The Ergodic Property And Positive Recurrence Of A Multi-group Lotka-Volterra Mutualistic System With Regime Switching

It is well known that population systems are often subject to environmental noise.The population is forced to expire when the perturbation is sufficiently large, while there are many nice properties when the perturbation is relatively small. For example, it can suppress potential explosions in population dynamics, promote or suppress exponential growth of population, different structure of white noise has different effects on the population system and so on.In this paper, we mainly consider a classic n-gram interacting stochastic multi-group Lotka-Volterra mutualistic system under regime switching() = (1(), 2(), · · ·, ())[((()) + (())()) + (())()],where () =(1(), 2(), · · ·, ())is a d-dimensional standard Brownian Motion.{(), ≥ 0} is a finite irreducible right-continuous Markov chain independent of the Brownian motion B, taking values in a finite state space = {1, 2, · · ·, }. Furthermore,{(), ≥ 0} may be affected by factors such as nutrition, rainfall, temperature, pressure and will be continuously reciprocating changes.The main aim here is to investigate the ergodic property and positive recurrence of the multi-group Lotka-Volterra mutualistic system by stochastic Lyapunov functions under small perturbation. The mean of the stationary distribution is also estimated which can be used to explain some recurring phenomena in practice and thus provide a good description of permanence.