The Stationary Distribution of Competitive Lotka-Volterra Population Systems with Jumps

Abstract

Dynamics of Lotka-Volterra population with jumps (LVWJ) have recently been established (see Bao et al., 2011, and Bao and Yuan, 2012). They provided some useful criteria on the existence of stationary distribution and some asymptotic properties for LVWJ. However, the uniqueness of stationary distribution for $n\ge \mathrm{2}$ and asymptotic pathwise estimation ${\lim }_{t\to +\mathrm{\infty }}\mathrm{(1}/\mathrm{t)}{\int }_{\mathrm{0}}^t\mathrm{‍}|X(s){|}^{p}d\mathrm{s\hspace{0.17em}}$ $(p>\mathrm{0})$ are still unknown for LVWJ. One of our aims in this paper is to show the uniqueness of stationary distribution and asymptotic pathwise estimation for LVWJ. Moreover, some characterizations for stationary distribution are provided.