This work is the continuation of our previous paper [6]. There, we dealt with the reaction-diffusion equation partial derivative(t)u = Delta u + f (x - cte, u), t > 0, x is an element of R-N, where e is an element of SN-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 vertical bar x vertical bar 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 - c t e) and the large time behaviour of solutions with arbitrary nonnegative bounded initial datum depend on the sign of the generalized principal eigenvalue 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 -> infinity is established next. In this paper, we also show that a bifurcation from the zero solution takes place as the principal eigenvalue 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't e, 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 derivative(t)u = Delta u + f (t, x - c t e, 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.
