Reactiondiffusion equations for population dynamics with forced speed I  The case of the whole space
Henri Berestycki  EHESS, CAMS, 54 Boulevard Raspail, F75006, Paris, France (email) Abstract: This paper is concerned with timedependent reactiondiffusion equations of the following type: $\partial_t u=$Δ$u+f(xcte,u),t>0,x\in\R^N.$
These kind of equations have been introduced in [1] in
the case $N=1$ for studying the impact of a climate shift on the
dynamics of a biological species.
$\partial_t u=$Δ$u+f(xcte,u)+g(x,u),t>0,x\in\R^N.$ Here, $g$ can be viewed as representing geographical characteristics of the territory which are not subject to shift. We derive analogous results as before, with $\l$ replaced by the generalized principal eigenvalue of the parabolic operator obtained by linearization about $u\equiv0$ in the whole space. In this framework, travelling waves are replaced by pulsating travelling waves, which are solutions of the form $U(t,xcte)$, with $U(t,x)$ periodic in $t$. These results still hold if the term $g$ is also subject to the shift, but on a different time scale, that is, if $g(x,u)$ is replaced by $g(xc'te,u)$, with $c'\in\R$.
Keywords: Reactiondiffusion equations, travelling waves, forced speed,
time periodic parabolic equations, principal eigenvalues, persistence, extinction.
Received: July 2007; Revised: October 2007; Available Online: February 2008. 
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