arXiv subjects

Henri Berestycki

Publications and source records attributed to Henri Berestycki.

At least 19 recordsLinked to original sources

arXiv2026

FKPP fronts in quenched random media

We study numerically the evolution of one-dimensional FKPP fronts initiated from steep initial conditions in the presence of a quenched random growth rate. Compared to both the homogeneous case (with velocity $v_0$) and deterministic disorder, quenched randomness increases the average propagation speed. We show that the velocity shift relative to the homogeneous case scales linearly with the disorder variance $\sigma^2$, with a universal prefactor -- independent of the specific distribution of the disorder -- such that $v = v_0 + a \sigma^2$, with $a \approx 0.02432 \pm 0.00002$. Moreover, the front position exhibits diffusive fluctuations across disorder realizations. The corresponding effective diffusion coefficient scales quadratically with $\sigma$, $D = \frac{b^2 \sigma^2}{2}$, with $b \approx 0.223 \pm 0.002$. These results suggest a universal statistical response of FKPP fronts to quenched heterogeneity.

cond-mat.dis-nn↗

arXiv2026

Range expansion by growth and congestion

We introduce here a nonlinear and nonlocal model that describes the range expansion of a population resulting from growth and competition for space. This type of phenomenon underlies the expansion of colonies of immotile cells which motivated this work. These colonies display long-range shuffling induced by congestion, and are also subject to jamming. We start by showing the well-posedness of the general evolution equation. We then derive a singular limit of this model corresponding to a regime where dispersal occurs only from saturated areas. The limiting model, which has the structure of an obstacle free boundary problem in time, provides an effective approach to the description of the range expansion of a population as a result of growth, saturation and dispersion. We establish the main mathematical properties of this singular problem by proving a comparison property and describing the dynamics of the free boundary that delimits the saturated area. We identify traveling wave solutions and characterize the asymptotic speed of spreading of compactly supported solutions.

math.AP↗

arXiv2026

The Influence of Exclusion Zones on the Coexistence of Predator and Prey with an Allee Effect

We propose a reaction--diffusion model of predator--prey interaction in which the predators occupy only a subset of the prey's territory, leaving a predator-free exclusion zone. Ecological examples include marine protected areas where it is illegal to fish, or buffer zones left between the territories of rival predators. The prey are subject to a strong Allee effect, so excessive predation may lead to the extinction of both species. The exclusion zone mitigates this problem by providing the prey with a refuge in which to proliferate without predation. Thus, paradoxically, a smaller predator territory may be able to support a more substantial population than a larger one. Using a topological degree argument, we show in any dimensions that, provided the exclusion zone is large enough, the system possesses spatially heterogeneous coexistence equilibria with positive populations of both species. This result is global in the sense that it does not rely on local bifurcations from semi-trivial stationary states. We also show that as the predator domain becomes asymptotically small, the total predator population does not vanish, and in some cases may actually be maximized in this limit of shrinking predation area. Conversely, we show that as the predator domain becomes large, it may exhibit thresholding behavior, passing suddenly from a regime with coexistence solutions to one in which extinction becomes unavoidable, highlighting the need for careful analysis in the management of predator--prey systems.

math.AP↗

arXiv2025

Uniqueness and multiplicity for semilinear elliptic problems in unbounded domains

We study the influence of geometry on semilinear elliptic equations of bistable or nonlinear-field type in unbounded domains. We discover a surprising dichotomy between epigraphs that are bounded from below and those that contain a cone of aperture greater than $\pi$: the former admit at most one positive bounded solution, while the latter support infinitely many. Nonetheless, we show that every epigraph admits at most one strictly stable solution. To prove uniqueness, we strengthen the method of moving planes by decomposing the domain into one region where solutions are stable and another where they enjoy a form of compactness. Our construction of many solutions exploits a connection with Delaunay surfaces in differential geometry, and extends to all domains containing a suitably wide cone, including exterior domains.

math.AP↗

arXiv2025

Generalized principal eigenvalues for parabolic operators in bounded domains

We introduce here new generalized principal eigenvalues for linear parabolic operators with heterogeneous coefficients in space and time. We consider a bounded spatial domain and an unbounded time interval $I$ : $I=\mathbb{R},\ \mathbb{R}^+$ or $\mathbb{R}^-$, and operators with coefficients having a fairly general dependence on space and time. The notions we introduce rely on the parabolic maximum principle and extend some earlier definitions introduced for elliptic operators [BNV]. We first show that these eigenvalues hold the key to understanding the large time behavior and entire solutions of heterogeneous Fisher-KPP type equations. We then describe the relation of these quantities with principal Floquet bundles for parabolic operators which provides further characterizations of the principal eigenvalues. These allow us to derive monotonicity properties and comparisons between generalized principal eigenvalues, as well as perturbation results and further properties involving limit operators. We show that the sign of these eigenvalues encodes different versions of the maximum principle for parabolic operators. Lastly, we explicitly compute the generalized principal eigenvalues for several classes of operators such as spatial-independent, periodic, almost periodic, uniquely ergodic or random stationary ergodic coefficients.

