A geometric method for asymptotic properties of the stochastic Lotka-Volterra model

In this paper, we use a geometric method to analyze the asymptotic properties of the stochastic Lokta-Volterra model, which mainly include two aspects: uniformly ultimate boundedness and almost sure permanence. The geometric method demonstrates that there exists a bounded region that lies in the interior of the first quadrant such that solutions of the stochastic model starting from the exterior of the region will almost surely go into the interior of it in finite time, meanwhile, solutions which start from the interior of the region will almost surely not leave the interior of it in any finite time. Firstly, we show that the stochastic competition, predator-prey and mutualism model are uniformly ultimately bounded and almost surely permanent by the geometric structures of invariant sets. Secondly we give numerical simulations to illustrate the geometric method as well as the conclusions.