arXiv subjects

Thomas Giletti

Publications and source records attributed to Thomas Giletti.

36 records · Page 2Linked to original sources

arXiv2019

Spreading speeds for multidimensional reaction-diffusion systems of the prey-predator type

We investigate spreading properties of solutions of a large class of two-component reaction-diffusion systems, including prey-predator systems as a special case. By spreading properties we mean the long time behaviour of solution fronts that start from localized (i.e. compactly supported) initial data. Though there are results in the literature on the existence of travelling waves for such systems, very little has been known-at least theoretically-about the spreading phenomena exhibited by solutions with compactly supported initial data. The main difficulty comes from the fact that the comparison principle does not hold for such systems. Furthermore, the techniques that are known for travelling waves such as fixed point theorems and phase portrait analysis do not apply to spreading fronts. In this paper, we first prove that spreading occurs with definite spreading speeds. Intriguingly, two separate fronts of different speeds may appear in one solution-one for the prey and the other for the predator-in some situations.

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arXiv2019

Existence and uniqueness of propagating terraces

This work focuses on dynamics arising from reaction-diffusion equations , where the profile of propagation is no longer characterized by a single front, but by a layer of several fronts which we call a propagating terrace. This means, intuitively, that transition from one equilibrium to another may occur in several steps, that is, successive phases between some intermediate stationary states. We establish a number of properties on such propagating terraces in a one-dimensional periodic environment, under very wide and generic conditions. We are especially concerned with their existence, uniqueness, and their spatial structure. Our goal is to provide insight into the intricate dynamics arising from multistable non-linearities.

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arXiv2019

Pulsating solutions for multidimensional bistable and multistable equations

We devote this paper to the issue of existence of pulsating travelling front solutions for spatially periodic heterogeneous reaction-diffusion equations in arbitrary dimension, in both bistable and more general multistable frameworks. In the multistable case, the notion of a single front is not sufficient to understand the dynamics of solutions, and we instead observe the appearance of a so-called propagating terrace. This roughly refers to a finite family of stacked fronts connecting intermediate stable steady states whose speeds are ordered. Surprisingly, for a given equation, the shape of this terrace (i.e., the involved intermediate states or even the cardinality of the family of fronts) may depend on the direction of propagation.

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arXiv2018

When fast diffusion and reactive growth both induce accelerating invasions

We focus on the spreading properties of solutions of monostable equations with fast diffusion. The nonlinear reaction term involves a weak Allee effect, which tends to slow down the propagation. We complete the picture of [3] by studying the subtle case where acceleration does occur and is induced by a combination of fast diffusion and of reactive growth. This requires the construction of new elaborate sub and supersolutions thanks to some underlying self-similar solutions.

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arXiv2017

Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions

We focus on the spreading properties of solutions of monostable equations with non-linear diffusion. We consider both the porous medium diffusion and the fast diffusion regimes. Initial data may have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity may involve a weak Allee effect, which tends to slow down the process. We study the balance between these three effects (nonlin-ear diffusion, initial tail, KPP nonlinearity/Allee effect), revealing the separation between "no acceleration" and "acceleration". In most of the cases where acceleration occurs, we also give an accurate estimate of the position of the level sets.

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arXiv2017

Travelling waves for a non-monotone bistable equation with delay: existence and oscillations

We consider a bistable ($0\textless{}θ\textless{}1$ being the three constant steady states) delayed reaction diffusion equation, which serves as a model in population dynamics. The problem does not admit any comparison principle. This prevents the use of classical technics and, as a consequence, it is far from obvious to understand the behaviour of a possible travelling wave in $+\infty$. Combining refined {\it a priori} estimates and a Leray Schauder topological degree argument, we construct a travelling wave connecting 0 in $-\infty$ to \lq\lq something" which is strictly above the unstable equilibrium $θ$ in $+\infty$. Furthemore, we present situations (additional bound on the nonlinearity or small delay) where the wave converges to 1 in $+\infty$, whereas the wave is shown to oscillate around 1 in $+\infty$ when, typically, the delay is large.

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arXiv2016

Extinction and spreading of a species under the joint influence of climate change and a weak Allee effect: a two-patch model

Many species see their range shifted poleward in response to global warming and need to keep pace in order to survive. To understand the effect of climate change on species ranges and its consequences on population dynamics, we consider a space-time heterogeneous reaction-diffusion equation in dimension 1, whose unknown~$u (t,x)$ stands for a population density. More precisely, the environment consists of two patches moving with a constant climate shift speed $c \geq 0$: in the invading patch $\{ t >0 , \, x \in \R \, | \ x < ct \}$ the growth rate is negative and, in the receding patch $\{ t >0 , \, x \in \R \, | \ x \geq ct \}$ it is of the classical monostable type. Our framework includes species subject to a weak Allee effect, meaning that there may be a positive correlation between population size and its per capita growth rate. We study the large-time behaviour of solutions in the moving frame and show that whether the population spreads or goes extinct depends not only on the speed $c$ but also, in some intermediate speed range, on the initial datum. This is in sharp contrast with the so-called `hair-trigger effect' in the homogeneous monostable equation, and suggests that the size of the population becomes a decisive factor under the joint influence of climate change and a weak Allee effect. Furthermore, our analysis exhibit sharp thresholds between spreading and extinction: in particular, we prove the existence of a threshold shifting speed which depends on the initial population, such that spreading occurs at lower speeds and extinction occurs at faster speeds.

