Differ Equ Dyn Syst DOI 10.1007/s12591-015-0244-z ORIGINAL RESEARCH A Note on Stochastic Gilpin–Ayala Population Model with Dispersal Aadil Lahrouz 1 · Adel Settati 2 © Foundation for Scientific Research and Technological Innovation 2015 Abstract A stochastic Gilpin–Ayala population model with diffusion between two patches is studied. A sufficient conditions for extinction and persistence are established. Furthermore, the existence of a stationary distribution is showed. The analytical results are illustrated by computer simulations. Keywords Stochastic model · Extinction · Persistence · Stationary distribution Introduction Population ecology is a sub-field of ecology that deals with the dynamics of species popu- lations and how these populations interact with the environment. It is the study of how the population sizes of species living together in groups change over time and space [3]. The Gilpin–Ayala population model is one of the most important and classic mathematical bio- economic models due to its theoretical and practical significance. In 1973, Gilpin and Ayala [6] claimed the following model: dx (t ) = x (t ) ( r − kx θ (t ) ) dt , (1) where x (t ) denotes the density of resource population at time t , r > 0 is called the intrinsic growth rate and k /r is the environmental carrying capacity. It is obvious that Eq. (1) becomes the classic logistic population model when θ = 1. B Aadil Lahrouz lahrouzadil@gmail.com Adel Settati settati_adel@yahoo.fr 1 Laboratory of Computer Sciences and Modeling, Department of Mathematics, Faculty of Sciences Dhar-Mehraz, B.P. 1796-Atlas, Fez, Morocco 2 Laboratory of Mathematics and applications, Department of Mathematics, Faculty of Sciences and techniques, B.P. 416-Tanger Principale, Tanger, Morocco 123