Paradox of Enrichment in a Stochastic Predator-Prey Model

Jawdat Alebraheem,Velusamy Vijayakumar

doi:10.1155/2020/8864999

Abstract

We propose a stochastic predator-prey model to study a novel idea that involves investigating random noises effects on the enrichment paradox phenomenon. Existence and stochastic boundedness of a unique positive solution with positive initial conditions are proved. The global asymptotic stability is studied to determine the occurrence of the enrichment paradox phenomenon. We show theoretically that intensive noises play an important role in the occurrence of the phenomenon, where increasing intensive noises lead to occurrence of the paradox of enrichment. We perform numerical simulations to verify and demonstrate the theoretical results. The new results in this study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.

Highlights

Theoretical ecology has grown to the extent that we regularly see papers in mathematical ecology discussing models in purely mathematical terms that from time to time bear little resemblance to real ecological processes
Applying the It􏽢o formula, as doing the same procedure for first equation of the system ((6a) and (6b)), we obtain the unique solution of stochastic differential equation (16) as t eα ξ (s)ds+σ 2 W2 ξ 2 (t) Y0 e 􏽒0 1
By applying the Ito formula, we have the unique solution of stochastic differential equation (20) as d ξ 1 􏼁 ρξ 1 dt + σ 1 ξ 1 dW1, H1 (t) ρ−σ 21 )t− α 􏽒

Summary

Introduction

Theoretical ecology has grown to the extent that we regularly see papers in mathematical ecology discussing models in purely mathematical terms that from time to time bear little resemblance to real ecological processes. Some recent studies investigated stochastic predator-prey models to study the random noise effects on the dynamic behaviors such as [24,25,26,27]. Motivated by the results in the previous study, we study a novel idea in this paper that involves investigating stochastic concept to study the random noise effects on the occurrence of the paradox of enrichment in a predator-prey model. This study may contribute to increasing attention to study the random noise effects on some ecological and biological phenomena as the paradox of enrichment.

Stochastic Predator-Prey Model

Global Asymptotic Stability and Paradox of Enrichment

Discussion and Conclusions