la capacidad de carga ambiental convencional de los depredadores K y es proporcional a la abundancia de presas x [6], es decir, K y = nx [6,7]. En este caso, se dice que el depredador es especialista [7]. ...
... Leslie asumió que la capacidad de carga ambiental convencional de los depredadores K y es proporcional a la abundancia de presas x [6], es decir, K y = nx [6,7]. En este caso, se dice que el depredador es especialista [7]. ...
... Un aspecto importante en la interacción depredador-presa es la respuesta funcional o función de consumo, la cual se refiere al cambio en la densidad de presas atacadas por un depredador en cada unidad de tiempo [8]. En el modelo de Leslie, la respuesta funcional del depredador se expresa mediante la función lineal h(x) = qx, tal como se utiliza en el modelo de Lotka-Volterra [6,7]. Es descrito por el sistema X ψ (x, y) : ...
This paper deals with a continuous-time predator-prey model of Leslie-Gower type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment. The conditions for the existence of equilibrium points and their local stability are established. In particular, it is shown that the point (0; 0) has a great importance in the dynamics of the model, since it determines a separating curve Σ that divides the behavior of the trajectories. Those trajectories that are above this curve have as their w-limit the point (0; 0), so the extinction of both populations may be possible depending on the initial conditions.
... In physics, a classical particle or an assembly of such particles evolving over time is described by an ordinary differential equation (ODE) called a "dynamical system" (e.g., [4][5][6][7]). This concept has been extended to biology, where the "particles" are replaced by living organisms that evolve over time due to their mutual interactions and interactions with the environment (e.g., [12,13]). There are different methods to establish an equation of motion for a given dynamical system, and one of the most powerful methods is to know a function called "Lagrangian". ...
... There have been attempts to establish theoretical biology by developing mathematical models of some biological systems (e.g., [12]). Such attempts include applications of the Lagrangian formalism to theoretical biology, or more specifically to population dynamics, which plays a special role in understanding competition for resources and predation in complex biological communities, and for preserving biodiversity [13]. The main purpose of this review article is to describe the current status of the Lagrangian formalism in theoretical biology and discuss its perspectives of becoming the main method to derive fundamental equations of theoretical biology. ...
... The choice of the four population dynamics models among a large variety of biological systems was justified by the fact that the population dynamics is the key to understanding the competition for resources and predation in complex communities, which becomes essential in preserving biodiversity (e.g., [13,[85][86][87]). In recent work described in the remaining sections of this article, the Lagrangian formalism in population dynamics is established by developing methods to construct the SLs, NSLs, and NLs for the considered models. ...
The Lagrangian formalism has attracted the attention of mathematicians and physicists for more than 250 years because of its significant roles in establishing modern theoretical physics. The history of the Lagrangian formalism in biology is much shorter, spanning only the last 50 years. In this article, a broad review of the Lagrangian formalism in biology is presented in the context of both its historical and modern developments. Detailed descriptions of different methods to derive Lagrangians for five selected population dynamics models are given, and the resulting Lagrangians are presented and discussed. The procedure to use the obtained Lagrangians to gain new biological insights into the evolution of the populations without solving the equations of motion is described and applied to the models. Finally, perspectives of the Lagrangian formalism in biology are discussed.
... Ecological models of population dynamics are well studied for non-human animals (Turchin, 2003a). Beyond immediate practical applications, they are of theoretical importance in understanding how the underlying interactions can lead to stable equilibria, oscillations, or even chaotic behavior (Strogatz, 2015). ...
... Understanding the constraints of ecological models enables a more nuanced understanding of where and how to integrate the complexities of social organization and group behavior into explanations. We model the interplay between human population and soil fertility through a dynamical system comprising two equations, capable of generating oscillations, either as a limit cycle or in response to stochastic fluctuations (Strogatz, 2015;Turchin, 2003a). Our research diverges from previous studies that often represent environmental resources using a logistic growth model, coupled with varied forms of interaction with human populations (Brander & Taylor, 1998;Janssen & Scheffer, 2004;Reuveny, 2012;Roman et al., 2017) since we are explicitly interested in how resource regeneration dynamics affect systemic outcomes. ...
... Our approach differs significantly from previous works that have characterized resources using the logistic equation (Brander & Taylor, 1998;Nagase & Uehara, 2011;Reuveny, 2012;Turchin, 2003a). The main difference comes from the assumed limitation on resource regeneration at low resource levels: most previous works focus on biotic resources that are characterized by an accelerating initial growth from low levels. ...
Soil fertility depletion presents a negative feedback mechanism that could have impacted early adopters of agriculture. We consider whether such feedback can lead to population cycles among early agriculturalists, such as the boom-and-bust patterns suggested by an increasing amount of evidence for Neolithic Europe. Using general mathematical arguments, we show that this is unlikely, due to the interplay of two factors. First, there is an important mathematical difference between biotic (i.e., logistic) and abiotic resource replenishment; soil nutrients are better modeled by the abiotic case, which leads to more stable dynamics. Second, under realistic conditions, the resource replenishment process operates on fast time scales compared to attainable population growth rates, reinforcing the tendency towards stable dynamics. Both these factors are relevant for early agricultural societies and imply that nutrient depletion is likely not the main contributing factor to boom-and-bust cycles observed in the archaeological record.
... There have been attempts to establish theoretical biology by developing mathematical models of some biological systems (e.g., [12]). Such attempts include applications of the Lagrangian formalism to theoretical biology, or more specifically to population dynamics, which plays a special role in understanding competition for resources and predation in complex biological communities, and for preserving biodiversity [13]. The main purpose of this review paper is to describe the current status of the Lagrangian formalism in theoretical biology and discuss its perspectives of becoming the main method to derive fundamental equations of theoretical biology. ...
... The choice of the four population dynamics models among a large variety of biological systems was justified by the fact that the population dynamics is the key to understanding the competition for resources and predation in complex communities, which becomes essential in preserving biodiversity (e.g., [13,[86][87][88]). In recent work described in the remaining sections of this paper, the Lagrangian formalism in population dynamics is established by developing methods to construct the SLs, NSLs and NLs for the considered models. ...
... Models of population dynamics can be classified as symmetric and asymmetric, where being symmetric means that the dependent variables and the constant coefficients in the equations of motion for these models can be replaced by each other; on the other hand, such replacements cannot be done for asymmetric models. In selecting the models, we follow [13,22,32,48,49,74] and consider the following three symmetric models: the Lotka-Volterra [76][77][78][79][80], Verhulst [81,82] and Gompertz [83,84] models that describe two interacting (preys and predators) species. The interaction is represented mathematically by two coupled, damped and nonlinear first-order ordinary differential equations (ODEs) as shown in Table 1 (e.g., [86][87][88]). ...
The Lagrangian formalism has attracted the attention of mathematicians and physicists for more than 250 years and has played significant roles in establishing modern theoretical physics. The history of the Lagrangian formalism in biology is much shorter, spanning only the last 50 years. In this paper, a broad review of the Lagrangian formalism in biology is presented in the context of both its historical and modern developments. Detailed descriptions of different methods to derive Lagrangians for five selected population dynamics models are given and the resulting Lagrangians are presented and discussed. The procedure to use the obtained Lagrangians to gain new biological insights into the evolution of the populations without solving the equations of motion is described and applied to the models. Finally, perspectives of the Lagrangian formalism in biology are discussed.
... The Malthusian model is useful for short-term projections in populations with abundant resources, while the logistic model is better suited for long-term projections in resource-limited environments (Murray, 2020). However, the logistic model has its limitations, particularly in its assumption of a constant carrying capacity and growth rate, which may not hold true in dynamic environments influenced by factors such as migration, policy changes, and socioeconomic shifts (Turchin, 2003). ...
... The Malthusian and logistic growth models are foundational in population dynamics, with their assumptions, derivations, and applications extensively documented in works by Murray (2020), May (2001), Turchin (2003), Shamsul (2024), Pierre (2022), and Gomez and Michael (2022). These studies provide detailed insights into the mathematical foundations, real-world applications, and limitations of these models. ...
Accurate population projection is essential for effective policy-making, resource allocation, and sustainable development, particularly in regions with rapid demographic changes such as Taraba State, Nigeria. This study employs an enhanced logistic model integrated with the Runge-Kutta-Fehlberg (RKF) method to project the population of Taraba State from 1992 to 2050. The enhanced model incorporates intrinsic growth rate, carrying capacity, birth rate, death rate, migration, and resource limitation factors, addressing the limitations of traditional Malthusian and logistic models. Using demographic data from the National Population Commission (NPC) and Taraba State Population Commission, the model was validated against historical population data, demonstrating high accuracy with minimal error margins. The results indicate a steady population growth, reaching approximately 8 million by 2050. This study highlights the effectiveness of advanced numerical techniques in population forecasting and provides valuable insights for policymakers and urban planners in Nigeria.
... Here we present a novel approach that overcomes some of the difficulties of modelling population dynamics -in particular for mutualistic interactions. We closely follow the ideas presented by Turchin [57], namely that models in theoretical ecology should be based on general, well established assumptions (akin to axioms) from which concrete models can be derived, e.g. by mathematical reasoning. Similarly, Metz [42] argues that specific models should be embedded "in a larger class of models, some members of which connect more directly to the real biological world." ...
... Overall, these assumptions are intended to assure that the population dynamics are predominantly affected by the potentially mutualistic interaction. Since populations are not able to grow indefinitely [57], we assume that births in all populations occur logistically. Assumption v is obviously unrealistic, but commonly used and keeps the model formulation feasible. ...
When formulating a model there is a trade-off between model complexity and (biological) realism. In the present paper we demonstrate how model reduction from a precise mechanistic "super model" to simpler conceptual models using Tikhonov-Fenichel reductions, an algebraic approach to singular perturbation theory, can mitigate this problem. Compared to traditional methods for time scale separations (Tikhonov's theorem, quasi-steady state assumption), Tikhonov-Fenichel reductions have the advantage that we can compute a reduction directly for a separation of rates into slow and fast ones instead of a separation of components of the system. Moreover, we can find all such reductions algorithmically. In the present paper we use Tikhonov-Fenichel reductions to analyse a mutualism model tailored towards lichens with an explicit description of the interaction. We find that (1) the implicit description of the interaction given in the reductions by interaction terms (functional responses) varies depending on the scenario, (2) there is a tendency for the mycobiont, an obligate mutualist, to always benefit from the interaction while it can be detrimental for the photobiont, a facultative mutualist, depending on the parameters, (3) our model is capable of describing the shift from mutualism to parasitism, (4) our model can produce bistability with multiple stable fixed points in the interior of the first orthant. To analyse the reductions we formalize and discuss a mathematical criterion that categorizes two-species interactions. Throughout the paper we focus on the relation between the mathematics behind Tikhonov-Fenichel reductions and their biological interpretation.
... These fluctuations arise from transient damped oscillations or from noise, which induces fluctuations on characteristic time scales and can interact with seasonal drivers to generate complex patterns [33][34][35][36]. Consumer-resource interactions [37][38][39] and patchy populations [40,41] demonstrate similar behavior. In systems with synchronized dynamics, the only demonstrated reliable criterion for causal inference is a negative cross-map lag [18]. ...
... More importantly, the existence of transient dynamics in a time series indicates insufficient observations. There is furthermore no guarantee any natural system will reach an attractor before going extinct or that the system's dynamics themselves do not evolve [39]. ...
