In this paper, we deal with the model with a very general growth law and an M - driven diffusion ∂u(t, x) ∂t = DΔ( u(t, x) M (t, x) )+ μ(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 T0-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).