Abstract
This paper proposes a stochastic three species food-chain model with harvesting and distributed delays. Some criteria for the global dynamics of all positive solutions, including the existence of global positive solutions, stochastic boundedness, extinction, global asymptotic stability in the mean, and the probability distribution, are established by using the stochastic integral inequalities, Lyapunov function method, and the inequality estimation technique. Furthermore, the effects of harvesting are discussed, the optimal harvesting strategy and the maximum of expectation of sustainable yield (MESY for short) are obtained. Finally, numerical examples are carried out to illustrate our main results.
Highlights
The notion of food-chain was first postulated by Eiton in 1927
To the best of our knowledge to date, the problem of a stochastic food-chain model with harvesting and distributed delays has not been studied in the past research
In this paper we firstly investigate the global dynamics of model (1), including the existence of global positive solutions, stochastic boundedness, extinction, global asymptotic stability in the mean, and the probability distribution, by using the stochastic integrals inequalities, Lyapunov function method, and the inequality estimation technique
Summary
The notion of food-chain was first postulated by Eiton in 1927 (see [1]). As he said, he proposed this idea due to the Chinese folk-adage: big fish eat small fish, small fish eat shrimps, shrimps eat mud. The following deterministic three species food-chain model has been investigated by many scholars (see [2,3,4,5]):. To the best of our knowledge to date, the problem of a stochastic food-chain model with harvesting and distributed delays has not been studied in the past research. Motivated by the above discussion, considering distributed time delays and white noises, in this paper, we establish the following stochastic three species food-chain model:. In this paper we firstly investigate the global dynamics of model (1), including the existence of global positive solutions, stochastic boundedness, extinction, global asymptotic stability in the mean, and the probability distribution, by using the stochastic integrals inequalities, Lyapunov function method, and the inequality estimation technique.
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