Spatial Properties of High-Mode Bifurcations of a Distributed Logistic Equation

We study the local dynamics of a solutions spatially distributed logistic equation in the case of a two-dimensional spatial variable. Two distribution functions important for applications are considered. It is shown, that the critical cases in the problem of equilibrium stability have an infinite dimention. For each critical case a special replacement is built, which reduces the original problem to a system of parabolic equations — a quasi-normal form, the solutions behavior of which defines the local dynamics. Some of the parameters in the quasi-normal form depend on a small parameter via a discontinuous function $\Theta(\varepsilon)$, which takes an infinite number of times all the values in the interval $[0,1)$ for $\varepsilon\to0$. This gives infinite alternation of forward and backward bifurcations in the initial boundary value problem. The obtained results are compared with those for the case of a one-dimensional spatial variable. New bifurcation phenomena which occur only in the case of a two-dimensional spatial variable are revealed.