Stationary distribution, extinction, density function and periodicity of an n-species competition system with infinite distributed delays and nonlinear perturbations

In this paper, we examine an n-species Lotka-Volterra competition system with general infinite distributed delays and nonlinear perturbations. The stochastic boundedness and extinction are first studied. Then we propose a new \begin{document}$ p $\end{document} -stochastic threshold method to establish sufficient conditions for the existence of stationary distribution \begin{document}$ \ell(\cdot) $\end{document} . By solving the corresponding Fokker–Planck equation, we derive the approximate expression of the distribution \begin{document}$ \ell(\cdot) $\end{document} around its quasi-positive equilibrium. For the stochastic system with periodic coefficients, we use the \begin{document}$ p $\end{document} -stochastic threshold method again to obtain the existence of positive periodic solution. Besides, the related competition exclusion and moment estimate of species are shown. Finally, some numerical simulations are provided to substantiate our analytical results.