Infectious diseases are notorious for their complex dynamics, which make it difficult to fit models to test hypotheses. Methods based on state-space reconstruction have been proposed to infer causal interactions in noisy, nonlinear dynamical systems. These "model-free" methods are collectively known as convergent cross-mapping (CCM). Although CCM has theoretical support, natural systems routinely violate its assumptions. To identify the practical limits of causal inference under CCM, we simulated the dynamics of two pathogen strains with varying interaction strengths. The original method of CCM is extremely sensitive to periodic fluctuations, inferring interactions between independent strains that oscillate with similar frequencies. This sensitivity vanishes with alternative criteria for inferring causality. However, CCM remains sensitive to high levels of process noise and changes to the deterministic attractor. This sensitivity is problematic because it remains challenging to gauge noise and dynamical changes in natural systems, including the quality of reconstructed attractors that underlie cross-mapping. We illustrate these challenges by analyzing time series of reportable childhood infections in New York City and Chicago during the pre-vaccine era. We comment on the statistical and conceptual challenges that currently limit the use of state-space reconstruction in causal inference.
... Stochastic populations such as probabilistic multi-object systems are of central importance in diverse applications areas, such as Systems Biology [1], Robotics [2] or Computer Vision [3]. In some cases, the sole interest for the practitioner is in their global characteristics, for instance in applications where only the cardinality of the considered population matters as in population dynamics [4,5] or where refined spatial information is not necessarily required. In some other cases, all the individuals of the population can be clearly identified and thus the way the population is represented becomes less important; indeed, problems of this type can be recast into a collection of individual elementary representations. ...
... This result highlights the structure of the equivalence relation ρ and of the mapping T ν in (5). ...
A representation of heterogeneous stochastic populations that are composed of sub-populations with different levels of distinguishability is introduced together with an analysis of its properties. It is demonstrated that any instance of this representation where individuals are independent can be related to a point process on the set of probability measures on the individual state space. The introduction of the proposed representation is fully constructive which ensures the meaningfulness of the approach.
... Population cycles are prevalent in ecosystems and play key roles in determining their functions 1,2 . While multiple mechanisms have been theoretically shown to generate population cycles [3][4][5][6] , there are limited examples of mutualisms driving self-sustained oscillations. Using an engineered 5 microbial community that cross-feeds essential amino acids, we experimentally demonstrate cycles in strain abundance that are robust across environmental conditions. ...
... Many mechanisms have been proposed to explain how internally generated oscillations can emerge from nonlinear interactions between populations. Beginning with Lotka and Volterra's foundational predator-prey model 7, 8 , 20 consumer-resource (+/-) interactions have been shown to be a common cause of population cycles 3 . Other theoretical mechanisms that can generate regular oscillations include age-and stage-structure 4 , intransitive competition 5 , and eco-evolutionary dynamics 6 . ...
Population cycles are prevalent in ecosystems and play key roles in determining their functions. While multiple mechanisms have been theoretically shown to generate population cycles, there are limited examples of mutualisms driving self-sustained oscillations. Using an engineered microbial community that cross-feeds essential amino acids, we experimentally demonstrate cycles in strain abundance that are robust across environmental conditions. A nonlinear dynamical model that incorporates the experimentally observed cross-inhibition of amino acid production recapitulates the population cycles. The model shows that the cycles represent internally generated relaxation oscillations, which emerge when fast resource dynamics with positive feedback drive slow changes in strain abundance. Our findings highlight the critical role of resource dynamics and feedback in shaping population cycles in microbial communities and have implications for biotechnology.
... The Lotka-Volterra model is a simple predator-prey model that assumes exponential growth of the prey population in the absence of predation. However, this assumption is not realistic in many ecological contexts (see in page 54 of given reference) [3]. A more realistic approach is to use a model that takes into account the carrying capacity of the environment and the logistic growth of the prey population. ...
... Existence of interior equilibria Stability R 1 Unique interior equilibrium point (E 1 * ) E 1 * is unstable surrounded by a limit cycle through Hopf bifurcation curve R 2 Two interior equilibrium points E 1 * and E 2 * E 1 * is unstable surrounded by a limit cycle and E 2 * is saddle (unstable) R 3 Two interior equilibrium points E 1 * and E 2 * E 1 * is unstable spiral and E 2 * is saddle (unstable) R 4 No interior equilibrium point R 5 Two interior equilibrium points E 1 * and E 2 * E 1 * is stable spiral and E 2 * is saddle which is unstable R 6 Unique interior equilibrium point E 1 * E 1 * is stable spiral The Hopf bifurcation curve in the two-dimensional plane bifurcates the region into two sub-regions: one where the solution exhibits stable behavior around the equilibrium point, and the other where the solution exhibits oscillatory behavior around the equilibrium point. Below this curve, there are stable equilibria, and above it, we get oscillatory or periodic solutions until reaching the homoclinic bifurcation curve. ...
In this article, we have investigated the temporal and spatiotemporal dynamics of the prey–predator model with hunting cooperation. Prey–predator models contribute to the field of mathematical biology by providing concrete examples of nonlinear dynamics, bifurcations, and chaos theory. These models explain the conditions under which populations remain stable or exhibit oscillatory behavior. They help us identify factors that lead to population cycles and stability. Predator control prey populations, preventing overgrazing or overpopulation, while prey availability influences the predator’s number. Cooperative hunting plays an important role in regulating prey–predator populations. It involves predator working in groups to hunt prey, improving their hunting efficiency compared to solitary hunting. In this paper, we consider a spatial and non-spatial prey–predator model with a Holling type IV functional response and cooperative hunting. The temporal model shows different kinds of bifurcations, such as Hopf, transcritical, homoclinic, saddle-node, and Bogdanov–Takens (BT) bifurcations. After examining the bifurcation properties, we proceed to enrich our model by introducing diffusion in both one and two dimensions, aiming to investigate the emergence of Turing and non-Turing patterns. They help us in understanding spatial patterns and structures that arise in nature, such as animal territories, vegetation patterns, and the distribution of organisms. The result of the numerical simulation shows that the model exhibits different kinds of Turing patterns, such as spots, a mixture of spots and stripes, and labyrinthine patterns in the pure Turing region, while spiral pattern in the non-Turing region.
... Detailed derivations, assumptions, and applications for both models are well-documented in works by Murray (2020), May (2001), Turchin (2003), Shamsul (2024), Pierre (2022), and Gomez and Michael (2022). These resources extensively cover the mathematical underpinnings, real-world applications, and limitations of Malthusian and logistic models. ...
Accurate forecasting of population is crucial for effective policy development, resource management, and sustainable growth. This research utilizes the Runge-Kutta-Fehlberg (RKF) technique to project the population increase in Taraba State, Nigeria. Conventional models, including the Malthusian and logistic growth approaches, provide basic insights but often do not adequately reflect the complexities of actual demographic behavior. The logistic model, which takes into account the limits of carrying capacity, presents a more plausible framework for forecasting population patterns. The rate of prediction and the carrying capacity were estimated using data through MATLAB analytics, yielding values of and . By applying the RKF method, this study improves the precision of numerical solutions to the logistic model, thereby minimizing errors in long-term forecasts. By utilizing census data from Nigeria for the years 1991 and 2006, alongside projected figures from the National Population Commission, Nigeria, from 2007 to 2022, the RKF method yields dependable estimates of population trends through 2050. By 2050, it is anticipated that Taraba State's population will approach 8 million individuals. The findings illustrate the RKF method’s capability to effectively capture demographic variations and enhance the accuracy of population forecasts. This study highlights the significance of sophisticated numerical approaches in demographic research and their potential to inform policy development and infrastructure planning in rapidly expanding areas like Taraba State.
... Density dependence refers to the dependency of per capita growth rate of a population on its own density (Sinclair and Pech 1996). These sorts of non-linear feedback mechanisms prevent populations from increasing exponentially when resources are limited and are important processes in determining patterns of competition and trophic interactions (Mueller and Joshi 2000;Turchin 2003). Investigating the interaction between the population genetics and ecological population dynamics of a gene drive is essential, as complex, non-linear mechanisms may arise and in turn influence the persistence, dynamics and ecology of gene drive systems. ...
Density dependence describes the regulation of population growth rate by population density. This process is widely observed in insect populations, including vectors such as mosquitoes and agricultural pests that are targets of genetic biocontrol using gene drive technologies. While there continues to be rapid advancement in gene drive molecular design, most studies prioritise gene drive efficacy over ecology, and the role of density‐dependent feedback on gene drives remains neglected. Furthermore, the details of density dependence experienced in these potential species of interest are usually poorly understood, creating additional constraints and challenges in evaluating the efficacy and efficiency of gene drive systems, especially those that promise local confinement after release. Here, we formulate and analyse a simple, non‐species‐specific mathematical model which integrates population dynamics by density dependence together with population genetics of a high‐threshold two‐locus underdominance system. Different models of density dependence and strengths of within‐species competition are investigated alongside other genetic and ecological parameters. Our results suggest that for an underdominance gene drive system, density dependence processes, by acting on births or deaths, influence the population dynamics by leading to significantly different population‐level suppression in the presence of a fitness cost. However, density dependence does not directly affect the fitness cost threshold for drive establishment. Moreover, we find that the magnitude and range of key ecological parameters (birth and death rates) could result in different outcomes depending on the type of density dependence employed. Our work highlights the importance of considering the ecological contexts in the design, development and deployment of gene drive molecular strategies.
... Currently, mathematical models in the form of differential and difference equations are widely used in the study of complex population dynamics processes [1][2][3]. Mathematical analysis of deformations in dynamic regimes in these nonlinear models is based on modern bifurcation theory [4][5][6]. In these circumstances, answering important biological questions involves studying bifurcations such as period doubling, Neimark-Sacker, transcritical, crisis, and flip. ...
The problem of mathematical modeling and analysis of stochastic phenomena in population systems with competition is considered. This problem is investigated based on a discrete system of two populations modeled by the Ricker map. We study the dependence of the joint dynamic behavior on the parameters of the growth rate and competition intensity. It is shown that, due to multistability, random perturbations can transfer the population system from one attractor to another, generating stochastic P-bifurcations and transformations of synchronization modes. The effectiveness of a mathematical approach, based on the stochastic sensitivity technique and the confidence domain method, in the parametric analysis of these stochastic effects is demonstrated. For monostability zones, the phenomenon of stochastic generation of the phantom attractor is found, in which the system enters the trigger mode with alternating transitions between states of almost complete extinction of one or the other population. It is shown that the noise-induced effects are accompanied by stochastic D-bifurcations with transitions from order to chaos.
... Food webs are very complex in nature. If we split into basic build blocks of two species then it can be show periodic or stable qualitative behavior [1]. This qualitative behavior fluctuated by prey & predator's individual traits. ...
... Peaks in the spectrum correspond to dominant frequencies, highlighting period-icities in predator-prey dynamics. This analysis quantitatively identifies cycles and their durations, offering predictive insights into population fluctuations and ecosystem behavior over time [15,16,10]. ...
... Functional response, which describes how predator species consume their focal prey per unit time, is crucial in determining the dynamic behaviour of predator and prey populations. Two main types of functional responses are identified in the literature: prey-dependent (such as Holling type I, II, III and IV) [13] and predator-dependent responses (including Beddington-DeAngelis, Crowley-Martin, Ivlev and Harrison type, among others) [14]. Among the Holling-type functional responses, Holling type IV is characterized by prey interference with predation, where the per capita predation rate increases to a maximum threshold prey density, beyond which it starts to decline [15]. ...
