Diffusion-mediated permanence problem for a heterogeneous Lotka-Volterra competition model

We analyse the dynamics of a prototype model for competing species with diffusion coefficients (d(1), d(2)) in a heterogeneous environment Omega. When diffusion is switched off, at each point x epsilon Omega we have a pair of ODE's: the kinetic. If for some x epsilon Omega the kinetic has a unique stable coexistence state, we show that there exist (d) over cap(1) > 0, (d) over cap(2) > 0 such that for every (d(1), d(2))epsilon (0,(d) over cap(1)) x (0, (d) over cap(2)) the RD-model is persistent, in the sense that it has a compact global attractor within the interior of the positive cone and has a stable coexistence state. The same result is true if there exist x(u), x(v), epsilon Omega such that the semitrivial coexistence states (u, 0) and (0, v) of the kinetic are globally asymptotically stable at x = x(u) and x = x(v), respectively. More generally, our main result shows that, for most kinetic patterns, stable coexistence of populations can be found for some range of the diffusion coefficients. Singular perturbation techniques, monotone schemes, fixed point index, global analysis of persistence curves, global continuation and singularity theory are some of the technical tools employed to get the previous results, among others. These techniques give us necessary and/or sufficient conditions for the existence and uniqueness of coexistence states, conditions which can be explicitly evaluated by estimating some principal eigenvalues of certain elliptic operators whose coefficients are solutions of semilinear boundary value problems. We also discuss counterexamples to the necessity of the sufficient conditions through the analysis of the local bifurcations from the semitrivial coexistence states at the principal eigenvalues. An easy consequence of our analysis is the existence of models having exactly two coexistence states, one of them stable and the other one unstable. We find that there are also cases for which the model has three or more coexistence states.