# American Institute of Mathematical Sciences

November  2017, 22(9): 3499-3528. doi: 10.3934/dcdsb.2017177

## On stochastic multi-group Lotka-Volterra ecosystems with regime switching

 School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China

* Corresponding author: Xiaoyue Li

Received  August 2016 Revised  May 2017 Published  July 2017

Fund Project: The second author is s supported by Natural Science Foundation of Jilin Province (No. 20170101044JC), the Education Department of Jilin Province (No.JJKH20170904KJ) and by the National Natural Science Foundation of China (11171056,11571065,11671072).

Focusing on stochastic dynamics involving continuous states as well as discrete events, this paper investigates dynamical behaviors of stochastic multi-group Lotka-Volterra model with regime switching. The contributions of the paper lie on: (a) giving the sufficient conditions of stochastic permanence for generic stochastic multi-group Lotka-Volterra model, which are much weaker than the existing results in the literature; (b) obtaining the stochastic strong permanence and ergodic property for the mutualistic systems; (c) establishing the almost surely asymptotic estimate of solutions. These can specify some realistic recurring phenomena and reveal the fact that regime switching can suppress the impermanence. A couple of examples and numerical simulations are given to illustrate our results.

Citation: Rui Wang, Xiaoyue Li, Denis S. Mukama. On stochastic multi-group Lotka-Volterra ecosystems with regime switching. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3499-3528. doi: 10.3934/dcdsb.2017177
##### References:

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##### References:
Sample paths of $|x(t)|$ of (4) (left) and (5) (right)
A sample path of $|x(t)|$ of the switching system
Sample paths of $|x(t)|$ of state-$1$ (left) and state-$2$ (right) in Example 7.1
Case 1. A sample path of $|x(t)|$ of the switching system in Example 7.1
Case 2. A sample path of $|x(t)|$ of the switching system in Example 7.1
A sample path of $x_1(t)$ and $x_2(t)$ of state-$1$, state-$2$ and state-$3$ in Example 7.2
Stationary distribution and scatter plot of a sample path of state-$1$ in Example 7.2
Stationary distribution and scatter plot of a sample path of state-$3$ in Example 7.2
Case 1. A sample path of $x_1(t)$ and $x_2(t)$ of the switching system in Example 7.2
Case 1. Stationary distribution and scatter plot of a sample path of the switching system in Example 7.2
Case 1. A sample path in time average of the switching system in Example 7.2
Case2. A sample path of $x_1(t)$ and $x_2(t)$ of the switching system in Example 7.2
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