The dynamics of a ratio-dependent interacting species system in space and time, concerning Beverton-Holt-like additional food for predators, are explored within the present framework. Initially, the non-spatial scenario is examined, revealing a variety of dynamical phenomena, including co-dimension 1 and 2 bifurcations in both local and global contexts, through the application of dynamical systems in biology and normal form theory. Additionally, bi-stability phenomena are observed in distinct regions under appropriate parameter values. In the two-phase mathematical analysis, the second phase focuses on the corresponding reaction-diffusion system, uncovering various rich spatiotemporal patterns. Comprehensive numerical simulations are subsequently performed to support the analytical results. The relationship between Beverton-Holt-like additional food and cross-diffusion structures gives rise to spatiotemporal complexity, as demonstrated by the simulation results that validate the behaviour of both temporal and spatiotemporal models.
... A plausible reason for sudden mass occurrences of animals is limit cycle dynamics, generated by exploitive consumerresource systems with a locally unstable equilibrium point (Gilpin and Rosenzweig 1972, Turchin 2003, Murdoch et al. 2003. Such systems are especially likely to emerge at high latitudes, where year-round active endotherms are usually dependent on a single functional group of resources during the long winters. ...
In his classical contributions, Olavi Kalela proposed that, due to the low primary productivity of the tundra, Norwegian lemmings are locked in a strong interaction with their winter forage plants. Proposedly, Norwegian lemmings respond to the threat of critical resource depletion by conducting long‐range migrations at their population peaks. A tacit premise of this conjecture is that predation pressure on the Fennoscandian tundra is too weak to prevent runaway increases of lemming populations, creating violent boom–crash dynamics. Our results on the dynamics of Norwegian lemmings on the Finnmarksvidda tundra during 1977–2017 are in line with the predictions of Kalela's hypothesis. In contrast to the Siberian and North American tundra, densities of avian predators in our study area have been low even during lemming years, and efficient ones have been lacking from lemming habitats. Lemmings have thus increased unhinged in peak summers and crashed to densities below the trappability threshold during post‐peak winters. Each lemming crash has been accompanied by massive habitat destruction. Indications of predator activity have been concentrated to productive shrublands, where lemmings have never reached high densities. Young lemmings have responded to high densities by becoming extremely mobile: they have been trapped in large numbers on islands, including a small island in the middle of Iešjávri, a 10 × 8 km tundra lake. Many lemmings have been seen swimming across the lake, and many drowned lemmings have been observed. The dynamics and behavior of Norwegian lemmings recorded by us differ radically from those of other Lemmus spp., indicating that cycles generated by lemming–vegetation interactions have two alternative states – one with and the other without intense summer predation. We propose that the cycles of Norwegian lemmings shifted to the latter state during their unique evolutionary history, when they survived the Last Glacial Maximum in a tiny refugium archipelago.
... Political parties outside India are also applying a change of mind strategy to online social network users in order to change their vote (Bhoopendra Soni 2018). When persons residing in opposite parties interact on social media, the occurrence of complexity in news dynamics may become a cause of tension among the users (Turchin 2013). As a result, identifying the pattern of disseminating contradictory information on social media and developing a system to manage complexity in news dynamics during emergencies is fascinating. ...
Multiple pieces of information regularly propagate in a social network. Different political party supporters utilize social systems not only for campaigning, publicity but also for opposing the opinions of other parties. They always try to create some agenda against the opposition. It is interesting to recognize the pattern when two conflicting pieces of information interact on social networks. Here, we present a nonlinear model of opposite information spread in a homogeneous network system. We considered two kinds of users, supporting two conflicting news stories at a time with the ability to protect their opinions from others. We obtained fixed points, their existence, and stability conditions. Here, we watch that social network system experience flip bifurcation and hopf bifurcation. We had chaos in the dynamics, which shows the uncertainty in the observation. Moreover, we suggested a strategy for controlling the complex dynamics of information spread on social networks in emergencies.
... To evaluate how the dynamics of the community abundance of each functional group were driven by climate-change-associated temporal changes in environmental variables, we performed multiple regression analyses. We assumed that the dynamics of community abundance followed a Gompertz-type autoregressive model (Royama, 1992;Turchin, 2003;Dennis et al., 2006). We used the annual change in community abundance as the response variable, and the log of abundance in year t, the temperature conditions (PC 1 and PC2) in year t, and pH of sea-surface water were treated as explanatory variables. ...
The influence of climate change on marine organism abundance has rarely been assessed (1) at the functional-group level; (2) simultaneously in major functional groups within the same ecosystem; (3) for >10 years; and (4) at metapopulation/community scales. A study simultaneously addressing these gaps would greatly enhance our understanding of the influence of climate change on marine ecosystems. Here, we analyzed 21 years of abundance data at the functional-group and species levels on a regional scale for four major functional groups (benthic algae, sessile animals, herbivorous benthos, and carnivorous benthos) in a rocky intertidal habitat along the northeastern Pacific coast of Japan. We aimed to examine the 21-year trends in regional abundance at both functional-group and species levels, plus their driving mechanisms and their dependence on species properties (thermal niche, calcification status, and vertical niche). Significant temporal trends in abundance were detected at functional-group levels for benthic algae (increasing) and herbivores and carnivores (both decreasing); they followed the temporal population trends of the dominant species. At species level, the metapopulation size of 12 of 31 species were increasing and 4 of those were decreasing, depending on the thermal niche and species calcification status. At both functional-group and species levels, temporal trends in abundance are caused by the direct or indirect influence of warming and ocean acidification. Comparing these results with community responses to marine heat waves in the same study area offered two implications: (1) long-term ecosystem changes associated with global warming will be unpredictable from the community response to marine heat waves, possibly owing to a lack of knowledge of the influence of calcifying status on species’ responses to climate change; and (2) thermal niches contribute greatly to predictions of the influence of warming on population size, regardless of the time scale.
... The interplay between prey and predator is crucial in preserving the ecosystem biodiversity. Mathematical Ecology is the branch of science in which dynamics of various interacting species with their surroundings is studied using analytical approaches to identify the species persistence as well as diversity (Turchin 2003;Roth 2016). These interactions may be different types likely: between individuals, populations with organisms and their environment. ...
In this paper, we explore the Leslie–Gower type prey–predator model with a Holling-IV functional response, examining both deterministic and stochastic environments. In the deterministic analysis, we establish the positivity and boundedness of solutions, as well as the stability criteria for various equilibria and different bifurcations, including transcritical, saddle-node, and Hopf bifurcations. For the stochastic component, we demonstrate the existence and uniqueness of global positive solutions and identify conditions for persistence in the mean. Additionally, we derive the stationary distribution and probability density function for the stochastic model. We conduct a stochastic sensitivity analysis by approximating the confidence domain, showing that the size of the confidence ellipse is influenced by the level of noise intensity. When the confidence ellipse intersects the separatrix, critical transitions or tipping points may occur. In such instances, the system may not revert to its previous state depending on the intensity of fluctuations. Most theoretical findings are supported by numerical simulations.
... Studies of small rodents have contributed greatly to our understanding of population dynamics (Stenseth 1999, Berryman 2002, Turchin 2003. In particular, geographically distributed long-term series have provided opportunities for macroecological studies , Kendall et al. 1998, Boonstra and Krebs 2012, Cornulier et al. 2013, Ehrich et al. 2020. ...
Long‐term studies of cyclic rodent populations have contributed fundamentally to the development of population ecology. Pioneering rodent studies have shown macroecological patterns of population dynamics in relation to latitude and have inspired similar studies in several other taxa. Nevertheless, such studies have not been able to disentangle the role of different environmental variables in shaping the macroecological patterns. We collected rodent time‐series from 26 locations spanning 10 latitudinal degrees in the tundra biome of Fennoscandia and assessed how population dynamics characteristics of the most prevalent species varied with latitude and environmental variables. While we found no relationship between latitude and population cycle peak interval, other characteristics of population dynamics showed latitudinal patterns. The environmental predictor variables provided insight into causes of these patterns, as 1) increased proportion of optimal habitat in the landscape led to higher density amplitudes in all species and 2) mid‐winter climate variability lowered the amplitude in Norwegian lemmings and grey‐sided voles. These results indicate that biome‐scale climate and landscape change can be expected to have profound impacts on rodent population cycles and that the macro‐ecology of such functionally important tundra ecosystem characteristics is likely to be subjected to transient dynamics.
... Although biological control can be an effective strategy in many cases, before introducing new organisms into an ecosystem, it is crucial to thoroughly investigate and carefully consider the potential consequences. The May-Holling-Tanner model [45,51,27], is partially studied in [5] with delay in [46], with proportion-dependent functional response in [30], and with an alternative food source for the predator in [18,19,44,43]. ...
This paper presents a detailed analysis of a bitrophic chain to understand the complex ecological behaviour applied to this specific model. The model is described by means of a two-dimensional system of ordinary differential equations. The existence and uniqueness of the solutions of this system are examined, as well as their boundedness and positivity. In addition, through a differentiable equivalence, conditions for local and global stability at biologically relevant critical points are established. Periodic solutions are also explored. Finally, the Python programming language is used to perform a quantitative analysis of these critical points, showing different scenarios of the qualitative analysis previously obtained.
... This is a major issue with natural populations wherein the carrying capacity of 15 the environment and the intrinsic growth rates of the same species are liable to vary between 16 populations, which implies that the value of the control threshold has to be determined on a case-17 by-case basis, through prior knowledge of the dynamics of the given population. Moreover, 18 natural populations might exhibit increasing / decreasing trends in size (Turchin, 2003) due to 19 extrinsic factors, which would make determination of the threshold even more problematic. This 20 problem is partially alleviated by another class of methods in which the perturbations are not 21 hard numbers, but proportionate to some quantity, usually the present population size (Doebeli 22 and Ruxton, 1997; Güémez and Matías, 1993;Solé et al., 1999) or the difference between the 23 present population size and some pre-determined threshold (Dattani et al., 2011). ...
Despite great interest in techniques for stabilizing the dynamics of biological populations and metapopulations, very few practicable methods have been developed. We propose an easily implementable method, Adaptive Limiter Control (ALC), for reducing extinction frequencies and the magnitude of fluctuation in population sizes and demonstrate its efficacy in stabilizing laboratory populations and metapopulations of Drosophila melanogaster. Metapopulation stability was attained through a combination of reduced size fluctuations and synchrony at the subpopulation level. Simulations indicated that ALC was effective over a wide range of maximal population growth rates, migration rates and population dynamics models. Since simulations using widely applicable, nonspecies-specific models of population dynamics were able to capture most features of the experimental data, we expect our results to be applicable to a wide range of species.
... where r is the intrinsic growth rate of the population of engineers and H t plays the role of a time-dependent carrying capacity for the population of engineers. The essential ingredient of the Gurney and Lawton model, which sets it apart from the other population dynamics models [21], is the requirement that usable habitats be created by engineers working on virgin habitats. In particular, if we assume that there are V t units of virgin habitats at generation t, then the fraction C (E t ) V t of them will be transformed in usable habitats at the next generation, t + 1. ...
Humans are the ultimate ecosystem engineers who have profoundly transformed the world's landscapes in order to enhance their survival. Somewhat paradoxically, however, sometimes the unforeseen effect of this ecosystem engineering is the very collapse of the population it intended to protect. Here we use a spatial version of a standard population dynamics model of ecosystem engineers to study the colonization of unexplored virgin territories by a small settlement of engineers. We find that during the expansion phase the population density reaches values much higher than those the environment can support in the equilibrium situation. When the colonization front reaches the boundary of the available space, the population density plunges sharply and attains its equilibrium value. The collapse takes place without warning and happens just after the population reaches its peak number. We conclude that overpopulation and the consequent collapse of an expanding population of ecosystem engineers is a natural consequence of the nonlinear feedback between the population and environment variables.
