Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion

Xinhong Zhang; Qing Yang

doi:10.3934/dcdsb.2021177

Article Contents

2022, Volume 27, Issue 6: 3155-3175. Doi: 10.3934/dcdsb.2021177

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Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion

Xinhong Zhang^, and
Qing Yang

College of Science, China University of Petroleum, Qingdao 266580, Shandong Province, China

^* Corresponding author: Xinhong Zhang
^* Corresponding author: Xinhong Zhang

Received: February 28, 2021

Revised: April 30, 2021

Published: June 2022

The first author is supported by National Natural Science Foundation of China Grant 11801566 and the Fundamental Research Funds for the Central Universities of China grant 19CX02059A

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Abstract

In this paper, we consider a stochastic predator-prey model with general functional response, which is perturbed by nonlinear Lévy jumps. Firstly, We show that this model has a unique global positive solution with uniform boundedness of $ \theta\in(0,1] $-th moment. Secondly, we obtain the threshold for extinction and exponential ergodicity of the one-dimensional Logistic system with nonlinear perturbations. Then based on the results of Logistic system, we introduce a new technique to study the ergodic stationary distribution for the stochastic predator-prey model with general functional response and nonlinear jump-diffusion, and derive the sufficient and almost necessary condition for extinction and ergodicity.

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Mathematics Subject Classification: Primary: 92B05, 60G51; Secondary: 60H10, 60G57.

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Figure 1. The left column shows the numbers of $ x(t),y(t) $ in system (26) with $ \alpha_{11} = \alpha_{12} = 0.2 $ and $ \alpha_{21} = \alpha_{22} = 0.1 $. The right column represents the histogram of the probability density functions of $ x,y $ individuals

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Figure 2. Simulations of the solution in stochastic system (26) with initial value $ \alpha_{11} = \alpha_{12} = 0.2 $ and $ \alpha_{21} = \alpha_{22} = 1.5 $

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