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arXiv2015

Sharp thresholds between finite spread and uniform convergence for a reaction-diffusion equation with oscillating initial data

We investigate the large-time dynamics of solutions of multi-dimensional reaction-diffusion equations with ignition type nonlinearities. We consider solutions which are in some sense locally persistent at large time and initial data which asymptotically oscillate around the ignition threshold. We show that, as time goes to infinity, any solution either converges uniformly in space to a constant state, or spreads with a finite speed uniformly in all directions. Furthermore, the transition between these two behaviors is sharp with respect to the period vector of the asymptotic profile of the initial data. We also show the convergence to planar fronts when the initial data are asymptotically periodic in one direction.

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arXiv2015

A KPP road-field system with spatially periodic exchange terms

We take interest in a reaction-diffusion system which has been recently proposed [11] as a model for the effect of a road on propagation phenomena arising in epidemiology and ecology. This system consists in coupling a classical Fisher-KPP equation in a half-plane with a line with fast diffusion accounting for a straight road. The effect of the line on spreading properties of solutions (with compactly supported initial data) was investigated in a series of works starting from [11]. We recover these earlier results in a more general spatially periodic framework by exhibiting a threshold for road diffusion above which the propagation is driven by the road and the global speed is accelerated. We also discuss further applications of our approach, which will rely on the construction of a suitable generalized principal eigenvalue, and investigate in particular the spreading of solutions with exponentially decaying initial data.

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arXiv2015

Asymptotic analysis of a monostable equation in periodic media

We consider a multidimensional monostable reaction-diffusion equation whose nonlinearity involves periodic heterogeneity. This serves as a model of invasion for a population facing spatial heterogeneities. As a rescaling parameter tends to zero, we prove the convergence to a limit interface, whose motion is governed by the minimal speed (in each direction) of the underlying pulsating fronts. This dependance of the speed on the (moving) normal direction is in contrast with the homogeneous case and makes the analysis quite involved. Key ingredients are the recent improvement \cite{A-Gil} of the well-known spreading properties \cite{Wein02}, \cite{Ber-Ham-02}, and the solution of a Hamilton-Jacobi equation.

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arXiv2015

Varying the direction of propagation in reaction-diffusion equations in periodic media

We consider a multidimensional reaction-diffusion equation of either ignition or monostable type, involving periodic heterogeneity, and analyze the dependence of the propagation phenomena on the direction. We prove that the (minimal) speed of the underlying pulsating fronts depends continuously on the direction of propagation, and so does its associated profile provided it is unique up to time shifts. We also prove that the spreading properties \cite{Wein02} are actually uniform with respect to the direction.

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arXiv2014

Long time behavior of solutions of a reaction-diffusion equation on unbounded intervals with Robin boundary conditions

We study the long time behavior, as $t\to\infty$, of solutions of $$ \left\{ \begin{array}{ll} u_t = u_{xx} + f(u), & x>0, \ t >0,\\ u(0,t) = b u_x(0,t), & t>0,\\ u(x,0) = u_0 (x)\geqslant 0 , & x\geqslant 0, \end{array} \right. $$ where $b\geqslant 0$ and $f$ is an unbalanced bistable nonlinearity. By investigating families of initial data of the type $\{ σϕ\}_{σ>0}$, where $ϕ$ belongs to an appropriate class of nonnegative compactly supported functions, we exhibit the sharp threshold between vanishing and spreading. More specifically, there exists some value $σ^*$ such that the solution converges uniformly to 0 for any $0 < σ< σ^*$, and locally uniformly to a positive stationary state for any $ σ> σ^*$. In the threshold case $σ= σ^*$, the profile of the solution approaches the symmetrically decreasing ground state with some shift, which may be either finite or infinite. In the latter case, the shift evolves as $C \ln t$ where~$C$ is a positive constant we compute explicitly, so that the solution is traveling with a pulse-like shape albeit with an asymptotically zero speed. Depending on $b$, but also in some cases on the choice of the initial datum, we prove that one or both of the situations may happen.