... The observation by May (1974) that simple models of population growth can generate irregular oscillations in population abundance has made chaos of considerable interest among ecologists (Costantino et al. 1997;Perry et al. 2000;Cushing et al. 2003;Turchin 2003;Becks et al. 2005). Chaotically fluctuating populations may reach low densities, where their probability of extinction must be high (Thomas et al. 1980;Berryman and Millstein 1989). ...
Interactions in ecological communities are inherently nonlinear and can lead to complex population dynamics including irregular fluctuations induced by chaos. Chaotic population dynamics can exhibit violent oscillations with extremely small or large population abundances that might cause extinction and recurrent outbreaks, respectively. We present a simple method that can guide management efforts to prevent crashes, peaks, or any other undesirable state. At the same time, the irregularity of the dynamics can be preserved when chaos is desirable for the population. The control scheme is easy to implement because it relies on time series information only. The method is illustrated by two examples: control of crashes in the Ricker map and control of outbreaks in a stage-structured model of the flour beetle Tribolium. It turns out to be effective even with few available data and in the presence of noise, as is typical for ecological settings.
... food search learning by predator, existence of alternative food, refuse for prey). It has certainly been useful in both theoretical investigations and practical applications (Turchin, 2003). Movement of phytoplankton and zooplankton population with different velocities can give rise to spatial patterns (Malchow, 2000). ...
In this paper, we investigate the complex dynamics of a spatial plankton-fish system with Holling type III functional responses. We have carried out the analytical study for both one and two dimensional system in details and found out a condition for diffusive instability of a locally stable equilibrium. Furthermore, we present a theoretical analysis of processes of pattern formation that involves organism distribution and their interaction of spatially distributed population with local diffusion. The results of numerical simulations reveal that, on increasing the value of the fish predation rates, the sequences spots spot-stripe mixtures stripes hole-stripe mixtures holes wave pattern is observed. Our study shows that the spatially extended model system has not only more complex dynamic patterns in the space, but also has spiral waves.
... In contrast to the assumption on the Gaussian nature of noise typically adopted in such autoregression models, we study the situation with bounded non-random perturbations. Such models are in common use in the population dynamics and macroeconomics, e.g., see [16] and [9], and the analysis of possible peak phenomena observed in these models is in demand. ...
We consider asymptotically stable scalar difference equations with unit-norm initial conditions. First, it is shown that the solution may happen to deviate far away from the equilibrium point at finite time instants prior to converging to zero. Second, for a number of root distributions and initial conditions, exact values of deviations or lower bounds are provided. Several specific difference equations known from the literature are also analyzed and estimates of deviations are proposed. Third, we consider difference equations with non-random noise (i.e., bounded-noise autoregression) and provide upper bounds on the solutions. Possible generalizations, e.g., to the vector case are discussed and directions for future research are outlined.
... The deterministic share of the observed fluctuations is usually assigned to nonlinear density-dependent processes, which create regulatory and stabilising forces. 1 Different theoretical and modelling frameworks have been used through the history of population dynamics, but time series analysis and autoregressive models are a frequent and natural choice, as the population size in the future is related to the population size in the past. [2][3][4] A variety of time series analysis methods have been used in population dynamics to diagnose their structure and density dependence, 5 particularly successful in the analysis of empirical data of long lived taxa such as mammals. [6][7][8] The approach proposed by Royama 9 combines diagnostic tools with the use of phenomenological models, and has increased the predictive power and understanding of the dynamics of intensively studied systems. ...
We study the population size time series of a Neotropical small mammal with the intent of detecting and modelling population regulation processes generated by density-dependent factors and their possible delayed effects. The application of analysis tools based on principles of statistical generality are nowadays a common practice for describing these phenomena, but, in general, they are more capable of generating clear diagnosis rather than granting valuable modelling. For this reason, in our approach, we detect the principal temporal structures on the bases of different correlation measures, and from these results we build an ad-hoc minimalist autoregressive model that incorporates the main drivers of the dynamics. Surprisingly our model is capable of reproducing very well the time patterns of the empirical series and, for the first time, clearly outlines the importance of the time of attaining sexual maturity as a central temporal scale for the dynamics of this species. In fact, an important advantage of this analysis scheme is that all the model parameters are directly biologically interpretable and potentially measurable, allowing a consistency check between model outputs and independent measurements.
... The mathematical characterization of population dynamics is well rooted within the field of ecology [12,13]. Recent works have extended this kind of analysis to different social aspects, including the evolution of speakers of different coexisting languages [14][15][16][17][18][19][20][21][22]. ...
We study the stability of two coexisting languages (Catalan and Spanish) in Catalonia (North-Eastern Spain), a key European region in political and economic terms. Our analysis relies on recent, abundant empirical data that is studied within an analytic model of population dynamics. This model contemplates the possibilities of long-term language coexistence or extinction. We establish that the most likely scenario is a sustained coexistence. The data needs to be interpreted under different circumstances, some of them leading to the asymptotic extinction of one of the languages involved. We delimit the cases in which this can happen. Asymptotic behavior is often unrealistic as a predictor for complex social systems, hence we make an attempt at forecasting trends of speakers towards 2030. These also suggest sustained coexistence between both tongues, but some counterintuitive dynamics are unveiled for extreme cases in which Catalan would be likely to lose an important fraction of speakers. As an intermediate step, model parameters are obtained that convey relevant information about the prestige and interlinguistic similarity of the tongues as perceived by the population. This is the first time that these parameters are quantified rigorously for this couple of languages. Remarkably, Spanish is found to have a larger prestige specially in areas which historically had larger communities of Catalan monolingual speakers. Limited, spatially-segregated data allows us to examine more fine grained dynamics, thus better addressing the likely coexistence or extinction. Variation of the model parameters across regions are informative about how the two languages are perceived in more urban or rural environments.
... The evolutionary dynamics from the ecological point of view has been explored in rather detail. The efforts have been summarized, for example, in a recent monograph by Turchin (74). This author takes the view that general laws should exist and should give rise to useful results relevant to ecology. ...
In the present article the recent works to formulate laws in Darwinian evolutionary dynamics are discussed. Although there is a strong consensus that general laws in biology may exist, opinions opposing such suggestion are abundant. Based on recent progress in both mathematics and biology, another attempt to address this issue is made in the present article. Specifically, three laws which form a mathematical framework for the evolutionary dynamics in biology are postulated. The second law is most quantitative and is explicitly expressed in the unique form of a stochastic differential equation. Salient features of Darwinian evolutionary dynamics are captured by this law: the probabilistic nature of evolution, ascendancy, and the adaptive landscape. Four dynamical elements are introduced in this formulation: the ascendant matrix, the transverse matrix, the Wright evolutionary potential, and the stochastic drive. The first law may be regarded as a special case of the second law. It gives the reference point to discuss the evolutionary dynamics. The third law describes the relationship between the focused level of description to its lower and higher ones, and defines the dichotomy of deterministic and stochastic drives. It is an acknowledgement of the hierarchical structure in biology. A new interpretation of Fisher's fundamental theorem of natural selection is provided in terms of the F-Theorem. The proposed laws are based on continuous representation in both time and population. Their generic nature is demonstrated through their equivalence to classical formulations. The present three laws appear to provide a coherent framework for the further development of the subject.
... A central question in ecology for last decades has been how to quantify the relative importance of stochastic and nonlinear factors for fluctuations in population size (Saether et al., 2000;Turchin, 2003;Hsieh et al., 2005). Characterizing the fluctuation patterns is the major challenge for time series analysis in fisheries and for modeling the process of fish production. ...
I address the question of the fluctuations in fishery landings. Using the fishery statistics time-series collected by the Food and Agriculture Organization of the United Nations since the early 1950s, I here analyze fishing activities and find two scaling features of capture fisheries production: (i) the standard deviation of growth rate of the domestically landed catches decays as a power-law function of country landings with an exponent of value 0.15; (ii) the average number of fishers in a country scales to the 0.7 power of country landings. I show how these socio-ecological patterns may be related, yielding a scaling relation between these exponents. The predicted scaling relation implies that the width of the annual per capita growth-rate distribution scales to the 0.2 power of country landings, i.e. annual fluctuations in per capita landed catches increase with increased per capita catches in highly producing countries. Beside the scaling behavior, I report that fluctuations in the annual domestic landings have increased in the last 30 years, while the mean of the annual growth rate declined significantly after 1972.
... Individuals die between t and ∆t if the number of sporangia on the frog exceeds the lethal threshold, 78 so that S i (t) > s max , at which point all of the sporangia on the individual frog die as well (so that the 79 sporangia load is defined to be zero). 80 The zoospore level at time t+∆t depends on the zoospore level in the pool and the sum of sporangia Many mechanistic models in ecology are formulated as deterministic differential equation models, which 87 are frequently fit using non-statistical techniques, such as non-linear least squares [19,43]. The use of 88 Approximate Bayesian Computation has also been proposed for deterministic models [42]. ...
Individual Based Models (IBMs) and Agent Based Models (ABMs) have become widely used tools to understand complex biological systems. However, general methods of parameter inference for IBMs are not available. In this paper we show that it is possible to address this problem with a traditional likelihood-based approach, using an example of an IBM developed to describe the spread of Chytridiomycosis in a population of frogs as a case study. We show that if the IBM satisfies certain criteria we can find the likelihood (or posterior) analytically, and use standard computational techniques, such as Markov Chain Monte Carlo (MCMC), for parameter inference.
... (Higham & Ryan, 2013) In this research, mathematical and computational techniques derived from the discipline of Cliodynamics are used to model the processes occurring during the early medieval age in England (Turchin 2003a). The techniques include using the Lotka-Volterra equations to model the evolution of competitive populations over time (Turchin 2003b). Insights from geopolitical theory are also used to inform the modelling process, particularly of the variables influencing the coefficients of the Lotka-Volterra equations (Collins & Sanderson 2009). ...
This paper examines England's early medieval period or the Dark Ages. Temporally, this period follows the Roman empire's collapse in 410 CE and extends until the end of the Anglo-Saxon period in 1066 CE with the Norman conquest. Spatially, it covers the geographical area of Britain south of Hadrian's Wall. Mathematical and computational techniques derived from the discipline of Cliodynamics are used to model the processes occurring during the early medieval age in England.
... The theories of population growth developed by many scientists, such as Malthus and Verhulst, paved the way to establish the first system in this field, which is the deterministic prey-predator system, the foundations of which were laid by the scientists Lotka and Volterra in the early 20th century (Turchin, 2013). ...
This thesis aims to study the stochastic differential system through one of its distinctive applications, which is the prey-predator model, that is considered one of the most important systems in the field of biomathematics. The study begins by presenting the biological and mathematical concepts related to ecosystems as well as explaining how population growth models develop over time. Ecosystems are affected by many influences, which leads to a change in their dynamics. On this basis, two new effects are proposed, Hide-and-Escape effect for the prey and Predation Skill Augmentation effect of the predator. These two effects are explained from the biological side and a detailed illustration is given on how to model them mathematically, also how to integrate them into prey-predator model in a competitive manner, which led to the formation of the deterministic prey-predator system. A comprehensive analysis of the resulting deterministic model is presented after adding the effects, starting with ensuring the boundedness and the stability of the system, studying the conditions of coexistence and extinction for Kolmogorov, and analyzing the bifurcation behavior at one of its parameters. Stochastic processes are also integrated into the system, as it is assumed that stochastic processes affect the growth rate of prey and predators, which led to the formation of a stochastic prey-predator system. The stochastic model is analyzed, and the existence and uniqueness of the solution is verified, and that the system is bounded. The permanence conditions of the system are also studied, as well as the conditions for persistence and extinction, using the stochastic analysis tools. Finally, numerical simulations of the two previous systems are presented, where the conditions and theories reached are verified using MATLAB.
