Asymptotic Behaviors for Delay Lotka–Volterra Model Disturbed by G-Brownian Motion

Ping He,Yong Ren,Defei Zhang

doi:10.1155/2020/3705325

Ping He, Yong Ren + Show 1 more

Open Access

https://doi.org/10.1155/2020/3705325

Copy DOI

Abstract

In this paper, we propose the stochastic Lotka–Volterra model with delay disturbed by G-Brownian motion dx=diagx1,x2,…,xnAxt−τ+bdBt+σxdBt. Under a natural assumption on noise, we study existence and uniqueness of the global positive solution for the system and its asymptotic pathwise moment behavior and prove that the solution does not explode to infinity in a finite time.

Highlights

Since the Lotka–Volterra model (LVM in short) was provided by Lotka [1] and Volterra [2], there were extensive works concerned with the dynamics of this system and global stability and the stochastic Lotka–Volterra population model, and in here, we only mention [3, 4] and [5,6,7,8]. e wellknown two-dimensional delay Lotka–Volterra ecological population model driven by Brownian motion is dx1 x1􏼂 + c11x1 +
We investigate a stochastic delay Lotka–Volterra model disturbed by G-Brownian motion: dx diag x1, x2, . . . , xn􏼁[(b + Ax(t − τ))d〈B〉(t) + σxdB(t)], (2)
Set max{ki(ω), i ∈ [1, n]} k0(ω), ∀ω ∈ ∩ni 1Ωi, t ∈ [k − 1, k], k ≥ k0(ω), it gets from (40): lim sup t⟶∞

Summary

Introduction

Since the Lotka–Volterra model (LVM in short) was provided by Lotka [1] and Volterra [2], there were extensive works concerned with the dynamics of this system and global stability and the stochastic Lotka–Volterra population model, and in here, we only mention [3, 4] (for deterministic situation) and [5,6,7,8] (for stochastic situation). e wellknown two-dimensional delay Lotka–Volterra ecological population model driven by Brownian motion is. Many important theoretical results in this field are obtained, for example, SLL for sublinear expectations are obtained in [15], capacity theory results are discussed in [16] and [17,18,19], and other related technologies in [20,21,22]. Inspired by these results, we investigate a stochastic delay Lotka–Volterra model disturbed by G-Brownian motion: dx diag x1, x2, . Some asymptotic pathwise moment estimations for the solutions of this system are presented

Stochastic Delay Lotka–Volterra Model Driven by G-Brownian Motion

Asymptotic Behaviors of the Solution

Asymptotic Moment Estimations