Global Asymptotic Stability of Stochastic Nonautonomous Lotka-Volterra Models with Infinite Delay

Abstract

A kind of general stochastic nonautonomous Lotka-Volterra models with infinite delay is investigated in this paper. By constructing several suitable Lyapunov functions, the existence and uniqueness of global positive solution and global asymptotic stability are obtained. Further, the solution asymptotically follows a normal distribution by means of linearizing stochastic differential equation. Moment estimations in time average are derived to improve the approximation distribution. Finally, numerical simulations are given to illustrate our conclusions.

Asymptotic properties of a stochastic Lotka–Volterra model with infinite delay and regime switching

“…Later, the stochastically ultimate boundedness will be derived when constructing a proper function therewith. Consequently, the moment estimation of the solution will be investigated for the stochastic Lotka-Volterra model (8) with Markovian chains in this paper.…”

“…We are about to discuss other properties of the solution to model (8) in this section, for instance, the absolute value of the solution will grow slower than time t when t goes to infinity. Moreover, the pth moment in time average of the solution to model (8) …”

“…We take the weight function k ij (t) = e -(λ+1)t in (8), then (H2) is always valid because ∞ 0 k ij (s)e λs ds = 1 < ∞. Therefore the existence and uniqueness of the positive solution is derived with probability one as time approaches infinity.…”

Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response

“…In recent years, models with time delay have been extensively studied by many authors [5,[16][17][18][19][20][21][22][23][24][25][26]. The authors of [1] discussed model (1) with a discrete delay:…”