... The development of theoretical justifications for the reasons for the appearance of oscillations in population systems began in the pioneering works of Lotka and Volterra, where the presence of a continuum family of periodic solutions was shown for a simple mathematical prey-predator model. For the two-dimensional models with more complex forms of functional response, namely Holling II-IV, an existence of the only one periodic solution associated with self-oscillatory regime was shown [Holling, 1959;Brauer & Castillo-Chávez, 2001;Turchin, 2003;Dawes & Souza, 2013;Hossie & Murray, 2023]. Note that self-oscillations appear as a result of various bifurcations, e.g. ...
The problem of analyzing the bifurcation mechanisms of complex stochastic oscillations in population dynamics is considered. We study this problem on the basis of the modified Leslie–Gower prey–predator model with randomly forced Holling-II functional response. The paper focuses on the effects of noise in the Canard explosion zone. The phenomenon of noise-induced splitting of Canard cycles is discovered and studied in terms of stochastic P-bifurcations. In parametric analysis, we use the stochastic sensitivity technique with the apparatus of confidence domains to find the most noise-sensitive Canard. For the phenomenon of stochastic splitting, an underlying deterministic mechanism using critical curves near the cycle orbit and sub-/super-critical zones is revealed.
... Verdy [32] introduced two predator-prey models incorporating an Allee effect for predator reproduction and conducted a numerical exploration of their bifurcation structures. Several studies have explored scenarios where the prey population exhibits logistic growth in the absence of predation, while the predator functional response follows the Holling type II model [36,37]. Researchers have also focused on examining the dynamics of predation models with an additional term for the self-limitation parameter [38]. ...
Authors have detected the importance of the Allee effect and have highlighted significant changes to system dynamics in the ecological environment. In this paper, using the theory of dynamical systems, we explore our investigation of a two-dimensional prey–predator model into two aspects: (i) we modify a competent Allee effect and the self-limitation term for the predator model system by incorporating the Crowley–Martin-type functional response, and (ii) we extend this modified model system by adding a strong Allee effect in prey growth. We report the behavior of the model system under the Crowley–Martin-type functional response and the impact of a strong Allee effect. We examine that an initial condition with high prey and low predator intensity always settles to predator extinction in the absence of a strong Allee effect. In the presence of a strong Allee effect, an initial condition with low prey and predator intensities leads the system to total extinction, while high prey and low predator intensity allows the model system to settle at predator extinction and high prey concentration. The addition of a strong Allee effect in the model system enriches the boundary equilibrium point. In attractor examination, we demonstrated that the model system without the Allee effect has attractors between boundary equilibrium and coexistence equilibrium, and between coexistence equilibria. The addition of a strong Allee effect produces attractors between coexistence equilibrium as well as between boundary equilibrium points. We deduce that both model systems experience enriched coexistence equilibria in a small parametric region. Modifying the model system without the Allee effect produces three attractors (one predator-free equilibrium point and two coexistence equilibrium points). The model system with a strong Allee effect gives four attractors (two predator-free equilibrium points and two coexistence equilibrium points). Our comprehensive bifurcation analysis reports both local and global bifurcations for both types of model systems. We explore bifurcation analysis through codimension-one and codimension-two bifurcations extensively by choosing the Allee effect as one of the key parameters. In this context, our modified model system exhibits saddle-node bifurcation, Hopf bifurcation, Bogdanov–Takens bifurcation and generalized Hopf bifurcation. We derive the sensitivity index of model system parameters for both types of model systems. We validated our analytical findings with the help of numerical simulations.
... In this case, the loss term in equation (3.2) reflects emigration outside the landscape, which is dynamically equivalent to death. The Volterra model, as is well known, is characterized by global stability, but with typically an oscillatory approach to the equilibrium [101]. Indeed, for a broad range of parameters, our warfare-refuge model is dynamically very similar to the Volterra model. ...
The impact of inter-group conflict on population dynamics has long been debated, especially for prehistoric and non-state societies. In this work, we consider that beyond direct battle casualties, conflicts can also create a ‘landscape of fear’ in which many non-combatants near theatres of conflict abandon their homes and migrate away. This process causes population decline in the abandoned regions and increased stress on local resources in better-protected areas that are targeted by refugees. By applying analytical and computational modelling, we demonstrate that these indirect effects of conflict are sufficient to produce substantial, long-term population boom-and-bust patterns in non-state societies, such as the case of Mid-Holocene Europe. We also demonstrate that greater availability of defensible locations act to protect and maintain the supply of combatants, increasing the permanence of the landscape of fear and the likelihood of endemic warfare.
... Just to mention a few, we have [41], where an impulse control model is used to manage the bird population and [42], where the authors study impulse dispersal in single-species models. Motivated by these works, we consider a specific example adapted from existing models (see [38,[43][44][45][46][47]) of bird population growth, which is represented by an ordinary differential equation system that describes and captures the complexity of bird population dynamics, considering both the influence of continuous factors and discrete events in a specific area. ...
This work considers the existence of solutions of the heteroclinic type in nonlinear second-order differential equations with ϕ-Laplacians, incorporating generalized impulsive conditions on the real line. For the construction of the results, it was only imposed that ϕ be a homeomorphism, using Schauder’s fixed-point theorem, coupled with concepts of L1-Carathéodory sequences and functions along with impulsive points equiconvergence and equiconvergence at infinity. Finally, a practical part illustrates the main theorem and a possible application to bird population growth.
Local populations of rare and declining species assume particular interest for their conservation value, as models for studying the dynamics of vulnerable populations and for insights into the successful maintenance of populations at risk. The Harris’ checkerspot Chlosyne harrisii (Nymphalidae) is a small orange and black butterfly whose populations have rapidly declined in recent years along the southern edge of its geographic range in New England, USA. Its larvae feed exclusively on the tall flat-topped white aster Doellingeria umbellata, a plant of old fields and roadsides. Here, I evaluate the dynamics of a population of this butterfly, whose numbers have fluctuated markedly over 19 years, with maximum and minimum yearly counts varying over 30-fold during that period, and exhibiting little suggestion of constancy. I focused on several life-cycle stages, including numbers and fate of egg masses and success in overwintering. Nearly two-thirds of the egg masses produced some young that entered hibernation, but only a few of these individuals survived the winter. Warm overwinter temperatures were associated with especially low overwintering success and wet, cold weather during the period of oviposition was associated with poor reproductive success. Little direct evidence existed in support of density dependence as a factor limiting population numbers over the period of this study. Checkerspots exhibited poor success as colonizers, for we found only one small satellite population likely emanating from the focal population.
We consider a modified Rosenzweig–MacArthur model that incorporates the negative impact of resource on the consumer. This negative effect of the resource has been empirically examined within various ecological systems. It plays a crucial role in driving transitions towards consumer extinction through multistability. Specifically, we show that the negative effect results in the bistability between two steady-state solutions for smaller values of the positive impact of resource on the consumer, whereas higher positive impact facilitates the coexistence of oscillatory behaviour and steady-state solution. We show that the presence of the predator’s negative efficiency facilitates abrupt transitions to distinct dynamical states in both forward and backward traces. We also show that the preferred state of the finite steady state for the persistence of both consumer and resource populations can be achieved for intermediate ranges of consumer’s positive and negative efficiency rates, carrying capacity and mortality rate of the consumer. We find that a large consumer’s negative efficiency rate always drives the system to the extinction of the consumer. We have derived analytical stability conditions for transcritical, Hopf and saddle-node bifurcations by a linear stability analysis, which agrees with the simulation results depicted in the two-parameter phase diagrams.
This paper investigates two-dimensional Gause-type prey-predator models with a strong Allee effect. By incorporating the Holling type-III ratio-dependent functional response, the model provides a complete view of prey-predator interactions in an ecosystem with different realistic phenomena. In this study, we mainly focus on how to influence the death rate of predators on system dynamics in the presence of a strong Allee effect. Here, the prey growth function is subjected to the strong Allee effect, which measures the correlation between the population size or density with a critical threshold value. In this manuscript, we have investigated the positivity of the solution, boundedness, stability, and sensitivity analysis of the proposed model. Apart from these, all the dynamical behaviours of the system have been captured through a comprehensive analysis of one and two-parameter bifurcation diagrams. In the course next outcomes of the local and global bifurcation analysis, we observed all possible bifurcations, such as the existence of saddle-node bifurcation, Hopf bifurcation, which are the local bifurcations. Moreover, we also observe that the system exhibits global bifurcation such as the Bogdanov–Takens bifurcation and homoclinic bifurcation. Finally, the analytical results are validated with numerical simulations.
En este artículo, abordamos la evolución de los modelos matemáticos para poblaciones, comenzando con el modelo malthusiano que describe el crecimiento de una población bajo condiciones favorables y pasando por el modelo de crecimiento logístico llegamos al modelo depredador presa Lotka-Volterra, donde el crecimiento malthusiano de la población presa es frenado por la presencia de una población depredadora, la cual entra en un equilibrio ecológico y ambas poblaciones coexisten.
This paper proposes a mathematical model based on a mechanistic approach and previous research findings for the bacterial canker disease development in kiwifruit vines. This disease is a leading cause of severe damage to kiwifruit vines, particularly in humid regions, and contributes to significant economic challenges for growers in many countries. The proposed model contains three parts. The first one is the model of the kiwifruit vine describing its light interception, its carbon acquisition, and the partitioning dynamics. The carbon resource represents the chemical energy required for maintaining the necessary respiration of the living organs and their growth processes. The second part of the model is the dynamics of the pathogenic bacterial population living within the vine’s tissues and competing with them for the carbon resource required for their proliferation. The third part of the model is the carbon dynamics described by a mass conservation formula which computes the remaining amount of carbon available for competition. The model was validated by comparing simulations with experimental results obtained from growth chambers. The results show that the proposed model can simulate reasonably well the functional part of the vine in both the healthy case and the disease case without plant defense mechanisms in which the bacteria are always dominant under favorable environmental conditions. They also show that the environmental effects on the vine’s growth and the infection progress are taken into account and align with the previous studies. The model can be used to simulate the infection process, predict its outcomes, test disease management techniques, and support experimental analyses.
In this paper, we consider a system of two coupled populations modeled by the Ricker map with Allee effect. It is shown that the system can exhibit various dynamic regimes with the change of intensity of migration flows between subpopulations. In the deterministic case, we localize parametric zones of multistability and explore different dynamic patterns of the system, such as equilibria, periodic, quasiperiodic, and chaotic oscillations. In this analysis, the apparatus of Lyapunov exponents is used. For oscillatory regimes, transitions between in-phase and anti-phase synchronization are discussed. In stochastic case, we study how intensity of migrations impacts the noise-induced partial and complete extinction of populations. For parametric analysis of noise-induced extinction, we use the method of confidence domains.
