Persistence and extinction for stochastic delay differential model of prey predator system with hunting cooperation in predators

Fathalla A Rihan,Hebatallah J Alsakaji

doi:10.1186/s13662-020-02579-z

Fathalla A Rihan, Hebatallah J Alsakaji

Open Access

https://doi.org/10.1186/s13662-020-02579-z

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Abstract

Stochastic differential models provide an additional degree of realism compared to their corresponding deterministic counterparts because of the randomness and stochasticity of real life. In this work, we study the dynamics of a stochastic delay differential model for prey–predator system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Illustrative examples and numerical simulations, using Milstein’s scheme, are carried out to validate our analytical findings. It is observed that a small scale of white noise can promote the survival of both species; while large noises can lead to extinction of the predator population.

Highlights

Prey–predator (PP) interaction is one of the most extensively studied issues in ecological and mathematical literature; see [1,2,3]
5 Discussion and conclusion In this paper, we studied the dynamics of stochastic delay differential equations (SDDEs) for a prey–predator system with hunting cooperation in predators
We investigated the effect of environmental fluctuations on the model

Summary

Introduction

Prey–predator (PP) interaction is one of the most extensively studied issues in ecological and mathematical literature; see [1,2,3]. In [27], the authors investigated the complex dynamics of a delayed PP system with cooperation among the prey species, they have considered time delays in the growth components for each of the species. Hattaf et al [14] studied the impact of random noise in the dynamics of delayed SIR epidemic model They deduced a threshold parameter to determine the extinction and persistence of the disease. In [32], the authors studied the effect of environmental fluctuations of a delayed Harrison-type PP model, they analyzed the impact of the combination of delay and noise in the dynamical behavior of the model. In [33], the authors studied the effect of environmental fluctuations on a competitive model for two phytoplankton species where one species liberate toxic substances by considering a discrete time delay parameter in the growth equations of both species.

Deterministic analysis

Discussion and conclusion