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arXiv2013

Convergence to pulsating traveling waves with minimal speed in some KPP heterogeneous problems

The notion of traveling wave, which typically refers to some particular spatio-temporal con- nections between two stationary states (typically, entire solutions keeping the same profile's shape through time), is essential in the mathematical analysis of propagation phenomena. They provide insight on the underlying dynamics, and an accurate description of large time behavior of large classes of solutions, as we will see in this paper. For instance, in an homogeneous framework, it is well-known that, given a fast decaying initial datum (for instance, compactly supported), the solution of a KPP type reaction-diffusion equation converges in both speed and shape to the traveling wave with minimal speed. The issue at stake in this paper is the gener- alization of this result to some one-dimensional heterogeneous environments, namely spatially periodic or converging to a spatially periodic medium. This result fairly improves our under- standing of the large-time behavior of solutions, as well as of the role of heterogeneity, which has become a crucial challenge in this field over the past few years.

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arXiv2012

Existence and convergence to a propagating terrace in one-dimensional reaction-diffusion equations

We consider one-dimensional reaction-diffusion equations for a large class of spatially periodic nonlinearities (including multistable ones) and study the asymptotic behavior of solutions with Heaviside type initial data. Our analysis reveals some new dynamics where the profile of the propagation is not characterized by a single front, but by a layer of several fronts which we call a terrace. Existence and convergence to such a terrace is proven by using an intersection number argument, without much relying on standard linear analysis. Hence, on top of the peculiar phenomenon of propagation that our work highlights, several corollaries will follow on the existence and convergence to pulsating traveling fronts even for highly degenerate nonlinearities that have not been treated before.

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arXiv2011

Maximal and minimal spreading speeds for reaction diffusion equations in nonperiodic slowly varying media

This paper investigates the asymptotic behavior of the solutions of the Fisher-KPP equation in a heterogeneous medium, $$\partial_t u = \partial_{xx} u + f(x,u),$$ associated with a compactly supported initial datum. A typical nonlinearity we consider is $f(x,u) = μ_0 (ϕ(x)) u(1-u)$, where $μ_0$ is a 1-periodic function and $ϕ$ is a $\mathcal{C}^1$ increasing function that satisfies $\lim_{x\to +\infty} ϕ(x) = +\infty$ and $\lim_{x\to +\infty} ϕ' (x) = 0$. Although quite specific, the choice of such a reaction term is motivated by its highly heterogeneous nature. We exhibit two different behaviors for $u$ for large times, depending on the speed of the convergence of $ϕ$ at infinity. If $ϕ$ grows sufficiently slowly, then we prove that the spreading speed of $u$ oscillates between two distinct values. If $ϕ$ grows rapidly, then we compute explicitly a unique and well determined speed of propagation $w_\infty$, arising from the limiting problem of an infinite period. We give a heuristic interpretation for these two behaviors.

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arXiv2011

Inside dynamics of pulled and pushed fronts

We investigate the inside structure of one-dimensional reaction-diffusion traveling fronts. The reaction terms are of the monostable, bistable or ignition types. Assuming that the fronts are made of several components with identical diffusion and growth rates, we analyze the spreading properties of each component. In the monostable case, the fronts are classified as pulled or pushed ones, depending on the propagation speed. We prove that any localized component of a pulled front converges locally to 0 at large times in the moving frame of the front, while any component of a pushed front converges to a well determined positive proportion of the front in the moving frame. These results give a new and more complete interpretation of the pulled/pushed terminology which extends the previous definitions to the case of general transition waves. In particular, in the bistable and ignition cases, the fronts are proved to be pushed as they share the same inside structure as the pushed monostable critical fronts. Uniform convergence results and precise estimates of the left and right spreading speeds of the components of pulled and pushed fronts are also established.

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arXiv2010

KPP reaction-diffusion systems with loss inside a cylinder: convergence toward the problem with Robin boundary conditions

We consider in this paper a reaction-diffusion system under a KPP hypothesis in a cylindrical domain in the presence of a shear flow. Such systems arise in predator-prey models as well as in combustion models with heat losses. Similarly to the single equation case, the existence of a minimal speed c* and of traveling front solutions for every speed c > c* has been shown both in the cases of heat losses distributed inside the domain or on the boundary. Here, we deal with the accordance between the two models by choosing heat losses inside the domain which tend to a Dirac mass located on the boundary. First, using the characterizations of the corresponding minimal speeds, we will see that they converge to the minimal speed of the limiting problem. Then, we will take interest in the convergence of the traveling front solutions of our reaction-diffusion systems. We will show the convergence under some assumptions on those solutions, which in particular can be satisfied in dimension 2.

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arXiv2010

KPP reaction-diffusion equations with a non-linear loss inside a cylinder

We consider in this paper a reaction-diffusion system in presence of a flow and under a KPP hypothesis. While the case of a single-equation has been extensively studied since the pioneering Kolmogorov-Petrovski-Piskunov paper, the study of the corresponding system with a Lewis number not equal to 1 is still quite open. Here, we will prove some results about the existence of travelling fronts and generalized travelling fronts solutions of such a system with the presence of a non-linear spacedependent loss term inside the domain. In particular, we will point out the existence of a minimal speed, above which any real value is an admissible speed. We will also give some spreading results for initial conditions decaying exponentially at infinity.

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