Conditions for Permanence and Ergodicity of Certain Stochastic Predator-Prey Models
TL;DR: In this paper, sufficient conditions for the permanence and ergodicity of a stochastic predator-prey model with a Beddington-DeAngelis functional response were derived.
Abstract: In this paper we derive sufficient conditions for the permanence and ergodicity of a stochastic predator–prey model with a Beddington–DeAngelis functional response. The conditions obtained are in fact very close to the necessary conditions. Both nondegenerate and degenerate diffusions are considered. One of the distinctive features of our results is that they enable the characterization of the support of a unique invariant probability measure. It proves the convergence in total variation norm of the transition probability to the invariant measure. Comparisons to the existing literature and matters related to other stochastic predator–prey models are also given.
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TL;DR: In this article, sufficient conditions for the coexistence and exclusion of a stochastic competitive Lotka-Volterra model are derived, and convergence in distribution of positive solutions of the model is also established.
87 citations
TL;DR: This work studies the permanence and extinction of a regime-switching predator-prey model with the Beddington--DeAngelis functional response with criteria that can justify whether a prey dies out in a random switching environment.
Abstract: In this work we study the permanence and extinction of a regime-switching predator-prey model with the Beddington--DeAngelis functional response. The switching process is used to describe the random changes of corresponding parameters such as birth and death rates of a species in different environments. When a prey will die out in some fixed environments and will not in others, our criteria can justify whether it dies out in a random switching environment. Our criteria are rather sharp, and they cover the known on-off type results on permanence of predator-prey models without switching. Our method relies on the recent study of ergodicity of regime-switching diffusion processes.
72 citations
Abstract: We study the persistence and extinction of species in a simple food chain that is modelled by a Lotka–Volterra system with environmental stochasticity. There exist sharp results for deterministic Lotka–Volterra systems in the literature but few for their stochastic counterparts. The food chain we analyze consists of one prey and $$n-1$$
predators. The jth predator eats the $$j-1$$
th species and is eaten by the $$j+1$$
th predator; this way each species only interacts with at most two other species—the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on an explicit quantity depending on the interaction coefficients of the system, which species go extinct and which converge to their unique invariant probability measure. Our work can be seen as a natural extension of the deterministic results of Gard and Hallam ’79 to a stochastic setting. As one consequence we show that environmental stochasticity makes species more likely to go extinct. However, if the environmental fluctuations are small, persistence in the deterministic setting is preserved in the stochastic system. Our analysis also shows that the addition of a new apex predator makes, as expected, the different species more prone to extinction. Another novelty of our analysis is the fact that we can describe the behavior of the system when the noise is degenerate. This is relevant because of the possibility of strong correlations between the effects of the environment on the different species.
54 citations
TL;DR: In this article, the authors studied the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal.
Abstract: This work is devoted to studying the dynamics of a structured population that is subject to the combined effects of environmental stochasticity, competition for resources, spatio-temporal heterogeneity and dispersal. The population is spread throughout n patches whose population abundances are modeled as the solutions of a system of nonlinear stochastic differential equations living on $$[0,\infty )^n$$
. We prove that r, the stochastic growth rate of the total population in the absence of competition, determines the long-term behaviour of the population. The parameter r can be expressed as the Lyapunov exponent of an associated linearized system of stochastic differential equations. Detailed analysis shows that if $$ r>0$$
, the population abundances converge polynomially fast to a unique invariant probability measure on $$(0,\infty )^n$$
, while when $$ r<0$$
, the population abundances of the patches converge almost surely to 0 exponentially fast. This generalizes and extends the results of Evans et al. (J Math Biol 66(3):423–476, 2013) and proves one of their conjectures. Compared to recent developments, our model incorporates very general density-dependent growth rates and competition terms. Furthermore, we prove that persistence is robust to small, possibly density dependent, perturbations of the growth rates, dispersal matrix and covariance matrix of the environmental noise. We also show that the stochastic growth rate depends continuously on the coefficients. Our work allows the environmental noise driving our system to be degenerate. This is relevant from a biological point of view since, for example, the environments of the different patches can be perfectly correlated. We show how one can adapt the nondegenerate results to the degenerate setting. As an example we fully analyze the two-patch case, $$n=2$$
, and show that the stochastic growth rate is a decreasing function of the dispersion rate. In particular, coupling two sink patches can never yield persistence, in contrast to the results from the non-degenerate setting treated by Evans et al. which show that sometimes coupling by dispersal can make the system persistent.
53 citations
Posted Content•
TL;DR: In this article, the permanence and extinction of a regime-switching predator-prey model with Beddington-DeAngelis functional response was studied. And the authors used the ergodicity of regime switching diffusion processes to justify whether a predator die out or not when it will die out in some environments and not in others.
