Abstract
We consider a stochastic one-predator-two-prey harvesting model with time delays and Lévy jumps in this paper. Using the comparison theorem of stochastic differential equations and asymptotic approaches, sufficient conditions for persistence in mean and extinction of three species are derived. By analyzing the asymptotic invariant distribution, we study the variation of the persistent level of a population. Then we obtain the conditions of global attractivity and stability in distribution. Furthermore, making use of Hessian matrix method and optimal harvesting theory of differential equations, the explicit forms of optimal harvesting effort and maximum expectation of sustainable yield are obtained. Some numerical simulations are given to illustrate the theoretical results.
Highlights
Many researchers are widely focused on the complex dynamics of biological systems such as delay population systems [1,2,3], stochastic population systems [4,5,6,7,8,9,10,11,12], and impulsive population systems [13,14,15]
Delayed differential equations can exhibit much more complex dynamics than differential equations without delay, and stable equilibrium can become unstable with the effects of a time delay
This paper is about a stochastic three-species population model with time delays and Levy jumps [49]
Summary
Many researchers are widely focused on the complex dynamics of biological systems such as delay population systems [1,2,3], stochastic population systems [4,5,6,7,8,9,10,11,12], and impulsive population systems [13,14,15]. Many researchers have studied the Lotka-Volterra time delay models with two competitive preys and one predator [22, 23]. Notice that the composite population systems with stochastic effects and time delays present some complex dynamics; this causes widespread researchers concern [24,25,26,27,28,29,30]. Applying the discontinuous stochastic process as Levy jump to model the abrupt nature phenomenon in ecosystem is necessary [33,34,35]. Considering the inevitable situations in the real world, we assume that the intrinsic growth rates a1 and a2 and the death rate a3 of the model are perturbed by the Levy jump to signify the sudden climate change, so we introduce the Levy jump into the underlying stochastic model (1).
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