Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains

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

where $e\in S^{N-1}$ and $c>0$ are given and $f(x,s)$ satisfies some usual assumptions in population dynamics, together with $f_s(x,0)<0$ for $|x|$ large. The interest for such equation comes from an ecological model introduced in [1] describing the effects of global warming on biological species. In [6],we proved that existence and uniqueness of travelling wave solutions of the type $u(x,t)=U(x-cte)$ and the large time behaviour of solutions with arbitrary nonnegative bounded initial datum depend on the sign of the generalized principal in $\R^N$ of an associated linear operator. Here, we establish analogous results for the Neumann problem in domains which are asymptotically cylindrical, as well as for the problem in the whole space with $f$ periodic in some space variables, orthogonal to the direction of the shift $e$.
   The $L^1$ convergence of solution $u(t,x)$ as $t\to\infty$ is established next. In this paper, we also show that a bifurcation from the zero solution takes place as the principal crosses $0$. We are able to describe the shape of solutions close to extinction thus answering a question raised by M.~Mimura. These two results are new even in the framework considered in [6].
   Another type of problem is obtained by adding to the previous one a term $g(x-c'te,u)$ periodic in $x$ in the direction $e$. Such a model arises when considering environmental change on two different scales. Lastly, we also solve the case of an equation

$\partial_t u=\Delta u+f(t,x-cte,u),$

when $f(t,x,s)$ is periodic in $t$. This for instance represents the seasonal dependence of $f$. In both cases, we obtain a necessary and sufficient condition for the existence, uniqueness and stability of pulsating travelling waves, which are solutions with a profile which is periodic in time.