 1153 Vitaly Volpert, Vitali Vougalter
 Stability and instability of solutions of a nonlocal reactiondiffusion
equation when the essential spectrum crosses the imaginary axis
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Apr 7, 11

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Abstract. The paper is devoted to an integrodifferential 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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