Hybrid competitive Lotka–Volterra ecosystems with a hidden Markov chain

Abstract

This work concerns Lotka–Volterra models that are formulated using stochastic differential equations with regime-switching. Distinct from the existing formulations, the Markov chain that models random environments is unobservable. For such partially observed systems, we use Wonham’s filter to estimate the Markov chain from the observable evolution of the population, and convert the original system to a completely observable one. We then show that the positive solution of our model does not explode in finite time with probability 1. Several properties including stochastic boundedness, finite moments, sample path continuity and large-time asymptotic behaviour are also obtained. Moreover, stochastic permanence, extinction and feedback controls are also investigated.