The grazing ecosystems of Kalmykia are extremely dynamic, this being associated with the high variability of grazing pressure in the region: a drastic reduction in the number of livestock in the 1990’s led to the restoration of pastures and the transition of their functioning from the “desert” to the “steppe” regime. Since the late 2000’s, new desertification processes have been gaining momentum, these being caused by increased grazing pressure and droughts. The vegetation cover of pastures quite quickly began to respond to the increase in livestock numbers by reducing the projective cover, while the species composition of plants sharply changed to “desert” only a few years later. Against the background of increasing desertification, the rodent community first collapsed and then recovered. Apparently, the collapse is a belated reaction to the steppefication of pastures that preceded desertification, this leading to a reduced number of desert species and an impoverished community, thereby slowing down its revival during a new cycle of desertification. Thus, both vegetation cover and rodent community demonstrated non-stationary dynamics with changing the regimes and a delayed response (inertia) in response to landscape desertification.
Rodent population cycles are observed in highly seasonal environments. As most rodents are herbivorous, the availability and the quality of their food resources varies greatly across seasons. Furthermore, it is well documented that herbivore densities have a measurable effect on vegetation and conversely. So, many studies investigated whether rodent population cycles could be induced by bottom-up regulation. A recent review summarized several sub-hypotheses leading to rodent population cycles: cycles may be due to inherent inter-annual variations of plant quantity, to overshoot of carrying capacity by overgrazing (i.e. lack of quantity), to changes in quality of food (decrease of quality of preferred food or switch towards less quality food) in response to rodent grazing (e.g. plant defences). If some sub-hypothesis seems to be more important than others, there is currently a prerequisite to construct scientific consensus: dietary description is still overlooked in many systems and should be more investigated.
This study focuses on fossorial water vole. It shows contrasted population dynamics depending on its geographical locations. It is known to be able to exhibit large outbreaks in grasslands in highly seasonal climate. It is thus a good model species to investigate plant hypotheses, first beginning by diet description.
The diet of water vole was investigated in and out of the outbreak area with a combination of approaches in the field, in different sampling sites and considering seasonality. We demonstrated that voles have a very large fundamental trophic niche, but strong behavioural selection, inducing a narrower realised niche, especially during winter. We created an experimental device based on camera trap and cafeteria tests. We observed a strong preference for dandelion ( Taraxacum officinale ) in wild water voles, that results in exclusive selection during winter for food stores. These preferences were constant across seasons, altitudes and grassland productivity gradients, despite the scarcity of this species in some experimental sites.
First, we conclude on the importance of using different methods to fully describe the diet of rodents Second, we assess that dandelion is a winter key resource for water vole. It thus might be interesting to investigate the role of dandelion in vole population dynamics.
The authors have synthesized the data on population dynamics and densities of rodents in seven biomes of the Palearctic, and related them to the data on standing crop of biomass and net productivity of ground vegetation (as rough indicators of food availability to rodents). Analysis of 44 long-term (≥5 yr) series of rodent trapping showed that there was a continuum from highly cyclic to non-cyclic populations. Standing crop of biomass of ground vegetation correlated positively with latitude; it was highest in the northern tundra and decreased towards South. Net productivity of ground vegetation (30 data points) did not show latitudinal trends. It was lowest in desert, tundra, and all types of forests, and highest in open habitats of the temperate zone and steppes. Mean densities of rodents were lowest in tundra, desert, and all types of forests (8-29 rodents/ha). The highest average densities were recorded in the farmlands of temperate zone and steppe (143-490 rodents/ha). Dichotomy between seasonal (non-cyclic) and multiannual (cyclic) fluctuations in rodent numbers was not found. Results of the long-term study on predation on rodents in the temperate deciduous forests did not support the hypothesis on the role of specialist and generalist predators in shaping rodent dynamics. Based on the observed vegetation-rodent correlations, the authors have proposed an interpretation of the mechanisms of rodent population dynamics in the Palearctic biomes. A prerequisite for rodent cycles to occur is abundant winter food, which enables rodents to continue an increase phase beyond one growing season (by winter breeding). Habitats with mean standing crop of ground vegetation of over 4000 kg dry weight/ha in summer are expected to harbour cyclic populations of rodents.
We re-examine the annual trappings of the Canadian lynx over the years 1821-1934 (inclusive), which have been reported and analysed extensively. For some references see Elton and Nicholson (1942), Rowan (1950), Moran (1952), Hannan (1960), Kashyap (1973) and Bulmer (1974). This paper shows that an autoregressive (AR) model of order eleven provides an acceptable alternative to the more widely adopted class of models with low order AR plus one harmonic component.
Changes in abundance, recruitment rates, and age-specific harvest rates of 3 populations of moose (Alces alces) in Newfoundland were estimated with cohort analysis. Populations declined during 1966-73 due to overharvesting, but recovered in all areas following reduction of license quotas in the mid-1970's. Age-specific vulnerability to hunting was highest for young adults and lowest for calves and mid-aged adults. Differences in vulnerability were possibly related to hunter preferences, previous experience of moose with hunters, and frequency of movements by moose. Data on kills/day and moose seen/day obtained from hunter questionaires were positively correlated with moose abundance but negatively correlated with the number of hunters. The latter suggested that interference had a strong influence on hunter success.
A modification of the sulfophosphovanillin photometric method for determining cholesterol levels in mammalian blood sera was adapted for determining the percentage of lipid in individual southern pine beetles, Dendroctonus frontalis Zimmermann. This method is simple and provided consistent results. It is ideally suited to ecological and physiological studies involving bark beetle behavior such as flight, orientation, oogenesis, and spermatogenesis. Using this modified method and flight mills, it was confirmed that lipids were metabolized during southern pine beetle flight and that females, the sex that initiates tree colonization, are heavier than males, contain a greater percentage of lipid, and are capable of flying longer and farther than male beetles. Regardless of sex, the greater the weight of the beetle, the greater its flight potential.
A mathematical model of dynamics of an undisturbed wild reindeer population in the Kola peninsula (the Laplandskii Reserve) is developed. The model describes relationships between the reindeer population and food resources (winter lichen pastures). The model displays the dynamics, consistent with that observed for the period of 60 years. The modeling showed that the natural dynamics of the reindeer population was represented by cycles of 35-40 years. The growth of the animal's number for 25-30 years alternates with its decrease for 10 years. The cyclic character of dynamics is determined by trophic relations and is dependent on the dynamics of lichen resources. The growth of the reindeer's number brings down the food supply up to the critical level resulting in a reduction of the population. When the population density falls by two times of the maximum, the food resources start growing. The fall in the number of reindeer starts earlier than the complete exhaustion of pastures takes place that provide conservation of lichens and the reindeer population.
The moose population has been intensively managed in Finland since the beginning of 1970s. However, recent decline in population sizes observed in many parts of the country was unexpected. In this study, the development of the Finnish moose population in 1974-1994 was analysed with a simulation model where the crucial factor was the annual hunting. The simulation model was also used to generate predictions of the future population size. The simulations for three game management districts (Varsinais Suomi, Etela Hame and Pohjois Savo) followed well the actual population data. In forecasts, the population size predictions began to become increasingly unreliable when the forecast horizon was extended to two or more years. The analysis revealed that a successful management strategy calls for information on spatial migration of the moose and more accurate population estimates.
Thanasimus dubius (F.) is an important predator of the southern pine beetle, Dendroctonus frontalis Zimmermann, a major pest of pine forests in the southern United States. We examined the development of T. dubius in the field using emergence traps, and by sampling the bark of trees previously attacked by D. frontalis. Over a 2-year period, several distinct episodes of T. dubius emergence occurred in trees enclosed by emergence traps, and bark sampling of other trees uncovered many T. dubius immatures almost 2 years after attack by D. frontalis. These results indicate that T. dubius development may be significantly longer and more variable under natural conditions than previously thought, and suggest that some individuals may undergo a diapause.
The author asked British Ecological Society members what they felt to be the most important current concepts in ecology. Analysis of the 645 returned questionnaires (a 14.7% response rate) is provided. Fifty of the most important concepts are listed in rank order (the top 10 were: the ecosystem, succession, energy flow, conservation of resources, competition, niche, materials cycling, the community, life-history strategies and ecosystem fragility), and ordinations of responses are presented and discussed. -P.J.Jarvis
Lepus americanus populations were studied at Rochester, Alberta, during the 2nd winter of a cyclic decline. Snowshoe populations on study areas with dense cover due to a burn in 1968 tended to have higher early-winter densities, lower overwinter survival, lower juvenile than adult survival, higher rates of dispersal, higher overwinter weight losses, and later onset of reproduction. Periods of high weight loss coincided with high rates of ingress that temporarily increased densities. Snowshoes on food-short areas exhibited typical demographic symptoms of malnutrition (marked differential loss of juveniles, high overwinter weight loss, and later onset of reproduction). Predation was the paramount (80-90%) immediate cause of hare mortality, but malnourished hares were significantly more vulnerable to predators, and losses rose sharply at temperatures below -30oC. Outright starvation is a short-term phenomenon confined largely to the first few months of a major decline from peak densities. This is followed by 1 or 2 winters when malnutrition and predation in concert produce high mortality, and probably by a further period when predation is solely responsible. -from Authors
The mechanisms driving short-term (3-5 yr) cyclic fluctuations in densities of boreal small rodents, and especially, those causing a crash in numbers, have remained a puzzle, although food shortage and predation have been proposed as the main factors causing these fluctuations. In the first large-scale vertebrate predator manipulation experiment with sufficient replication, densities of small mustelids (the least weasel Mustela nivalis and the stoat M. erminea) and avian predators (mainly the Eurasian Kestrel Falco tinnunculus and Tengmalm's Owl Aegolius funereus) were reduced in six different areas, 2-3 km2 each, in two crash phases (1992 and 1995) of the 3-yr cycle of voles (field vole Microtus agrestis, sibling vole M. rossiaemeridionalis, and bank vole Clethrionomys glareolus). The reduction of all main predators reversed the decline in density of small rodents in the subsequent summer, whereas in areas with least weasel reduction and in control areas without predator manipulation, small rodent densities continued to decline. That only reduction of all main predators was sufficient to prevent this summer crash was apparently because least weasels represent <40% of vole-eating predators in western Finland. These results provide novel evidence for the hypothesis that specialist predators drive a summer decline of cyclic rodent populations in northern Europe.
Models are useful when they are compared with data. Whether this comparison should be qualitative or quantitative depends on circumstances, but in many cases some statistical comparison of model and data is useful and enhances objectivity. Unfortunately, ecological dynamic models tend to contain assumptions and simplifications that enhance tractability, promote insight, but spoil model fit, and this can cause difficulties when adopting a statistical approach. Furthermore, the arcane numerical analysis required to fit dynamic models reliably presents an impediment to objective model testing by fitting. This paper presents methods for formulating and fitting partially specified models, which aim to achieve a measure of generality by avoiding some of the irrelevant incidental assumptions that are inevitable in more traditional approaches. This is done by allowing delay differential equation models, difference equation models, and differential equation models to be constructed with part of their structure represented by unknown functions, while part of the structure may contain conventional model elements that contain only unknown parameters. An integrated practical methodology for using such models is presented along with several examples, which include use of models formulated using delay differential equations, discrete difference equations/matrix models, ordinary differential equations, and partial differential equations. The methods also allow better estimation from ecological data by model fitting, since models can be formulated to include fewer unjustified assumptions than would usually be the case if more traditional models were used, while still including as much structure as the modeler believes can be justified by biological knowledge: model structure improves precision, while fewer extraneous assumptions reduce unquantifiable bias.
