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# Simple Mathematical Models With Very Complicated Dynamics

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## Abstract

First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hiearchy of stable cycles, to apparently random fluctuations. There are consequently many fascinating problems, some concerned with delicate mathematical aspects of the fine structure of the trajectories, and some concerned with the practical implications and applications. This is an interpretive review of them.
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
© Nature Publishing Group1976
... In the theoretical study of epidemics, mathematical models with discrete time are effectively used [3][4][5][6]. In population dynamics, an important role is played by the simple conceptual models with logistic-and Rickertype maps [4,7,8]. Due to the strong nonlinearity, these models exhibit complex regular and chaotic regimes. ...
... In the study of population systems, it is generally accepted to take into account the impact of unavoidable random perturbations [8,[15][16][17][18][19][20]. Even small stochastic disturbances in biological systems can dramatically change its dynamics [21][22][23]. ...
... Logistic-type models are widely used in the study of population dynamics. The logistic map combines simplicity and representativeness, allowing even in the one-dimensional case to simulate equilibrium regimes, as well as regular and chaotic oscillations [8,34,35]. ...
Article
Motivated by the important problem of analyzing and predicting the spread of epidemics, we propose and study a discrete susceptible-infected model. This logistic-type model accounts such significant parameters as the rate of infection spread due to contacts, mortality caused by disease, and the rate of recovery. We present results of the bifurcation analysis of regular and chaotic survival regimes for interacting susceptible and infected subpopulations. Parametric zones of multistability are found and basins of coexisting attractors are determined. We also discuss the particular role of specific transients. In-phase and anti-phase synchronization in the oscillations of the susceptible and infected parts of the population is studied. An impact of inevitably present random disturbances is studied numerically and by the analytical method of confidence domains. Various mechanisms of noise-induced extinction in this epidemiological model are discussed.
... Over the last few years, the study of population dynamics through mathematical modelling becomes more interesting to the theoretical ecologists and mathematical biologists due to its rich dynamics and wide applications (May, 1976;Hastings and Powell, 1991). For modelling the population dynamics, two types of mathematical models are popular, namely, continuous-time models, represented by differential equations and discrete-time models, represented by difference equations. ...
... Moreover, the discrete-time model gives rich and complex dynamics. For instance, a single-species discrete-time model can exhibit chaos (May, 1976) and more complex dynamical behaviour whereas in a continuous-time autonomous model minimum three-species are needed for exhibiting chaos (Hastings and Powell, 1991). ...
Thesis
Full-text available
This thesis entitled “Hunting cooperation among predators: A mathematical study of ecological models” attempts to study the effect of hunting cooperation among predators in ecological systems. In nature, many predator species cooperate during hunting for a variety of reasons, such as (i) hunting and killing large prey, (ii) searching for food, (iii) increasing the ability to capture (or subdue) prey, (iv) attacking a herd of prey, (v) increasing vigilance and protection against other predators, (vi) preventing the theft of corpses by other hunters, (vii) reducing the distance of chasing, etc. Thus how predators enhance their biomass by group hunting, and as a result, how they impact prey biomass is a natural issue. Although, the study of cooperative behaviour during hunting is not new, in the mathematical modelling approach there are only a few studies. Thus the main objective of the study is to observe the dynamics of two-species predator-prey models in the presence of hunting cooperation among predators. In Chapter 1, first, we briefly described the evolution of mathematical models in ecology with the emphasis of two-species interaction and different types of functional response. Then characterize the cooperative hunting phenomenon in an ecological context. Here, we mainly concern with the following ecological issues: (1) why predators cooperate during hunting? (2) example of different cooperative hunting species and their strategies, (3) how to incorporate cooperative hunting phenomenon in ecological models? In Chapter 2, we consider a discrete-time predator-prey model with logistic type prey growth to study the impact of hunting cooperation. We investigate the basic analysis of the discrete model such as fixed points, stability, and bifurcation analysis, and explore that hunting cooperation has the potential to modify the well-known period-doubling route to chaos by reverse period-halving bifurcations and makes the system stable. Also, for an additional increase of the strength of hunting cooperation, the system exhibits chaotic oscillations via Neimark-Sacker bifurcation. However, very high hunting cooperation can be detrimental for the system and populations go to extinction. This is because of the overexploitation of the prey populations by their predators. The discrete system shows bistability behaviour between prey only fixed point and interior fixed point, and the basin of attraction of the interior fixed point increases with the strength of hunting cooperation. Moreover, hunting cooperation induces a strong demographic Allee effect in the discrete system, where predator populations persist due to cooperation during hunting and would go to extinction without hunting cooperation. In Chapter 3, we extend the continuous version of the model studied in the previous chapter, by incorporating predator induce fear in the birth rate of the prey population. We observe that without hunting cooperation, the unique coexistence equilibrium point is globally asymptotically stable. However, an increase in the hunting cooperation induce fear may destabilize the system and produce periodic solution via Hopf bifurcation. We explore very rich dynamics such as both supercritical and subcritical Hopf bifurcations, Bogdanov-Takens bifurcation, backward bifurcation, and different types of bistabilities. This model also generates strong demographic Allee effect in predator species. In