Abstract
A stochastic four species food-chain model is proposed in this paper. Here, artificial harvest in each species and the effect of time delay for interaction between species are considered, which makes the model more applicable in real situations. Specifically, we address the stochastic global dynamics behavior, including the existence of global positive solutions, stochastic ultimate boundedness, extinction with probability one, persistence in mean and global stability. The asymptotic stability in the probability distribution is obtained, and the criterion for the existence and non-existence of the optimal harvesting strategy is also derived. Furthermore, this paper can provide reference for the research of general n-species stochastic food-chain models.
Highlights
Ecosystem of one species is very rare in nature
Over the last few decades, two or three species systems such as predator–prey and food-chain systems have long been the main topic of mathematical ecology and ecology
We find that there exists the larger difficulty to straightway investigate the dynamical behavior of the general n-species stochastic food-chain model at present
Summary
Ecosystem of one species is very rare in nature. In natural ecosystems, the coexistence of a large number of species is almost universal (see [1]). We focus on the following stochastic four species food-chain model with harvesting and distributed delay: Remark 1 Observing the proof process from Lemma 4, we find that Lemma 4 seemingly can be extended to the general n-species stochastic food-chain system with distributed delay and harvesting.
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