11-53 Vitaly Volpert, Vitali Vougalter

Stability and instability of solutions of a nonlocal reaction-diffusion equation when the essential spectrum crosses the imaginary axis (705K, pdf) Apr 7, 11

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Abstract. The paper is devoted to an integro-differential equation arising in population dynamics where the integral term describes nonlocal consumption of resources. This equation can have several stationary points and, as it is already well known, a travelling wave solution which provides a transition between them. It is also possible that one of these stationary points loses its stability resulting in appearance of a stationary periodic in space structure. In this case, we can expect a possible transition between a stationary point and a periodic structure. The main goal of this work is to study such transitions. The loss of stability of the stationary point signifies that the essential spectrum of the operator linearized about the wave intersects the imaginary axis. Contrary to the usual Hopf bifurcation where a pair of isolated complex conjugate eigenvalues crosses the imaginary axis, here a periodic solution may not necessarily emerge. To describe dynamics of solutions, we need to consider two transitions: a steady wave with a constant speed between two stationary points, and a periodic wave between the stationary point which loses its stability and the periodic structure which appears around it. Both of these waves propagate in space, each one with its own speed. If the speed of the steady wave is greater, then it runs away from the periodic wave, and they propagate independently one after another.

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