Chapter 4, we study another type of model namely, the modified Leslie-Gower model with the same phenomena, hunting cooperation in predators and fear effect in prey. The main feature of Leslie-Gower model with compare to the Lotka-Volterra model is the logistic growth of predators, where the carrying capacity of the predator species is proportional to the prey biomass. We observe that the fear factor can stabilize the predator-prey system by excluding the existence of periodic solutions and makes the system more robust compared to hunting cooperation. This system also shows very rich dynamics such as Hopf bifurcation, Bogdanov-Takens bifurcation, and multi-type bistabilities. Chapter 5 is devoted to exploring the impact of time delay during cooperative hunting in a predator-prey model. Cooperative hunting predators do not aggregate in a group instantly but individuals use different stages and strategies such as tactile, visual, vocal cues/signals, or a suitable combination of these to communicate with each other. It is indeed plausible to add some time delay representing the delay in forming a group and ready for attack. Generally, delay has a destabilizing effect on predator-prey dynamics, but in our model, delay has both stabilizing as well as destabilizing effects. Also, for an increase in the strength of the delay, then system dynamics switch multiple times and finally become chaotic. We see that depending on the threshold of time delay, the system may restore its original state or may go far away from its original state and unable to recollect its memory. We also observe different kinds of multistability behaviours, coexistence of multiple attractors, and interesting changes in the basins of attractions of the system. We infer that depending on the initial population size and the strength of cooperation delay, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species. Hunting cooperation has both stabilizing and destabilizing effects on the dynamics of the systems depending on the values of model parameters. It can mediate the survival of predators, where predators go extinct without cooperation. Mathematical models with hunting cooperation among predators may exhibit different types of bistabilities (node-node/node-cycle). This can be biologically interpreted as, depending on the initial population size, the populations can exhibit stable coexistence, oscillating coexistence, or extinction of the predator species (for the case of specialist predators). Predator-prey models with hunting cooperation among predators exhibit rich dynamical behaviours. We believe that, this research work will definitely enrich the existing knowledge about the impact of hunting cooperation in predators on various mathematical models in ecology.
... In particular, it captures both the stable and chaotic aspects of population growth. The logistic map has been extensively studied since the seminal work of Robert May [11]. For example, many protozoan populations such as yeast and grain beetles illustrate logistic growth in laboratory studies [12]. ...
... Here we recall a few properties of the deterministic logistic map, S λ (x). For a more complete discussion see, for example, [11,5,14]. ...
Preprint
Full-text available
The logistic map is a nonlinear difference equation well studied in the literature, used to model self-limiting growth in certain populations. It is known that, under certain regularity conditions, the stochastic logistic map, where the parameter is varied according to a specified distribution, has a unique invariant distribution. In these cases we can compare the long-term behavior of the stochastic system with that of the deterministic system evaluated at the average parameter value. Here we examine the relationship between the mean of the stochastic logistic equation and the mean of orbits of the deterministic logistic equation at the expected value of the parameter. We show that, in some cases, the addition of noise is beneficial to the populations, in the sense that it increases the mean, while for other ranges of parameters it is detrimental. A conjecture based on numerical evidence presented at the end.
... However, our results clearly show that scalar population models typically mischaracterize dynamics, treating complexity as noise and leading to the conclusion that chaos is rare [17][18][19]55 . As noted by Robert May, such models "do great violence to reality" 56 . More flexible methods (for example, refs. ...
Article
Full-text available
Chaotic dynamics are thought to be rare in natural populations but this may be due to methodological and data limitations, rather than the inherent stability of ecosystems. Following extensive simulation testing, we applied multiple chaos detection methods to a global database of 172 population time series and found evidence for chaos in >30%. In contrast, fitting traditional one-dimensional models identified <10% as chaotic. Chaos was most prevalent among plankton and insects and least among birds and mammals. Lyapunov exponents declined with generation time and scaled as the −1/6 power of body mass among chaotic populations. These results demonstrate that chaos is not rare in natural populations, indicating that there may be intrinsic limits to ecological forecasting and cautioning against the use of steady-state approaches to conservation and management.
... This is the most commonly used chaotic map and, being one-dimensional, it is among the simplest maps as well. Biologist Robert May first introduced this map in 1976 [45]. ...
Article
In this paper, we propose an adaptive encryption scheme for color images using Multiple Distinct Chaotic Maps (MDCM) and DNA computing. We have chosen three distinct chaotic maps, including a 2D-Henon map, a Tent map, and a Logistic map, to separately encrypt the red, green, and blue channels of the original image. The proposed scheme adaptively modifies the parameters of the maps, utilizing various statistical characteristics such as mean, variance, and median of the image to be encrypted. Thus, whenever there is a change in the plain image, the secret keys also change. This makes the proposed scheme robust against the chosen and known plaintext attacks. DNA encoding has also been used to add another layer of security. The experimental analysis of the proposed scheme shows that the average value of entropy is approximately eight, the Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are 99.61% and 33%, respectively, and correlation coefficients close to zero, making the scheme not only reliable but also resilient against many attacks. Moreover, the use of low-dimensional maps reduces the computational costs of the scheme to a large extent.
