Dynamical Analysis and Optimal Harvesting Strategy for a Stochastic Delayed Predator-Prey Competitive System with Lévy Jumps

Guodong Liu,Kaiyuan Liu,Xinzhu Meng

doi:10.1155/2019/2187274

Abstract

This paper develops a theoretical framework to investigate optimal harvesting control for stochastic delay differential systems. We first propose a novel stochastic two-predator and one-prey competitive system subject to time delays and Lévy jumps. Then we obtain sufficient conditions for persistence in mean and extinction of three species by using the stochastic qualitative analysis method. Finally, the optimal harvesting effort and the maximum of expectation of sustainable yield (ESY) are derived from Hessian matrix method and optimal harvesting theory of delay differential equations. Moreover, some numerical simulations are given to illustrate the theoretical results.

Highlights

Optimal control problem in the field of biological mathematics has been widely concerned by researchers
Resource exploitation always aims to maximum sustainable yield (MSY) or the profit associated with the maximum economic yield (MEY) [1]
The above two classes of models can not describe some natural phenomena completely and it is believed that models with three or more species can explain the dynamical behaviors of the population accurately [5,6,7,8]

Summary

Introduction

Optimal control problem in the field of biological mathematics has been widely concerned by researchers. Motivated by the above discussion, we can assume that the intrinsic growth rate r1 and the death rates r2 and r3 of model (1) perturbed by the Levy jump to signify the sudden climate change, ri 󳨀→ ri + γidLi(t) [42, 46], and we can obtain the following stochastic model incorporating Levy jump: dx (t) = x (t) [r1 − a11x (t) − a12y1 (t − τ12) − a13y2 (t − τ13)] dt + σ1x (t) dB1 (t) + x (t). We devote our main attention to obtain the optimal harvesting control strategy of system (5) To this end, we firstly investigate the dynamical behavior of the three species including persistence in mean and extinction and asymptotically stable distribution. We conclude our results by numerical simulations and discussions in the last section

Persistence in Mean and Extinction

Optimal Harvesting

Numerical Simulations and Discussions