ArticlePDF Available

Bifurcations and Dynamic Complexity in Simple Ecological Models

Authors:

Abstract

Many biological populations breed seasonally and have nonoverlapping generations, so that their dynamics are described by first-order difference equations, Nt+1 = F (Nt). In many cases, F(N) as a function of N will have a hump. We show, very generally, that as such a hump steepens, the dynamics goes from a stable point, to a bifurcating hierarchy of stable cycles of period 2n, into a region of chaotic behavior where the population exhibits an apparently random sequence of "outbreaks" followed by "crashes." We give a detailed account of the underlying mathematics of this process and review other situations (in two- and higher dimensional systems, or in differential equation systems) where apparently random dynamics can arise from bifurcation processes. This complicated behavior, in simple deterministic models, can have disturbing implications for the analysis and interpretation of biological data.
... The complexity of ecological dynamics has been a challenge for mathematical and theoretical ecologists for several decades. Population oscillations of variable amplitude, chaos, spatiotemporal pattern formation and regime shifts are just a few examples of commonly observed dynamical complexity [1][2][3]. It has long been understood that, to produce an equally complex behaviour, mathematical models of ecological dynamics must not necessarily be complex [4]. ...
... Factors and processes that define and shape the complexity of ecological dynamics have been a focus of research for several decades [1][2][3][4][5]18,19,21], yet there are many open questions. In particular, whether the dynamical complexity of ecosystems originates in their structural complexity (e.g. the complexity of corresponding foodwebs) is a controversial issue. ...
Article
Full-text available
The prey-predator system is an elementary building block in many complicated models of ecological dynamics that exhibit complex dynamical behaviour and as such it remains to be a focus of intense research. Here we consider a new model that incorporates multiple timescales (in the form of a 'slow-fast' system) with the ratio-dependent predator response. In the nonspatial case, we study this model exhaustively using an array of analytical tools to demonstrate the existence of canard cycles and relaxation oscillation passing through the close vicinity of the complicated singular point of the system. In the spatial case, the model exhibits a wide variety of spatio-temporal patterns that have been studied using extensive numerical simulations. In particular, we reveal the explicit dependence of the slow-fast timescale parameter on the Turing instability threshold and show how this affects the properties of the emerging stationary population patches. We also show that the self-organized spatial heterogeneity of the species can reduce the risk of extinction and can stabilize the chaotic oscillations. We argue that incorporating multiple timescales can enhance our understanding for studying realistic scenarios of local extinction and periodic outbreaks of the species. Our results suggest that the ubiquitous complexity of ecosystem dynamics observed in nature stems from the elementary level of ecological interactions such as the prey-predator system.
... Closed systems can be considered as autonomous units. If an extinction occurs it is irreversible (May & Oster, 1976;Caswell, 1978;Begon et ah, 1990). In closed systems population growth depends only upon the rate at which the individuals within the population can give rise to new offspring (e.g. ...
Thesis
p>This thesis examines the influence of recruitment on the persistence of populations in open and closed systems. Three systems were modelled, addressing questions concerning the influences on consumer dynamics of prey recruitment, consumer behaviour, and prey behaviour. Theoretical concepts were developed with mathematical models, and tested on observations from field experiments and empirical data collected from the literature. The following focal questions were addressed: 1. How do consumer populations respond to migration of their limiting prey into or out of the system? A Lotka-Volterra type model revealed that even a small prey outflux had catastrophic consequence for predator persistence. In contrast, the predator population was stabilised by subsidising it with prey influx. These outcomes for prey subsidy were compared to those for prey enrichment, which is known to destabilise populations (Rosenzweig's 'paradox of enrichment'). This research has implications for conservation and pest management. An influx to conserve a threatened population or an outflux to eliminate a pest population may have more effect when applied to the limiting resource than to the focal species itself. 2. How do populations of colonists respond to conspecifics in an open system? Recruitment of barnacles to rocky shores was monitored in experimental tests of alternative models for recruitment dynamics. 3. How do prey respond to dietary switching by their predators? A two-prey model was applied to tundra microtine populations (the 'predators') eating vegetation (the 'prey') with wound-induced toxic defences to predation. Parameter values were gathered from the literature, and model outputs were tested against five empirically derived criteria that characterise population cycles in this group. For plants without chemical defences the model met only four of these criteria.</p
... The dynamics of host and parasitoid populations can be modelled by the two-dimensional map introduced by Beddington (1975), Kot (2001), and Edelstein-Keshet (2005) as an extension of the Nicholson-Bailey (NB) model (Nicholson and Bailey, 1935). Since the original NB model predicts unstable oscillations for both populations, Beddington implemented a stabilizing factor, expressing the host reproductive rate as the exponential of the logistic increment (Moran, 1950;Ricker, 1954;May and Oster, 1976) ...