Chapter
The transmission of images covers the highest percentage of data, and thus, the protection of the image becomes one of the key security concerns. The chaotic cryptographic algorithms forms an efficient way to achieve secure image encryption because of its unique properties like, sensitivity to initial conditions, and randomlike behavior. In this paper, an encryption scheme based on two chaotic maps, a 1-D Logistic map and a 2-D Difference map, is proposed which makes use of a secret key, generated from the maps. The image encryption and decryption are implemented using MATLAB, and its performance is analyzed using various tests which includes key sensitivity analysis, PSNR, correlation coefficient analysis, key space analysis, and statistical analysis using NPCR, UACI, and MAE tests. The uniform distribution of the histogram and the simulation results indicate a reduction in the possibility of static attacks, high accuracy, and resistance against noise.
Article
In this paper, we propose to use a computational method of chaos control to simulate complex experimental spectra. This computational chaos control technique is based on the Ott–Grebogi–York (OGY) method. We chose the logistic map as the base mathematical model for the development of our work. For the numeric part, we created arbitrary precision algorithms to generate the solutions. This way, we completely eliminated any degradation of chaos from our results. These algorithms were also necessary for the proper perturbation process that the computational chaos control method requires. We control the chaos of the logistic map in two cases of Period 1 and one case of Period 2 to demonstrate that our control method works. The behavior of a complex experimental spectrum was taken and numerically simulated. The simulated spectrum was obtained by controlling the chaos of the logistic map in a variable way with the methods proposed in this work. Our results show that it is possible to simulate very complicated experimental spectra by computationally controlling the chaos of an equation unrelated to the experimental system.
Article
Mathematical theories of populations have been derived and effectively used in many contexts in the last two hundred years They have appeared both implicitly and explicitly in many important studies of populations: human populations as well as populations of animals, cells and viruses. Several features of populations can be analyzed. First, growth and age structure can be studied by considering birth and death forces acting to change them. In particular, the long time state of the population and its sensitivity to changes in birth and death schedules can be determined. Another population phenomenon which is amenable to analysis is the way various individual traits are propagated from one generation to the next. An extensive theory for this has evolved from the first observations of inheritance by Gregor Mendel. In addition, the spread of a contagious phenomenon, such as disease, rumor, fad or information, can be studied by means of mathematical analysis. Of particular importance in this area are studies of the dependence of contagion on parameters such as contact and quarantine rates. Finally, the dynamics of several interacting populations can be analyzed. Theories of interaction have become useful with recent studies of ecological systems and economic and social structures. The most direct approach to the study of populations is the collection and analysis of data. However, serious questions arise about how this should be done. For one thing, facilities can be swamped by even simple manipulation and analysis of vast amounts of data. In addition, there are the questions: Which data should be collected? Which data adequately describe the phenomenon, in particular which are sensitive indicators for detecting the presence of a phenomenon? A study of the population's underlying structure is essential for answering these questions, and mathematical theories provide a systematic way for doing this. Many techniques for analyzing complicated physical problems can be applied to population problems. In addition, many new techniques peculiar to these problems must be developed. Several population problems will be analyzed here which illustrate these methods and techniques. The monograph begins with a study of population age structure. A basic model is derived first, and it reappears frequently throughout the remainder. Various extensions and modifications of the basic model are then applied to several population phenomena, such as stable age distributions, self-limiting effects and two-sex populations.
Article
This paper is concerned with certain qualitative aspects of the sensitivity problem in relation to small variations of a parameter of a system, the behaviour of which can be described by an autonomous recurrence relation: Vn+1 = F(Vn, λ) (1) V being a vector, λ the parameter. The problem consists in the determination of the bifurcation values λ0 of λ, i.e. values such that the qualitative behaviour of a solution of (1) should be different for λ = λ0 ± ε where ε is a small quantity. Bifurcations that correspond to a critical case in the Liapunov sense, and the crossing through this critical case, are considered. Examples of bifurcations, not connected with the presence of a critical case, and which correspond to a large deformation of the stability domain boundary of an equilibrium point, a fixed point of (1), under the effect of a parameter variation, are given where V is a two dimensional vector.
Article
By taking account of obvious and inescapable limitations on the functioning of the accelerator, we explain some of the chief characteristics of the cycle, notably its failure to die away, along with the fact that capital stock is usually either in excess or in short supply. By a succession of increasingly complex models, the nature and methods of analysing non-linear cycle models is developed. The roles of lags and of secular evolution are illustrated. In each case the system’s equilibrium position is unstable, but there exists a stable limit cycle toward which all motions tend.