Hybrid competitive Lotka–Volterra ecosystems: countable switching states and two-time-scale models

Abstract

This work is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. Our main objective of the paper is to reduce the computational complexity by using the two-time-scale systems. Because existence and uniqueness as well as continuity of solutions for Lotka-Volterra ecosystems with Markovian switching in which the switching takes place in a countable set are not available, such properties are studied first. The two-time scale feature is highlighted by introducing a small parameter into the generator of the Markov chain. When the small parameter goes to 0, there is a limit system or reduced system. It is established in this paper that if the reduced system possesses certain properties such as permanence and extinction, etc., then the complex system also has the same properties when the parameter is sufficiently small. These results are obtained by using the perturbed Lyapunov function methods. Lotka [17] and Volterra [29], the well-known Lotka-Volterra models have been investigated extensively in the literature and used widely in ecological and population dynamics, among others. When two or more species live in close proximity and share the same basic requirements, they usually compete for resources, food, habitat, or territory. Initially being posed as a deterministic model, subsequent study has taken randomness into consideration; see [6,10,24] and references therein. Recent effort on the so-called hybrid systems has much enlarged the applicability of Lotka-Volterra systems. One class of such hybrid systems uses a continuous-time Markov chain to model environmental changes and other random factors not represented in the usual stochastic differential equations; see [33] for a comprehensive study of switching diffusions. Introduced byDeterministic Lotka-Volterra systems have been studied by many people. A number of important results were obtained. A set of sufficient conditions for the existence of a globally stable equilibrium point in various models of the n-dimensional Lotka-Volterra system was obtained in [16]; limit cycles for some deterministic competitive three-dimensional Lotka-Volterra systems were treated in [30]. Random perturbations to the Lotka-Volterra model were considered in the literature; see for example [3,12] and many references therein, and also [11,13] for up-to-dated progress on stochastic replicator dynamics. Recently, much effort has been devoted to studying the stochastic Lotka-Volterra with regime-switching; see [7,18,27,34,35]. While most recent works focus on Markov chain with a finite state space, to take into consideration of various factors, it is also natural to consider the Markov chain with a countable state space, which is the effort of the current paper.One of our main aims here is to reduce the computational complexity. Assuming that the Markov chain has a two-time-scale structure so that it has a fast changing part and a slowly varying part, we show that the systems under consi...

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