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

This work is the continuation of our previous paper[6]. There, we dealt with the reaction-diffusionequation∂ₜ u=Δ u+f(x-cte,u), t>0, x∈N,where e∈ SN-1 and $c>0$ are given and $f(x,s)$ satisfiessome usual assumptions in population dynamics, together withfₛ(x,0) TheL^1$ convergence of solution $u(t,x)$ as $t→∞$ is establishednext. In this paper, we also showthat a bifurcation from the zero solution takes place as the principal crosses $0$. We areable to describe the shape of solutions close to extinctionthus answering a question raised by M.~Mimura.These two results are new even in the frameworkconsidered 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 consideringenvironmental change on two different scales.Lastly, we also solve the case of an equation∂ₜ u=Δ u+f(t,x-cte,u),when $f(t,x,s)$ is periodic in t.This for instance represents theseasonal dependence of f.In both cases, we obtain a necessary and sufficient condition for theexistence,uniqueness and stability of pulsating travelling waves, which aresolutions with a profile which is periodic in time.