Article
Full-text available
Environmental stochasticity affects population dynamics in a variety of ways, including the possibility of drastic modifications in the stability properties of the ecosystem. In this work, we investigate a case of coupled host-parasitoid dynamics adopting Beddington’s conceptual two-dimensional map. We stochastically perturb some of the parameters controlling either the host dynamics or the host-parasitoid interaction, observing a dramatic change in the system dynamics with the emergence of on-off intermittency, a behavior characterized by the irregular alternation between quiescent phases and sudden population bursts. This phenomenon is herein offered as a qualitative, environmental-based description of population outbreaks.
... The growth of a population of bacteria, or any other type of organism, perfectly exemplifies this statement: The growth is initially exponential and approximates logistics when, with the increase in population density, the ecological conditions are changed. It has been shown that feedback, which also acts on the ecological level of biocenosis, is the connecting element between the theory of evolution and chaos theory (May and Oster 1976;May 1979). ...
Article
Full-text available
Contemporary scientific knowledge is built on both methodological and epistemological reductionism. The discovery of the limitations of the reductionist paradigm in the mathematical treatment of certain physical phenomena originated the notion of complexity, both as a pattern and process. After clarifying some very general terms and ideas on biological evolution and biological complexity, the article will tackle to seek to summarize the debate on biological complexity and discuss the difference between complexities of living and inert matter. Some examples of the major successes of mathematics applied to biological problems will follow; the notion of an intrinsic limitation in the application of mathematics to biological complexity as a global, relational, and historical phenomenon at the individual and species level will also be advanced.
... Zones with a higher density of dots are highlighted by lighter blue colors (Figure 11). Note also the bifurcation of the model output at r ≈ 2. This behavior is the result of the discretization of the logistic (May and Oster 1976). At r > 2, a cycle of length 2 emerges, followed as We can also avoid overplotting in plot_multiscatter() by randomly sampling and displaying only n simulations. ...
Article
Full-text available
The R package sensobol provides several functions to conduct variance-based uncertainty and sensitivity analysis, from the estimation of sensitivity indices to the visual representation of the results. It implements several state-of-the-art first and total-order estimators and allows the computation of up to fourth-order effects, as well as of the approximation error, in a swift and user-friendly way. Its flexibility makes it also appropriate for models with either a scalar or a multivariate output. We illustrate its functionality by conducting a variance-based sensitivity analysis of three classic models: the Sobol' (1998) G function, the logistic population growth model of Verhulst (1845), and the spruce budworm and forest model of Ludwig, Jones, and Holling (1976).
Article
Full-text available
An understanding of interactions between anthropogenic stressors and intrinsic population drivers is needed to fully understand wildlife population declines. Density dependence is a key aspect of population regulation for many species, especially for species that have high reproductive potential, such as amphibians. However, patterns of density dependence have been characterized for only a few species and little work has evaluated how density‐dependent interactions may be altered by anthropogenic stressors. We combined the results of a mesocosm experiment with demographic population modeling to investigate how the conversion of native prairie to agricultural grasslands dominated by Tall Fescue grass (Lolium arundinacea) affected larval density dependence and adult population size of an imperiled amphibian, Lithobates areolatus (Crawfish Frog). Overall, density dependence was overcompensatory, suggesting that L. areolatus exhibits scramble competition as larvae. Both vegetation treatments had low survival at high densities, but more individuals survived to metamorphosis at moderate densities in Fescue treatments compared to Prairie treatments. We evaluated the implications of our experimental results using a stochastic density‐dependent matrix population model to project long‐term population dynamics. Simulated populations breeding in Fescue‐dominated wetlands had a more variable population size and up to 400% higher probability of quasi‐extinction within 200 years, compared to populations breeding in ponds with prairie vegetation. Without varying density in experimental treatments and using mathematical models to project emergent population dynamics, our mesocosm experiment results would have suggested a slightly positive effect of Fescue grass on amphibian development and survival. Vegetation changes surrounding breeding wetlands might play an important role in the decline of amphibian populations persisting in low‐intensity agricultural areas. We combined the results of a mesocosm experiment with demographic population modeling to investigate how the conversion of native prairie to agricultural grasslands dominated by Tall Fescue grass (Lolium arundinacea) affected larval density dependence and adult population size of an imperiled amphibian, Lithobates areolatus (Crawfish Frog). We found that population size was more variable and that extinction risk was up to 4 times higher in simulated populations breeding in Fescue‐dominated wetlands. Our results show the importance of scaling individual‐level effects to the population level for understanding the effects of anthropogenic stressors.
Article
The periodic Ricker equation has been studied by several authors, including the present one. However, the periodic model derived from the original one has not been studied in detail. We show that the model often taken as a periodic Ricker model is a particular case of the original one and compare their dynamics. In particular, we characterize the parameter region where the model has a periodic point of period two, which is globally stable. We also compute the parameter regions where complex behavior is exhibited.
Article
Different from the discrete-time population models based on evolution of generations or life cycles, we formulate discrete-time homogeneous and stage-structured models with time steps in more general settings such that survivals are included at each time step. We assume that sterile mosquitoes are released and their number in the field is kept at a constant level. We study the interactive dynamics of wild and sterile mosquitoes where only sexually active sterile mosquitoes are considered. We determine threshold values of releases and investigate the interactive dynamics for both homogeneous and stage-structured populations. Numerical examples are provided to confirm and demonstrate the obtained theoretical results.
