An invariant measure for a system of stochastic equations for a prey-predator model with spatial diffusion (Q624846)

This paper considers a stochastic prey-predator model with spatial diffusion in a Hilbert abstract space: \[ dN_1(t) = [(\alpha - \beta N_2(t) - \mu N_1(t))N_1(t) + \kappa_1 \Delta N_1(t)]dt + \rho_1 N_1(t)dW(t), \] \[ dN_2(t) = [(-\gamma + \delta N_1(t) - \nu N_2(t))N_2(t) + \kappa_2 \Delta N_2(t)]dt + \rho_2 N_2(t)dW(t). \] Here, \(N_1\) and \(N_2\) are the population densities of the two species, \(\alpha,\;\beta,\;\gamma,\;\delta,\;\mu,\;\nu,\;\rho_i\) and \(\kappa_i\) are some positive constants and \(W(t)\) is a Brownian motion. The authors prove the existence of an invariant measure for this system of stochastic equations. The proof of this result is based on a theorem of Krylov-Bogoliubov and estimation of the solution.