Dynamics of a spatially distributed logistic equation with small diffusion and small delay

For the spatially distributed Hutchinson equation with transport and small diffusion constant, we show that the loss of stability of the equilibrium can occur even for asymptotically small values of the delay coefficient. Here infinitely many roots of the characteristic quasipolynomial tend to the imaginary axis as the small parameter, the diffusion constant, tends to zero. Thus, the critical (in the problem on the equilibrium stability) case of infinite dimension is realized. We construct special quasinormal forms, namely, nonlinear parabolic systems and families of degenerate parabolic systems whose nonlocal dynamics describes the behavior of solutions of the original equation in a small neighborhood of the equilibrium. These quasinormal forms can have a rather complicated dynamics; moreover, the onset and disappearance of steady-state modes as the small parameter tends to zero is a typical phenomenon. Therefore, the local dynamics of the Hutchinson equation with and without transport are very distinct.