A famous nonlinear stochastic equation (Lotka-Volterra model with diffusion)
type: Item
instance of: scholarly article
title: A famous nonlinear stochastic equation (Lotka-Volterra model with diffusion)
zbMATH Open document ID: 1049.92030
DOI: 10.1016/S0895-7177(03)90056-2
author: Sumit K. Garg
published in: Mathematical and Computer Modelling
publication date: 2004-08-06T00:00:00Z
review text: The paper is devoted to a survey of several probabilistic models related to the classical Lotka-Volterra equation (predator-prey systems). A typical two-dimensional model is described by a system of stochastic differential equations \[ \begin{align*}{ dX_1(t)=&X_1(t)(b_1-a_{11}X_1(t)-a_{12}X_2(t))dt+G_1X_1(t)dw_1(t),\cr dX_2(t)=&X_2(t)(b_2-a_{21}X_1(t)-a_{22}X_2(t))dt+G_2X_2(t)dw_2(t),\cr }\end{align*} \] where \(w_1,w_2\) are independent standard Brownian motions. Using the Itô formula this system can be given an exponential form and the long-time behavior of the solutions can be analyzed. Also, some rather well-known facts on absolute continuity and stability of solutions of SDEs as well as on filtering of linear Gaussian systems are recalled.
reviewed by: Tomasz Bojdecki
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 60G35
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 2082393
zbMATH Keywords: Lotka-Volterra equation
zbMATH Keywords: predator-prey model
zbMATH Keywords: stochastic differential equation
zbMATH Keywords: stability
zbMATH Keywords: filtering equation
MaRDI profile type: Publication
cites work: The Influence of External Real and White Noise on the L<scp>OTKA</scp>‐V<scp>OLTERRA</scp> Model
cites work: Linear stochastic systems with constant coefficients. A statistical approach
cites work: Q3685040
cites work: On contiguity of probability measures corresponding to semimartingales
cites work: Q5509397
cites work: The effect of diffusion on the time varying logistic population growth
cites work: Permanence and asymptotic behavior of the \(N\)-species nonautonomous Lotka-Volterra competitive systems
cites work: Periodic solutions of single-species nonautonomous diffusion models with continuous time delays
cites work: Q3973634
cites work: Q5616513
cites work: Q4765883
cites work: Q5556832
description: scientific article; zbMATH DE number 2082393