Evaluates the hypothesis that if a climatic anomaly that favours an increase in fecundity and/or survival persists over several consecutive generations, its effects on a forest insect population may be multiplicative, and after some years of continuous increase the "released' population will cause noticeable defoliation. Mechanisms by which weather might cause changes in forest insect abundance are outlined; indirect effects are more likely to be significant, eg by influencing the level of stress in the host plant, which in turn affects its nutritional quality, chemical defences or digestibility. Outbreaks of spruce budworm Choristoneura fumiferana, forest tent caterpillar Malacosoma disstria and southern pine beetle Dendroctonus frontalis are used as examples. The nature of temporal and spatial variation in climatic patterns also needs to be introduced into analysis and interpretation. On balance, there is probably a rather low upper limit on the amount of variability in pest population levels that can be explained by weather variables. -P.J.Jarvis
During summer 1991, lemmings occurred at high densities whereas, in 1992, densities were substantially lower and decreased further during the summer. In 1991, avian predators such as snowy owls Nyctea scandiaca, gulls and skuas bred well; Arctic foxes Alopex lagopus were rarely observed in the study area but bred in the immediate vicinity. In both years there was a late thaw, but this did not deter breeding by birds. The insect food supply for waders showed similar patterns of abundance in both years. In 1991, 73 nests of nine species of wader were found within a 14-km2 study area, and dark-bellied brent geese nested in association with snowy owls. The overall density of wader nests was 7 per km2. Clutches disappeared at only two waders nests and no brent goose nests, and the Mayfield estimate of the daily probability of predation for waders was 0.0022. In contrast, the daily probability of predation was 0.20 in 1992, when there was a similar breeding density of waders. Arctic foxes were seen searching for food daily within the study area, and fox droppings were associated with nests taken by predators. The predicted scenarios for peak and decreasing lemming years, low predation and high nest success in the peak year and high predation and low nest success in the decreasing year, therefore occurred. -Authors
Largely synchronous population fluctuations of Clethrionomys glareolus, C. rufocanus, and Microtus agrestis were monitored by snap-trapping in spring and autumn in 1971-1988 in a strongly seasonal environment near UmeA, northern Sweden. All species were cyclic in the sense that they showed fairly regular (3-4 yr) fluctuations, but amplitudes ((max)/n(min)) varied, averaging almost-equal-to 200-fold in each species. This conclusion was supported by autocorrelation and spectral analysis, and by fitting time series data to a model for phase-forgetting cycles. By contrast, data did not conform to a model for phase-remembering cycles (with fixed period and amplitude). The transition between cycles, i.e., from the low to increase phase, was characterized by a distinct shift in rate of change in numbers from low to high or markedly higher values both in summer and winter. Generally, rate of change in summer declined continuously from the increase phase through each cycle. Moreover, there was a similar decrease of rate of change in winter, although rate of change (mainly in C. rufocanus and M. agrestis) first frequently increased early in the cycle. Rate of change was delayed density dependent in all species both in summer and winter, as revealed by high negative correlations with density in previous autumn and spring for summer and winter changes, respectively. These new findings of delayed density dependence (DDD) support the suggestion that vole cycles are generated by a time-lag mechanism. Possible mechanisms of the DDD are discussed. Regression analyses of rate of change in the different voles suggest that, besides the strong dependence on previous density, rate of change in numbers was also affected by current seed supply (in C glareolus) and/or weather variables (temperature and precipitation sums) that may have affected the quantity or quality of food.
We studied moose (Alces alces) browsing during two successive winters in six different over-wintering sites (yards) in Matane Fish and Game Reserve, Quebec. This region is one of comparatively high moose density for the province (0.8 moose/km2). Four plant species provided the bulk of the browse: mountain maple (Acer spicatum), balsam fir (Abies balsamea), beaked hazelnut (Corylus cornuta), and paper birch (Betula papyrifera). In 4 of 6 studied yards, 30 percent of all stems had received some browsing, whereas in the remaining 2 this amounted to 20 and 10 percent. Studies of the weight of twigs eaten using regression curves relating diameter and weight showed that in 5 of 6 yards, less than 15 percent of all accessible browse was effectively removed by moose. This percentage was lower than anticipated for an area of high moose density. Pellet group counts showed that the biomass removed by a single moose during a day varied between 1.9 and 5.7 kg (dry weight) with a mean estimate of 2.5 kg/moose/day.
Pre-adult survival decreased with pre-adult density. Adult fecundity decreased with both pre-adult density and adult crowding. These effects were used to construct a model of density-dependent population dynamics, which predicts the number of pre-adults for each generation in a population with 2 life stages and non-overlapping generations. Populations of small size and non-overlapping generations were maintained for 9 generations. Numbers of eggs and adults showed sustained, erratic fluctuations. A deterministic version of the dynamical model predicts rapid approach to equilibrium and cannot explain such fluctuations. Predictions of a stochastic version of the dynamical model, however, agree with the observed behaviour of the experimental populations. Oscillations in numbers are interpreted as a consequence of random fluctuations. -from Author
It has recently been claimed that the functional response of predators depends on the numbers of prey and predators only via their ratio. Contrary to this claim, I argue that such "ratio-dependent" functional responses are very unlikely to occur on theoretical grounds and have not been shown empirically. Previous justifications for ratio-dependent functional responses ignore the role of these responses in determining prey population dynamics. Patterns in the abundances of trophic levels and experimental results that have been cited as support for "ratio-dependent" predation are consistent with numerous other explanations. These other explanation do not suffer from the pathological behaviors and lack of a plausible mechanism that afflict ratio-dependent models. Furthermore, ratio-dependent functional responses, because of their lack of mechanistic basis, do not provide a good basis for developing more detailed models. Nor can the use of ratio-dependent models be justified as a limiting case of more general predator-dependent models.
Some practical techniques are discussed for analyzing time series whose statistical properties are changing with time. We first consider how principal component analysis can reduce the multidimensional nature of certain series and, in particular, apply this technique to the analysis of changing seasonal patterns. Discussions of trend, changes in oscillatory behavior, and "unusual" events follow. The problem of making inferences regarding causation is briefly considered. We conclude with a call for flexibility in approach.
Data are conventionally considered as the given objective basis for knowledge, whereas theories, being mental constructions have to conform to this basis, or be ruled out. As a corollary, when mathematics is invoked it is assumed to play the role solely of providing an accurate rewording of ideas that could as well have been expressed verbally. It is here argued that this is untenable within ecology: 1) because most ecological data are anything but 'hard facts'; 2) because (as a matter of historical record) it is standard scientific practice to reject ecological data that are in conflict with established theories; 3) because many contemporary ecological theories are in fact retained (and rightly so) although they are demonstrably wrong; and 4) because it is by no means obvious that the proximate goal of theorizing should be to find the truth, hence an important role of mathematics is to provide productive lies. -from Author
Red Grouse (Lagopus lagopus scoticus) show unstable population dynamics. The number shot for sport at Rickarton moor in northeast Scotland, for example, has cycled with 10-11-yr periodicity since 1946. Here, demographic and other causes of a population cycle were documented from 1979-1989, and an experiment tested the prediction that removing some cocks during the increase phase would prevent a subsequent cyclic decline. Throughout the study, sport shooting was stopped on the area where the main work was done. During 1979-1982, before the experimental removal of cocks began, numbers over the whole moor rose from a trough at the start of the study. On the control area, the cyclic peak in 1983 was followed by a decline until 1988, as predicted in advance from models derived from a previous study elsewhere. On the experimental area, enough territorial cocks were removed each spring from 1982 to 1986 to prevent the population from attaining peak densities for five successive years, and no cyclic decline occurred. The removal of cocks resulted in similar numbers of hens being lost from the breeding population. The main demographic cause of population change on control (cycling) and experimental (cycle broken) areas was variation in the recruitment of young cocks to the spring population. On the control area, recruitment was related to cycle phase and breeding success. Changes in food, nitrogen metabolism, and parasite burdens could not explain the cycle. Demographic patterns were consistent with a model in which changes in age structure affected recruitment. These and previous results refute four hypotheses as necessary causes of population cycles in Red Grouse: (1) maternal nutrition, (2) a version of Chilly's genetic hypothesis, (3) host-parasite (caecal threadworm), and (4) predator-prey relationships. The hypothesis that age structure changes and associated behavior cause cycles by affecting recruitment and, thus, population change, remains unrefuted.
We developed an empirically based model of density-dependent vole population growth based on experimental data on population dynamics of Microtus pennsylvanicus in large field enclosures. Statistical analysis of the data indicated that both density-dependent regulation and seasonal effects were important in influencing vole population growth. Together, these two factors explained approximately one-half of variance in the realized per capita rate of change exhibited by experimental vole populations. A population model assuming simple functional forms (linear for population density and sine for seasonality) provided an adequate description of the data, with more complex functional forms leading to at best minimal improvements. The natural rate of population increase, averaged over all seasons, was estimated as (mean ± SE) r(max) = 6.0(±0.4) yr-1. This estimate suggests an impressive power of population increase, implying that each female vole could be replaced by about 400 daughters a year later (assuming density-independent growth). A survey of literature, however, indicates that this is by no means the largest rate of increase observed in a vole population.
(1) The death rate imposed by the predator population on the prey population is, inter alia, linked to the rate at which individual predators locate and consume prey, and hence to the rate of increase of the predator population. The overall rate of increase will depend, we argue, on three main components. These are (i) the duration of each predator instar, (ii) survival rates within instars and (iii) the fecundity of the adults. (2) Because the number of prey eaten per predator is defined by a complex surface involving prey density, predator density and their relative distributions, expressions for the components of the predator rate of increase are derived, for simplicity, as a slice through this complex surface in the prey-density plain for a single predator. (3) The effects of prey density on the development rates of particular predator instars and fecundity of adult predators are defined from arguments based on simple energetic considerations. The predicted relationships are in general agreement with experimental data reported from a wide range of predatory arthropods. (4) The effects of prey density on survival within particular predator instars are defined from arguments based on the assumption that mortality through food shortage will occur at some characteristic mean ingestion rate (mu I), with deaths in the population as a whole being approximately normally distributed about this mean. Again, agreement between models and experimental data is satisfactory. (5) Unfortunately, although these relationships have been documented for a wide variety of organisms, concentration on one species and all its components has not occurred. This represents a major gap in experimental ecological work.
Linear regression was used to relate snow accumulation during single and consecutive winters with white-tailed deer Odocoileus virginianus fawn:doe ratios, moose Alces alces twinning rates and calf:cow ratios, and annual changes in deer and moose populations. Significant relationships were found between snow accumulation during individual winters and these dependent variables during the following year, but strongest relationships were between the dependent variables and the sums of the snow accumulations over the previous three winters. The percentage of the variability explained was 36-51%. Significant relationships were also found between winter vulnerability of moose calves and the sum of the snow accumulations in the current, and up to 7 previous, winters, with 49% of the variability explained. These relationships imply that winter influences on maternal nutrition can accumulate for several years and that this cumulative effect strongly determines fecundity and/or calf and fawn survivability. Although wolf Canis lupus predation is the main direct mortality agent on fawns and calves, wolf density itself appears to be secondary to winter weather in influencing the deer and moose populations. -from Authors
The reproductive effort as a function of the reproducing individual's weight and number of fetuses has been estimated. This estimate is based on models for energy requirements of reproducing females of the common vole Microtus arvalis (Pall.). This function forms the basis for construction of age (or weight) specific demographic fitness sets. Based on these, optimal reproductive rates are predicted for stable and cyclic populations. The predictions thus deduced find support in available field information. For instance, the model explains the larger litters and earlier age of maturation in cyclic microtine populations as compared to stable populations; further, it explains the increased litter size with increasing parity. On the basis of our analysis a new pattern is revealed; the maximal realized litter size is reached earlier in fluctuating populations than in more stable populations; thus, the ratio between litter size of old and young females is predicted to be largest in the most cyclic populations. This compares favourably with field data on several species. /// На основе моделей энергетических потребностей размножающихся самок обыкновенной полевки Microtus arvalis Pall. определяли эффективностъ размножения как функцию веса размножающейся особи и числа зародышей (эта функция образует основу конструкции возрастной или бесовой демографической пирамиды вида). На этой основе предсказана оптималъная скоростъ размножения для стабилъных и циклических популяций. Полученные таким образом предсказания подтвердилисъ полевыми данными. Например, моделъ объясняет появление более многочисленного помета и более ранние сроки созревания в циклических популяциях полевок в сравнении со стабилъными; далее она объясняет увеличение размеров детенышей с повышением деторождаемости. На основе наших исследований выявлена новая закономерностъ; максималъная реализуемая численностъ помета достигается ранъше во флуктуирующих популяциях, чем в более стабилъных. Таким образом, отношения между величиной помета у старых и молодых самок должно бытъ наиболъшим у популяций с ярки выраженными циклами. Это хорошо сочетается с полевыми наблюдениями на некоторых видах.
