Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space

$\partial_t u=$Δ$u+f(x-cte,u),t>0,x\in\R^N.$

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.
    In the present paper, we first extend the results of [1] to arbitrary dimension $N$ and to a greater generality in the assumptions on $f$. We establish a necessary and sufficient condition for the existence of travelling wave solutions, that is, solutions of the type $u(t,x)=U(x-cte)$. This is expressed in terms of the sign of the generalized principal eigenvalue $\l$ of an associated linear elliptic operator in $\R^N$. With this criterion, we then completely describe the large time dynamics for this equation. In particular, we characterize situations in which there is either extinction or persistence.
    Moreover, we consider the problem obtained by adding a term $g(x,u)$ periodic in $x$ in the direction $e$:

$\partial_t u=$Δ$u+f(x-cte,u)+g(x,u),t>0,x\in\R^N.$

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 $U(t,x-cte)$, 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 $g(x-c'te,u)$, with $c'\in\R$.