# American Institute of Mathematical Sciences

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

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

* Corresponding author: Xinhong Zhang

Received  March 2021 Revised  May 2021 Early access July 2021

Fund Project: 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

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.

Citation: Xinhong Zhang, Qing Yang. Dynamical behavior of a stochastic predator-prey model with general functional response and nonlinear jump-diffusion. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021177
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##### References:
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
Simulations of the solution in stochastic system (26) with initial value $\alpha_{11} = \alpha_{12} = 0.2$ and $\alpha_{21} = \alpha_{22} = 1.5$