(1) We studied the population dynamics of snowshoe hares (Lepus americanus) in the Kluane Lake region of the Yukon by live-trapping nine areas year-round. We provided rabbit chow as winter food to three of these populations from September to May, 1977 to 1984. (2) Peak densities were reached in 1980 and 1981 on all areas except Jacquot Island. Two areas with extra winter food maintained densities three times that of their controls, while one food-supplemented area on Jacquot Island showed little effect of improved feeding on hare density. (3) Supplementary food did not prevent the cyclic decline and all areas reached low densities by 1984. The beginning of the decline was delayed 6 months on one food area but not delayed on the other. (4) The amplitude of the cycle was 141-fold based on spring numbers and 268-fold based on August numbers. This is considerably higher than the amplitudes measured in Minnesota and Alberta. (5) The hare cycle was caused by changes in recruitment (probably determined by losses during the first 8 weeks of life), juvenile survival in autumn, and adult survival in autumn. Survival rates gradually decreased during the cycle, and were lowest in its decline phase. (6) Extra winter food did not prevent the drop in survival that occurs during the decline phase of the hare cycle, nor did it prevent the low recruitment rate that occurs during the decline.
This paper presents some concepts and methodology essential for the analysis of population dynamics of univoltine species. Simple stochastic difference equations, comprised of endogenous and exogenous components, are introduced to provide a basic structure for density-dependent population processes. The endogenous component of a population process is modelled as a function of density in the past p generations, including the most recent. The exogenous component of the process consists of all density-independent components of the ecological factors involved, including enhance variations. The model is called a p^t^h order density-dependent process. For a successful analysis of a population process by the above model, it is important that the process be in a state of statistical equilibrium, or stationarity. The simplest notion of stationarity is introduced, and the average behavior of the process, under this assumption, discussed. The order of density dependence in the population process of a given species depends on its interaction with other species involved in the food web. Considering certain attributes of the food web, in particular the limited number of trophic levels, the pyrmaid of numbers, the linear linkages between closely interacting species, and niche separation among competing species, it is argued that the order of density dependence is probably not much higher than three. A second-order model is perhaps adequate in many practical cases. The dynamics of some lower order density-dependent processes are compared by simulations, with a view to showing the effect of density-dependent and density-independent components at different orders. Several types of density dependence are discussed. If a given factor influencing the temporal variation in density is by itself influenced by density, it is called @'causally density-dependent,@' which may reveal itself by some degree of correlation with density. A density-independent factor, however, may also show some sort of correlation with density in the recent past. This is called @'statistically density-dependent.@' Such statistical density dependence may be due to: (1) spurious correlation, (2) bias in an estimator of the correlation coefficient, (3) autocorrelations in the density-independent factor, and (4) an intriguing mathematical property of the stochastic process. Particularly because of the last two reasons, it is often difficult to distinguish, by correlation method, between causal and statistical density dependence. Distinction also exists between temporal and spatial density dependence, the latter not necessarily implying the involvement of the former. The importance of the distinction between these types of density dependence is discussed in relation to the data analysis and model building. A Statistical analysis of the effect of ecological factors on population dynamics is attempted. Since it is often difficult to determine, by correlation, the causally density-dependent structure of a population process under the influence of some unknown density-independent factors, it is suggested that we reverse the procedure to determine the effect of the density-independent factors first. To confirm the involvement of some suspected density-independent factors in the species dynamics, I propose several methods of correlation between annual fluctuations in some population parameters, such as density, rate of change in density, and their transforms, and those in suitable indices of the suspected factors. Merits, demerits, and limitations of these methods are also discussed. To simplify the arguments, the correlation models are set up first without stage division, and then are elaborated to those in which the whole generation span is divided into several life-cycle stages, so that life table information can be used effectively for the identification of the density-independent factors involved in each stage. A set of life tables of the spruce budworm, Choristoneura fumiferana (Clem.), is analyzed to provide an example of the application of the above concepts and methods. Concluding remarks include some notes on designing life table studies.
The Charnov-Finerty hypothesis for explaining the microtine density cycle is discussed. Analyses of several numerical models lead us to conclude that the hypothesis is implausible; at best the assumed mechanisms will only lead to damped oscillations.
This report involves time-series statistical analysis (including concurrent physical and community variables) of the population dynamics of 4737 voles (Microtus californicus) trapped over 19 yr while emigrating from two study enclosures on a Northern California grassland. Population fluctuation of voles, as documented in the literature as well as in this study, generally cannot be described by periodic or regularly cyclic equations, but rather are either random or occasionally pseudoperiodic where autoregressive correlation explains irregular cycles on the basis of the population's previous demographic history. For these two California vole populations, an autoregressive component accounted for approximately a third of the total variability in the population, while random extrinsic environmental variation explained almost all of the remaining variation. Weather played a key determinant role, influencing microtine populations both directly, and indirectly through an effect on vegetation. The distinction between periodic, pseudoperiodic, and random fluctuations in vole populations, and indeed in wildlife populations in general, cannot be dismissed as merely a question of semantics, because each entails a specific ecologic interpretation. The demographic characteristics of microtine rodents, although fluctuating more dramatically, were similar to those of other rodents.
(1) The main features of the ten-year cycle are the regularity of the period and the irregularity of the amplitude of the oscillations; these features are obvious in data on the lynx cycle, and in the correlogram and periodogram calculated from the data. (2) A statistical model is proposed for the analysis of the 10-year cycle which takes these features into account by including both a strictly periodic term and an autoregressive term depending on the size of the population in the previous year; it is shown that this model fits the data better than the second-order autoregressive model used by Moran (1953). (3) All data on the ten-year cycle in Canada have the same period of about 9.63 years. The other parameters in the model can be estimated by the method of least squares, and estimates of the phase and the average amplitude of the oscillations can then be obtained. (4) This method is used to analyse data on the fur-bearing mammals of Canada between 1751 and 1969. A 10-year cycle exists during at least part of this time in the following species: coyote, fisher, red fox, lynx, marten, mink, muskrat, skunk, wolf, wolverine and snowshoe rabbit. A cycle has also been confirmed in the Atlantic salmon. There is a tendency for the cycle to be most pronounced in the midwest of Canada and to become weaker and later as one moves away from this region. A similar cycle exists in the taiga zone of Russia. (5) The simplest theory is that the cycle in all other species is caused, directly or indirectly, by the cycle in the snowshoe rabbit. The food habits of all the cyclic species are reviewed with this theory in mind. There is considerable difficulty in linking some of the cyclic species convincingly with the snowshoe rabbit, but this is nevertheless still thought to be the most likely explanation since no cyclic meteorological factor has been discovered.
Winter feeding experiments with captive snowshoe hares (Lepus americanus) indicated a mean daily requirement of about 300 g of woody browse having a maximum diameter of 3-4 mm. Food supplies were measured near Rochester, Alberta, during 6 consecutive winters, (1970-71 to 1975-76). Food was insufficient for hare populations on 2 study areas during a cyclic peak (winter 1970-71) and the following winter; food was still in short supply on a 3rd study area 2 winters after the peak. Twenty hare exclosures and their control plots, established in 1968 and sampled in summer 1971 and 1972, showed that hares had reduced the total biomass of woody stems ≤1.5 cm in diameter by more than 50%. Browsing-intensity surveys conducted in 1971, at the end of the peak winter, disclosed that almost 50% of the woody stems had been severely or heavily browsed; 2 years later, less than 2% were so intensively browsed. Marked changes occurred in nutrient levels of 6 common browse species monitored during 1969-74, but these appeared unrelated in any causal way to changes in hare population parameters. Mortality of malnourished captive hares was significantly related to ambient winter temperatures. Results of these field observations and feeding trials are discussed in relation to the hypothesis that cyclic declines of snowshoe hare populations are initiated by overwinter food shortage.
This paper reformulates the notion of density dependence and shows how this notion plays an important role in constructing appropriate models for data analysis. The regulation and persistence of population processes are interpreted as a close resemblance to the behavior of a series of random variables in which the second moments are bounded. On this basis the formal criteria of persistence are deduced. General structural models of population processes are set up and translated into discrete single-variable difference equations, ranging from the simplest linear first-order process to more complex nonlinear second-order processes. The discussion includes the derivation of general conditions for the second-order limit cycles, a reanalysis of the Canadian lynx 10-yr cycle, and models for population outbreaks. Based on the results of the preceding study of models, the notion of density dependence is reformulated. First, the meaning of the word 'dependence' is discussed. In the context of 'density dependence,' the word has two meanings; the causal dependence of a factor on density, and the statistical dependence. Statistical dependence is defined as a converse of statistical independence, the latter being a process in which the rate of change in density has zero correlation with density; this is a very special class of processes and is unlikely to occur in natural population processes. Therefore, the test of density dependence against the null hypothesis of statistical independence will not provide much insight. It is also argued that a deduction from the persistence criteria shows that a negative correlation between density and its rate of change is a necessary outcome of regulation and hence that the notion of 'density-dependent regulation' in statistical dependence is an uninspiring tautology. As opposed to statistical density independence, which necessarily generates an unbounded population process, causal density independence may satisfy the persistence conditions and hence may regulate populations. However, such a causally 'density-independent regulation' tends to be 'fragile' against perturbations by random exogenous factors. It is a particular class of causally density-dependent processes that can ensure regulation more durable against such perturbations. The inference of generating mechanism from observation is discussed. Although regression analysis is an essential method of inference, simple regression analysis will not work unless the observed processes are known to be a simple Markov chain. Statistical inference of generating mechanisms in observed systems depends largely on the choice of appropriate models, and it is in the construction of such models that the notion of causal density dependence plays an important role.
(1) From 1961 to 1988 small mammals were censused on four trap lines at the Boda Research Station in central Sweden. Clethrionomys glareolus was the dominant small mammal, comprising 85% of the total catch of 1955 small rodents. (2) Clethrionomys glareolus fluctuated cyclically with peaks averaging 3.6 years apart over the 28 years of the study and an average amplitude of 55-fold. Microtus agrestis and Apodemus flavicollis, although much less common, showed cyclic fluctuations in close synchrony with Clethrionomys glareolus. (3) Sorex araneus was much more common in the Boda area in the 1960s than it is today, but it also has fluctuated in synchrony with the Clethrionomys cycle. (4) Clethrionomys glareolus showed the Chitty Effect of increased body weights of adults in the peak density years.