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.

Keywords:

• Lotka–Volterra model
• perturbed Lyapunov method
• Markov chain
• singular perturbation
• extinction
• permanence
• switching diffusion

Mathematics Subject Classification:

• 60J60
• 60J27
• 92D25
• 92D40

Additional information

Funding

This research was supported in part by the National Science Foundation under grant DMS-1207667.