On Some Nonlinear Diffusion Models in Population Dynamics

Reaction-diffusion models in population dynamics have been widely investigated in recent years [6]. In particular, quasilinear models have been recently suggested [11] to deal with situations where diffusivity depends on crowding (see also [7]). In the case of two interacting species such models have the general form (1)$$\begin{array}{*{20}{c}} {{u_t} = {\Delta _y}\left( {u,v} \right) + u\;f\left( {x,u,v} \right)} \\ {{v_t} = {\Delta _j}\left( {u,v} \right) + v\;g\left( {x,u,v} \right)} \\ \end{array}$$ the solution satisfying suitable initial and boundary conditions in a bounded domain (here t is the time variable, x the space variable, Δ the diffusion operator, u=u(t,x), v=v(t,x) are population densities, see [6,7,11]).