NHM
Preface
Henri Berestycki Danielle Hilhorst Frank Merle Masayasu Mimura Khashayar Pakdaman
Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He has been a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.

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keywords:
DCDS
Reaction-diffusion equations for population dynamics with forced speed II - cylindrical-type domains
Henri Berestycki Luca Rossi
This work is the continuation of our previous paper [6]. There, we dealt with the reaction-diffusion equation

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

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.

keywords: Reaction-diffusion equations bifurcation. travelling waves forced speed principal eigenvalues asymptotically cylindrical domains
DCDS
On least energy solutions to a semilinear elliptic equation in a strip
Henri Berestycki Juncheng Wei
We consider the following semilinear elliptic equation on a strip:

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

keywords: Critical Sobolev Exponent. Strip Semilinear Elliptic Equations Least Energy Solutions Unbounded Domains
NHM
Traveling fronts guided by the environment for reaction-diffusion equations
Henri Berestycki Guillemette Chapuisat
This paper deals with the existence of traveling fronts for the reaction-diffusion equation: $$ \frac{\partial u}{\partial t} - \Delta u =h(u,y) \qquad t\in \mathbb{R}, \; x=(x_1,y)\in \mathbb{R}^N. $$ We first consider the case $h(u,y)=f(u)-\alpha g(y)u$ where $f$ is of KPP or bistable type and $\lim_{|y|\rightarrow +\infty}g(y)=+\infty$. This equation comes from a model in population dynamics in which there is spatial spreading as well as phenotypic mutation of a quantitative phenotypic trait that has a locally preferred value. The goal is to understand spreading and invasions in this heterogeneous context. We prove the existence of threshold value $\alpha_0$ and of a nonzero asymptotic profile (a stationary limiting solution) $V(y)$ if and only if $\alpha<\alpha_0$. When this condition is met, we prove the existence of a traveling front. This allows us to completely identify the behavior of the solution of the parabolic problem in the KPP case.
    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.
keywords: reaction-diffusion population dynamics. Traveling front KPP
NHM
Preface
Henri Berestycki Danielle Hilhorst Frank Merle Masayasu Mimura Khashayar Pakdaman
Professor Hiroshi Matano was born in Kyoto, Japan, on July 28th, 1952. He studied at Kyoto University, where he prepared his doctoral thesis under the supervision of Professor Masaya Yamaguti. He obtained his first academic position as a research associate at the University of Tokyo. He then moved to Hiroshima University in 1982 and came back to Tokyo in 1988. He is a Professor at the Graduate School of Mathematical Sciences at the University of Tokyo since 1991.

For more information please click the “Full Text” above.
keywords:
DCDS
A non-local bistable reaction-diffusion equation with a gap
Henri Berestycki Nancy Rodríguez

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 [5]. As in the local diffusion case, we prove that obstruction is possible if the gap is sufficiently large. An interesting difference between the local dispersal and the non-local dispersal is that in the latter the obstructing steady states are discontinuous. We characterize these jump discontinuities and discuss the scaling between the range of the dispersal and the critical length of the gap observed numerically. We further explore other differences between the local and the non-local dispersal cases. In this paper, we illustrate these properties by numerical simulations and we state a series of open problems.

keywords: Entire solution gap problem non-local diffusion comparison principle propagation
DCDS
Reaction-diffusion equations for population dynamics with forced speed I - The case of the whole space
Henri Berestycki Luca Rossi
This paper is concerned with time-dependent reaction-diffusion equations of the following type:

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

keywords: principal eigenvalues travelling waves Reaction-diffusion equations extinction. time periodic parabolic equations persistence forced speed
DCDS-S
The periodic patch model for population dynamics with fractional diffusion
Henri Berestycki Jean-Michel Roquejoffre Luca Rossi
Fractional diffusions arise in the study of models from population dynamics. In this paper, we derive a class of integro-differential reaction-diffusion equations from simple principles. We then prove an approximation result for the first eigenvalue of linear integro-differential operators of the fractional diffusion type, and we study from that the dynamics of a population in a fragmented environment with fractional diffusion.
keywords: Fractional diffusion reaction-diffusion equation KPP nonlinearity persistence.
NHM
A model of riots dynamics: Shocks, diffusion and thresholds
Henri Berestycki Jean-Pierre Nadal Nancy Rodíguez
We introduce and analyze several variants of a system of differential equations which model the dynamics of social outbursts, such as riots. The systems involve the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. Our models include the effects of exogenous and endogenous factors as well as various propagation mechanisms. From numerical and mathematical analysis of these models we show that the assumptions made on how different locations influence one another and how the tension in the system disperses play a major role on the qualitative behavior of bursts of social unrest. Furthermore, we analyze here various properties of these systems, such as the existence of traveling wave solutions, and formulate some new open mathematical problems which arise from our work.
keywords: non-local diffusion. traveling wave solutions Mathematical modeling numerical solutions partial differential equations

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