Families of equilibria and dynamics in a population kinetics model with cosymmetry

We consider a system of nonlinear parabolic equations with an additional property-the so-called cosymmetry-which implies the appearance of a nontrivial family of equilibria. By nontrivial we mean that the stability spectrum is not constant along the family of stationary states. The present system generalizes a special case of a distributed population model discussed in [Computing 16 (Suppl.) (2002) 67] from two to three species. The components of the system have the interpretation of interacting populations which inhabit a common domain. For this Letter we concentrate on the 1D case and apply a finite-difference scheme which respects the cosymmetry. We describe the scenario of instability for the state of rest and observe a rich palette of regimes depending on model parameters and on the initial state.