Abstract
Under the assumption that stochastic white noise perturbations are directly proportional to the deviation of the state from the equilibrium states of a continuous mutualistic model, we use the Euler–Maruyama discretization method to obtain a two-species stochastic discrete mutualism model. For this stochastic model, we establish conditions on the asymptotic mean square stability of the positive equilibrium state and the almost sure asymptotic stability of the three boundary equilibrium states. The theoretical results are supported with numerical simulations.
Highlights
Any species in nature is not isolated, always related to other species in the community
We study the almost sure asymptotic stability
4 Conclusion In this paper, we proposed a two-species stochastic discrete mutualism model where the stochastic white noise perturbation is directly proportional to the deviation of the state from the equilibrium staes
Summary
Any species in nature is not isolated, always related to other species in the community. The relationships among them are generally divided into four types: mutualism, parasitism, competition, and predation. Mutualism is one of the ubiquitous interactions among species, which is beneficial for all the species involved [1]. Mutualism plays an important role in all ecosystems; in theoretical biology, it has received much attention of many scholars (see, e.g., [2,3,4,5,6,7,8,9,10]). The simplest mutualistic model was proposed by May [11]. May’s equations for two species can be written as du dt. Lots of modifications of May’s model have been proposed to better understand mutualism
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