Article
Supertrack orbits are used to investigate boundary crises in an one-dimensional, two-parameter (ν,β), nonlinear Gauss map. After the crises, the time evolution of the orbit is shown to be pseudo-chaotic. We investigate the chaotic transient, that is, the time an orbit spends in a region where the chaotic attractor existed prior to the crisis, and confirm it decays exponentially with time. The relaxation time is given by a power-law τ∝μγ with μ=|β-βc| corresponding to the distance measured in the parameter where the crises are observed. βc is the parameter that characterizes the occurrence of a boundary crisis and the numerical value of the power measured was γ=1/2.
Article
Full-text available
Some of the simplest nonlinear difference equations describing the growth of biological populations with nonoverlapping generations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between 2 population points, to stable cycles with 4, 8, 16, . . . points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but boundedpopulation fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized analyses; its existence in such fully deterministic nonlinear difference equations is a fact of considerable mathematical and ecological interest.
Article
A biological population can be viewed as a distributed parameter dynamical system. The dynamics of host-parasite systems are investigated by coupling populations via age specific interactions; analysis and simulation results are compared with experiments performed on a laboratory ecosystem. A number of novel dynamical effects emerge from the model which have some interesting ecological consequences.
Article
THE logistic curve is often used in teaching ecology as a first description of growth of an animal population. For many reasons, frequency related to age structure and time-lag effects, it does not usually fit in practice; and a population may undergo oscillations of one type or another. The causes of oscillations have been discussed in detail by numerous authors (see refs. 1–5), some of whom propose more or less complex conditions which would generate them. Nevertheless, the logistic growth curve is a useful starting point in the study of population ecology, so that the following simple relation between, the rate of increase, generation time and the type of approach to the ‘saturation’ level may be of interest.
Article
Identify the conditions giving rise to stability and oscillations in single species populations interacting with a maintained resource.
Article
A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.
Article
It has been theoretically assumed that the population density at the equilibrium oscillates with damping from generation to generation. In the adult population of the southern cowpea weevil,Callosobruchus maculatus, it was exemplified. But, it was not so clear in the adult population of the azuki bean weevil,C. chinensis as seen in that ofC. maculatus. This difference seems to be due to the scramble type of competition that occurs in larval stage inC. maculatus, instead of in the egg stage asC. chinensis. Comparing with the oscillation from generation to generation obtained in the present experiment to that ofLucilia population found byNicholson, the oscillation inLucilia population is composed of the cycle in a generation and the descending phase of each cycle of it is not regulated density-dependently. The present result seems to be more appropriate for the demonstration of the theory of self-adjustment of population.
Article
The way phenomena or processes evolve or change in time is often described by differential equations or difference equations. One of the simplest mathematical situations occurs when the phenomenon can be described by a single number as, for example, when the number of children susceptible to some disease at the beginning of a school year can be estimated purely as a function of the number for the previous year. That is, when the number x n+1, at the beginning of the n + 1st year (or time period) can be written $${x_{n + 1}} = F({x_n}),$$ (1.1) where F maps an interval J into itself. Of course such a model for the year by year progress of the disease would be very simplistic and would contain only a shadow of the more complicated phenomena. For other phenomena this model might be more accurate. This equation has been used successfully to model the distribution of points of impact on a spinning bit for oil well drilling, as mentioned if [8, 11] knowing this distribution is helpful in predicting uneven wear of the bit. For another example, if a population of insects has discrete generations, the size of the n + 1st generation will be a function of the nth. A reasonable model would then be a generalized logistic equation $${x_{n + 1}} = r{x_n}[1 - {x_n}/K].$$ (1.2)
Article
This paper gives some history of the problem, and includes superperiod data for quadratic and cubic nonlinear terms, together with a computation for a prime number of particles in the string. Extension of the problem to a circular array is discussed, and there is a biliography.
Article
For biological populations with nonoverlapping generations, population growth takes place in discrete time steps and is described by difference equations. Some of the simplest such nonlinear difference equations can exhibit a remarkable spectrum of dynamical behavior, from stable equilibrium points, to stable cyclic oscillations between two population points, to stable cycles with four points, then eight, 16, etc., points, through to a chaotic regime in which (depending on the initial population value) cycles of any period, or even totally aperiodic but bounded population fluctuations, can occur. This rich dynamical structure is overlooked in conventional linearized stability analyses; its existence in the simplest and fully deterministic nonlinear (“density dependent”) difference equations is a fact of considerable mathematical and ecological interest.
Article
Differences in age specific demographic characteristics can considerably alter the behaviour of the population dynamics of a species or community of species. In this analysis techniques are developed which enable the stability of the equilibria of a set of models involving age structure to be investigated. The underlying model in all cases is a simple matrix representation of first order difference equations. The analysis enables the results of computer investigations of various population models of this type to be explained.