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Non-local reaction-diffusion equations arise naturally to account for diffusions involving jumps rather than local diffusions related to Brownian motion. In ecology, long distance dispersal require such frameworks. In this work we study a one-dimensional non-local reaction-diffusion equation with bistable type reaction. The heterogeneity here is due to a gap, some finite region where there is decay. Outside this gap region the equation is a classical homogeneous (space independent) non-local reaction-diffusion equation. This type of problem is motivated by applications in ecology, sociology, and physiology. We first establish the existence of a *generalized traveling front* that approaches a traveling wave solution as *t*-∞, propagating in a heterogeneous environment. We then study the problem of obstruction of solutions. In particular, we study the propagation properties of the generalized traveling front with significant use of the work of Bates, Fife, Ren and Wang in [

$\partial_t u=$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 $c'\in\R$.

For more information please click the “Full Text” above.

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

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.

$\Delta u-u + u^p=0 \ \mbox{in} \ \R^{N-1} \times (0, L),$

$ u>0, \frac{\partial u}{\partial \nu}=0 \ \mbox{on} \ \partial (\R^{N-1} \times (0, L)) $

where $ 1< p\leq \frac{N+2}{N-2}$. When $ 1 < p <\frac{N+2}{N-2}$, it is shown that there exists a unique L _{*} >0 such that for L $\leq $L _{*} , the least energy solution is trivial, i.e., doesn't depend on $x_N$, and for L >L _{*} , the least energy solution is nontrivial. When $N \geq 4, p=\frac{N+2}{N-2}$, it is shown that there are two numbers L _{*} < L _{**} such that the least energy solution is trivial when L $\leq$L _{*}, the least energy solution is nontrivial when L $\in$(L _{*},L _{**}], and the least energy solution does not exist when L >L _{**}. A connection with Delaunay surfaces in CMC theory is also made.

We also study here the case where $h(y,u)=f(u)$ for $|y|\leq L_1$ and $h(y,u) \approx - \alpha u$ for $|y|>L_2\geq L_1$. This equation provides a general framework for a model of cortical spreading depressions in the brain. We prove the existence of traveling front if $L_1$ is large enough and the non-existence if $L_2$ is too small.

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