Dynamical behavior in a hybrid stochastic triple delayed prey predator bioeconomic system with Lévy jumps

Abstract

In this paper, a hybrid triple delayed prey predator bioeconomic system with prey refuge and Lévy jumps is established, where both maturation delay for prey and predator population and gestation delay for predator population are considered. For deterministic system, positivity and uniform permanence of solution are discussed. Local stability of deterministic system around interior equilibrium is investigated due to variations of triple time delays. For stochastic system without time delay, sufficient conditions for stochastically ultimate boundedness and stochastic permanence are discussed. Existence of stochastic Hopf bifurcation and stochastic stability are investigated. For stochastic system with triple time delays, existence and uniqueness of global positive solution are studied. Finally, combined dynamic effects of triple time delays and Lévy jumps on the hybrid stochastic system are discussed by constructing appropriate Lyapunov functions. Numerical simulations are supported to illustrate theoretical analysis.