Analysis of stochastic disease including predator-prey model with fear factor and Lévy jump

Xueqing He; Ming Liu; Xiaofeng Xu; Xueqing He; Ming Liu; Xiaofeng Xu

doi:10.3934/mbe.2023080

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 1750-1773. doi: 10.3934/mbe.2023080

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Research article

Analysis of stochastic disease including predator-prey model with fear factor and Lévy jump

1.
Department of Mathematics, Northeast Forestry University, Harbin, Heilongjiang 150040, China
2.
School of Mathematical Sciences, Heilongjiang University, Harbin, Heilongjiang 150080, China

Academic Editor: Troy Day

Received: 24 August 2022 Revised: 18 October 2022 Accepted: 24 October 2022 Published: 07 November 2022

In this paper, we investigate the dynamical properties of a stochastic predator-prey model with a fear effect. We also introduce infectious disease factors into prey populations and distinguish prey populations into susceptible prey and infected prey populations. Then, we discuss the effect of Lévy noise on the population considering extreme environmental situations. First of all, we prove the existence of a unique global positive solution for this system. Second, we demonstrate the conditions for the extinction of three populations. Under the conditions that infectious diseases are effectively prevented, the conditions for the existence and extinction of susceptible prey populations and predator populations are explored. Third, the stochastic ultimate boundedness of system and the ergodic stationary distribution without Lévy noise are also demonstrated. Finally, we use numerical simulations to verify the conclusions obtained and summarize the work of the paper.
- Lévy noise,
- fear effect,
- stationary distribution,
- stochastic predator-prey model
Citation: Xueqing He, Ming Liu, Xiaofeng Xu. Analysis of stochastic disease including predator-prey model with fear factor and Lévy jump[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 1750-1773. doi: 10.3934/mbe.2023080

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Abstract

In this paper, we investigate the dynamical properties of a stochastic predator-prey model with a fear effect. We also introduce infectious disease factors into prey populations and distinguish prey populations into susceptible prey and infected prey populations. Then, we discuss the effect of Lévy noise on the population considering extreme environmental situations. First of all, we prove the existence of a unique global positive solution for this system. Second, we demonstrate the conditions for the extinction of three populations. Under the conditions that infectious diseases are effectively prevented, the conditions for the existence and extinction of susceptible prey populations and predator populations are explored. Third, the stochastic ultimate boundedness of system and the ergodic stationary distribution without Lévy noise are also demonstrated. Finally, we use numerical simulations to verify the conclusions obtained and summarize the work of the paper.

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