Abstract: In this work we study the permanence and extinction of a regime-switching predator-prey model with Beddington-DeAngelis functional response. The switching process is used to describe the random changing of corresponding parameters such as birth and death rates of a species in different environments. Our criteria can justify whether a prey die out or not when it will die out in some environments and will not in others. Our criteria are rather sharp, and they cover the known on-off type results on permanence of predator-prey models without switching. Our method relies on the recent study of ergodicity of regime-switching diffusion processes.
52 citations
References
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Book•
01 Jan 1981
TL;DR: In this article, Stochastic Differential Equations and Diffusion Processes are used to model the diffusion process in stochastic differential equations. But they do not consider the nonlinearity of diffusion processes.
Abstract: (1983). Stochastic Differential Equations and Diffusion Processes. Technometrics: Vol. 25, No. 2, pp. 208-208.
4,691 citations
Book•
01 Jan 1998TL;DR: In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities.
Abstract: Every form of behavior is shaped by trial and error. Such stepwise adaptation can occur through individual learning or through natural selection, the basis of evolution. Since the work of Maynard Smith and others, it has been realized how game theory can model this process. Evolutionary game theory replaces the static solutions of classical game theory by a dynamical approach centered not on the concept of rational players but on the population dynamics of behavioral programs. In this book the authors investigate the nonlinear dynamics of the self-regulation of social and economic behavior, and of the closely related interactions among species in ecological communities. Replicator equations describe how successful strategies spread and thereby create new conditions that can alter the basis of their success, i.e., to enable us to understand the strategic and genetic foundations of the endless chronicle of invasions and extinctions that punctuate evolution. In short, evolutionary game theory describes when to escalate a conflict, how to elicit cooperation, why to expect a balance of the sexes, and how to understand natural selection in mathematical terms.
Comprehensive treatment of ecological and game theoretic dynamics
Invasion dynamics and permanence as key concepts
Explanation in terms of games of things like competition between species
4,480 citations
TL;DR: Predation, one such process that affects numbers, forms the subject of the present paper and is based on the density-dependence concept of Smith ( 1955) and the competition theory of Nicholson (1933).
Abstract: The fluctuation of an animal's numbers between restricted limits is determined by a balance between that animal's capacity to increase and the environmenta1 cheks to this increase. Many authors have indulged in the calculating the propressive increase of a population when no checks nrerc operating. Thus Huxley calculated that the progeny of a single Aphis in the course of 10 generations, supposing all survived,would “contain more ponderable substance than five hundred millions of stout men; that is, more than the whole population of China”, (in Thompson, 1929). Checks, however, do occur and it has been the subject of much controversy to determine how these checks operate. Certain general principles—the density-dependence concept of Smith ( 1955) , the competition theory of Nicholson (1933)—have been proposed both verbally and mathematically, but because they have been based in part upon untested and restrictive assumptions they have been severelv criticized (e.g. Andrewartha and Birch 1954). These problems could be considerably clarified if we knew the mode of operation of each process that affects numbers, if we knew its basic and subsidiary components. predation, one such process, forms the subject of the present paper.
3,087 citations
TL;DR: A number of experiments indicate that a basic assumption implicit in early population models of predator-prey and parasite-host interactions is false.
Abstract: A number of experiments (Burnett 1956; Hassell & Huffaker 1969; Holling 1959; Ullyett 1949a,b, 1950) indicate that a basic assumption implicit in early population models of predator-prey and parasite-host interactions (Lotka 1925; Yolterra 1926; Nicholson & Bailey 1935) is false. This assumption requires the number of hosts parasitized (or prey attacked) to be proportional to the density of hosts and the density of parasites*. It is possible to formalize this assumption in terms of searching efflciency (E) where E is deSned by E Na (1)
1,424 citations
Book•
01 Dec 1996TL;DR: In this paper, the authors present a generalization of the Hamilton-Jacobi theory for systems on Lie groups and homogenous spaces, including linear and polynomial control systems with quadratic costs.
Abstract: Introduction Acknowledgments Part I. Reachable Sets and Controllability: 1. Basic formalism and typical problems 2. Orbits of families of vector fields 3. Reachable sets of Lie-determined systems 4. Control affine systems 5. Linear and polynomial control systems 6. Systems on Lie groups and homogenous spaces Part II. Optimal Control Theory: 7. Linear systems with quadratic costs 8. The Riccati equation and quadratic systems 9. Singular linear quadratic problems 10. Time-optimal problems and Fuller's phenomenon 11. The maximum principle 12. Optimal problems on Lie groups 13. Symmetry, integrability and the Hamilton-Jacobi theory 14. Integrable Hamiltonian systems on Lie groups: the elastic problem, its non-Euclidean analogues and the rolling-sphere problem References Index.
1,066 citations