Asymptotic stability of a stochastic May mutualism system
type: Item
instance of: scholarly article
title: Asymptotic stability of a stochastic May mutualism system
zbMATH Open document ID: 1451.92260
author: Guodong Liu
author: Haokun Qi
author: Zhengbo Chang
author: Xinzhu Meng
publication date: 2020-10-07T00:00:00Z
review text: The main purpose of this article is to investigate the population dynamics of a typical system of mutualism. The authors investigate the type of stochastic mutualism system proposed by May. In comparison with the Lotka-Volterra mutualism system, the cooperation of this system is realized by changing the carrying capacity of each other. It is shown that the system of stochastic mutualism has a unique positive solution. However, the influence of environmental fluctuations on the population cannot be ignored: if the random perturbations are so small that the probability density of the distributions for the solutions is absolutely continuous, and the densities converge to an invariant density or converge to a singular measure weakly. First, the existence of a continuous density $k$ is proved using the method of the Lie bracket, which implies that $\{P (t)\}t\geq 0$ is an entire Markov semigroup with continuous kernel $k$. Then the positiveness of the density $k$ is checked by determining the control functions $\varphi$. As a result, we see that the semigroup $\{P(t)\}t\geq 0$ is asymptotically stable. Mathematical methods related to the theory of Markov semigroups and the Fokker-Planck equation can enrich the study of asymptotic population behavior.
reviewed by: Fatima T. Adylova
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34F05
Mathematics Subject Classification ID: 35Q84
zbMATH DE Number: 7256623
zbMATH Keywords: stochastic mutualism model
zbMATH Keywords: asymptotic stability
zbMATH Keywords: Markov semigroup
zbMATH Keywords: Fokker-Planck equation
zbMATH Keywords: stable stationary distribution
MaRDI profile type: Publication
full work available at URL: j.camwa.2019.07.022
OpenAlex ID: W2969076740
Wikidata QID: Q111265731
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DOI: 10.1016/J.CAMWA.2019.07.022
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 7256623