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

Rui Wang; Xiaoyue Li; Denis S. Mukama

doi:10.3934/dcdsb.2017177

Article Contents

2017, Volume 22, Issue 9: 3499-3528. Doi: 10.3934/dcdsb.2017177

This issue Previous Article Dynamic behavior of a stochastic predator-prey system under regime switching Next Article A preconditioned fast Hermite finite element method for space-fractional diffusion equations

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
* Corresponding author: Xiaoyue Li

Received: July 31, 2016

Revised: April 30, 2017

Published: October 2017

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)

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Abstract

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.

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Mathematics Subject Classification: Primary:60H10;Secondary:92B05.

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Figure 1. Sample paths of $|x(t)|$ of (4) (left) and (5) (right)

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Figure 2. A sample path of $|x(t)|$ of the switching system

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Figure 3. Sample paths of $|x(t)|$ of state-$1$ (left) and state-$2$ (right) in Example 7.1

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Figure 4. Case 1. A sample path of $|x(t)|$ of the switching system in Example 7.1

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Figure 5. Case 2. A sample path of $|x(t)|$ of the switching system in Example 7.1

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Figure 6. A sample path of $ x_1(t) $ and $ x_2(t) $ of state-$1$, state-$2$ and state-$3$ in Example 7.2

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Figure 7. Stationary distribution and scatter plot of a sample path of state-$1$ in Example 7.2

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Figure 8. Stationary distribution and scatter plot of a sample path of state-$3$ in Example 7.2

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Figure 9. Case 1. A sample path of $ x_1(t) $ and $x_2(t)$ of the switching system in Example 7.2

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Figure 10. Case 1. Stationary distribution and scatter plot of a sample path of the switching system in Example 7.2

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Figure 11. Case 1. A sample path in time average of the switching system in Example 7.2

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Figure 12. Case2. A sample path of $x_1(t)$ and $x_2(t)$ of the switching system in Example 7.2

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