A reaction-diffusion model with nonlinearity driven diffusion

In this paper, we deal with the model with a very general growth law and an M-driven diffusion partial derivative u(t, x)/partial derivative t = D Delta(u(t, x)/M(t, x)) + mu(t, x) f (u(t, x), M(t, x)). For the general case of time dependent functions M and A mu, the existence and uniqueness for positive solution is obtained. If M and A mu are T (0)-periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and A mu are time-independent, then the non-constant stationary solution M(x) is globally stable. Thus, we can easily formulate the conditions deriving the above behaviors for specific population models with the logistic growth law, Gilpin-Ayala growth law and Gompertz growth law, respectively. We answer an open problem proposed by L. Korobenko and E. Braverman in [Can. Appl. Math. Quart. 17(2009) 85-104].