On limit systems for some population models with cross-diffusion (Q713235)

This paper studies the stationary solutions of the reaction-diffusion system \[ u_t=\Delta [ (1+\alpha v +\gamma u)u]+u(a-u-cv) \] \[ v_t=\Delta [ (1+\beta u +\delta v)v]+v(b-du-v), \] in a bounded domain in \(\mathbb R^N\), which models the density of two species which compete (as described by the terms \(cv\) and \(du\)) and whose diffusion is influenced by a repulsive force due to the presence of other individuals (which accounts for the nonlinear diffusion terms). This sytem originates by \textit{N. Shigesada}, \textit{K. Kawasaki} and \textit{E. Teramoto} [``Spatial segregation of interacting species'', J. Theor. Biol. 79, No. 1, 83--99 (1979; \url{doi:10.1016/0022-5193(79)90258-3})] and has been studied by several authors since. The system is studied both in the case of Neumann (no-flux) and of Dirichlet boundary conditions. The main results concern the behavior of positive stationary solutions when one of the cross-diffusion coefficients (\(\alpha\) or \(\beta\)) goes to infinity. Under appropriate conditions, it is proved that the stationary solutions then converge to the solutions of certain limiting systems. An essential role in these proof is played by a-priori estimates on the solutions which are independent of the size of the cross-diffusion parameter, which are proved here. Some results on the solutions of the limiting systems are provided. Several open problems are presented in the concluding section.