Abstract
A double delayed hybrid stochastic prey-predator bioeconomic system with Lévy jumps is established and analyzed, where commercial harvesting on prey and environmental stochasticity on population dynamics are considered. Two discrete time delays are utilized to represent the maturation delay of prey and gestation delay of predator, respectively. For a deterministic system, positivity of solutions and uniform persistence of system are discussed. Some sufficient conditions associated with double time delays are derived to discuss asymptotic stability of interior equilibrium. For a stochastic system, existence and uniqueness of a global positive solution are studied. By using the invariant measure theory and singular boundary theory of diffusion process, existence of stochastic Hopf bifurcation and stochastic stability are investigated. By constructing appropriate Lyapunov functions, asymptotic dynamic behavior of the proposed hybrid stochastic system with double time delays and Lévy jumps is discussed. Numerical simulations are provided to show consistency with theoretical analysis.
Highlights
It is well known that a harvest effort has a strong dynamical impact on the prey-predator system [1,2,3,4,5], which plays a significant role in bioeconomics management among various species in a harvested prey-predator system [6,7,8]
The dynamic effects of double time delays have been investigated in [30], the combined dynamic effects of multiple time delays and Lévy jumps on population dynamics have not been investigated in [30]
Compared with the system established in [18], an algebraic equation is introduced into system (3), which concentrates on the dynamic effect of the economic interest of commercial harvesting on population dynamics and provides a straightforward way to investigate complex dynamics due to the variation of economic interests
Summary
It is well known that a harvest effort has a strong dynamical impact on the prey-predator system [1,2,3,4,5], which plays a significant role in bioeconomics management among various species in a harvested prey-predator system [6,7,8]. In this paper, keeping all these Assumptions 1–3 in mind, a double delayed hybrid stochastic prey-predator bioeconomic system with Lévy jumps is established as follows: dx t = x1 t − τ1. Some hybrid bioeconomic systems with time delay and stochastic fluctuations are established in [27,28,29,30] to investigate the combined dynamic effects of stochastic fluctuation and commercial harvesting on population dynamics These proposed systems in [27,28,29,30] are constructed by several differential equations with stochastic fluctuations and an algebraic equation.
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