Asymptotic stability of a stochastic May mutualism system
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.
scientific article; zbMATH DE number 7256623
scientific article