A famous nonlinear stochastic equation (Lotka-Volterra model with diffusion)
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.
scientific article; zbMATH DE number 2082393