Stability in distribution of competitive Lotka–Volterra system with Markovian switching

In this paper, we consider a stochastic Lotka–Volterra competitive system dxi(t)=xi(t){[bi(ξ(t))-∑j=1naij(ξ(t))xj(t)]dt+σi((ξ(t)dwi(t)}, where wi(t) (i = 1, 2, … , n) are independent standard Brownian motions and ξ  (·) is Markov chain taking values in a finite space M={1,2,…,m}M={1,2,…,m}. Global attractivity, upper boundedness and other properties are obtained. In addition, asymptotically stable in distribution as the main result of our paper is derived under some conditions. We gave a numerical simulation for invariant distribution of an example by using the Monte Carlo simulation method at the end.