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 $$\frac{{\partial u(t,x)}} {{\partial t}} = D\Delta (\frac{{u(t,x)}} {{M(t,x)}}) + \mu (t,x)f(u(t,x),M(t,x)).$$ For the general case of time dependent functions M and µ, the existence and uniqueness for positive solution is obtained. If M and µ are T ₀-periodic functions in t, then there is an attractive positive periodic solution. Furthermore, if M and µ 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].