A Stochastic Lotka-Volterra System And Its Asymptotic Behavior

Lotka-Volterra systems are used to study the interactions between populations,ranging from deterministic Lotka-Volterra system to stochastic Lotka-Volterra system.This thesis mainly focus on dx(t)= diag(x1(t),...,xn(t))[(b + Ax(t))dt + ?x(t)pdw(t))].where ?(t)is a scalar standard Brown motion,and b =(b1,b2,…,bn)',bi is endogenous rate of the i-th population,A =(aij),i,j=1,2,...,n,and aij describes the impact of the i-th population with respect to the j-th population.Population systems perturbed by the white noise have been studied by many authors recently in case of p = 0 and p = 1.The purpose of the thesis is to find out what happens when p>1/2.We will show that the system has positive and global solution,and examine the asymptotic behavior of this system such as boundedness,moment estimation and path estimation.In addition,we discuss the following dynamics of the Lotka-Volterra multi-mutualism system,where the growth rate is nonlinear.dxn(t)= xn(t)[(bn+an1x1(t)p+an2x2(t)p)dt+?ndwn(t)],To solve this problem,we apply Ito formula to transform this nonlinear growth rate to linear case.