math.AP↗

arXiv2024

Diffusion of knowledge and the lottery society

The Lucas-Moll system is a mean-field game type model describing the growth of an economy by means of diffusion of knowledge. The individual agents in the economy advance their knowledge by learning from each other and via internal innovation. Their cumulative distribution function satisfies a forward in time nonlinear non-local reaction-diffusion type equation. On the other hand, the learning strategy of the agents is based on the solution to a backward in time nonlocal Hamilton-Jacobi-Bellman equation that is coupled to the aforementioned equation for the agents density. Together, these equations form a system of the mean-field game type. When the learning rate is sufficiently large, existence of balanced growth path solutions to the Lucas-Moll system was proved in~\cite{PRV,Porretta-Rossi}. Here, we analyze a complementary regime where the balanced growth paths do not exist. The main result is a long time convergence theorem. Namely, the solution to the initial-terminal value problem behaves in such a way that at large times an overwhelming majority of the agents spend no time producing at all and are only learning. In particular, the agents density propagates at the Fisher-KPP speed. We name this type of solutions a lottery society.

math.AP↗

arXiv2024

Front propagation through a perforated wall

We consider a bistable reaction-diffusion equation $u_t=Δu +f(u)$ on $\mathbb{R}^N$ in the presence of an obstacle $K$, which is a wall of infinite span with many holes. More precisely, $K$ is a closed subset of $\mathbb{R}^N$ with smooth boundary such that its projection onto the $x_1$-axis is bounded and that $\mathbb{R}^N \setminus K$ is connected. Our goal is to study what happens when a planar traveling front coming from $x_1 = -\infty$ meets the wall $K$.We first show that there is clear dichotomy between "propagation" and "blocking". In other words, the traveling front either passes through the wall and propagates toward $x_1=+\infty$ (propagation) or is trapped around the wall (blocking), and that there is no intermediate behavior. This dichotomy holds for any type of walls of finite thickness. Next we discuss sufficient conditions for blocking and propagation. For blocking, assuming either that $K$ is periodic in $y:=(x_2,\ldots, x_N)$ or that the holes are localized within a bounded area, we show that blocking occurs if the holes are sufficiently narrow. For propagation, three different types of sufficient conditions for propagation will be presented, namely "walls with large holes", "small-capacity walls", and "parallel-blade walls". We also discuss complete and incomplete invasions.

math.AP↗

arXiv2024

Biological invasions and epidemics with nonlocal diffusion along a line

The goal of this work is to understand and quantify how a line with nonlocal diffusion given by an integral enhances a reaction-diffusion process occurring in the surrounding plane. This is part of a long term programme where we aim at modelling, in a mathematically rigorous way, the effect of transportation networks on the speed of biological invasions or propagation of epidemics. We prove the existence of a global propagation speed and characterise in terms of the parameters of the system the situations where such a speed is boosted by the presence of the line. In the course of the study we also uncover unexpected regularity properties of the model. On the quantitative side, the two main parameters are the intensity of the diffusion kernel and the characteristic size of its support. One outcome of this work is that the propagation speed will significantly be enhanced even if only one of the two is large, thus broadening the picture that we have already drawn in our previous works on the subject, with local diffusion modelled by a standard Laplacian. We further investigate the role of the other parameters, enlightening some subtle effects due to the interplay between the diffusion in the half plane and that on the line. Lastly, in the context of propagation of epidemics, we also discuss the model where, instead of a diffusion, displacement on the line comes from a pure transport term.

math.AP↗

arXiv2023

A stable-compact method for qualitative properties of semilinear elliptic equations

We study the uniqueness of reaction-diffusion steady states in general domains with Dirichlet boundary data. Here we consider "positive" (monostable) reactions. We describe geometric conditions on the domain that ensure uniqueness and we provide complementary examples of nonuniqueness. Along the way, we formulate a number of open problems and conjectures. To derive our results, we develop a general framework, the stable-compact method, to study qualitative properties of nonlinear elliptic equations.

math.AP↗

arXiv2022

The steady states of strong-KPP reactions in general domains

We study the uniqueness of steady states of strong-KPP reaction--diffusion equations in general domains under various boundary conditions. We show that positive bounded steady states are unique provided the domain satisfies a certain spectral nondegeneracy condition. We also formulate a number of open problems and conjectures.

math.AP↗

arXiv2021

Modeling the propagation of riots, collective behaviors, and epidemics

This paper is concerned with a family of Reaction-Diffusion systems that we introduced in [15], and that generalizes the SIR type models from epidemiology. Such systems are now also used to describe collective behaviors.In this paper, we propose a modeling approach for these apparently diverse phenomena through the example of the dynamics of social unrest. The model involves two quantities: the level of social unrest, or more generally activity, u, and a field of social tension v, which play asymmetric roles. We think of u as the actually observed or explicit quantity while v is an ambiant, sometimes implicit, field of susceptibility that modulates the dynamics of u. In this article, we explore this class of model and prove several theoretical results based on the framework developed in [15], of which the present work is a companion paper. We particularly emphasize here two subclasses of systems: tension inhibiting and tension enhancing. These are characterized by respectively a negative or a positivefeedback of the unrest on social tension. We establish several properties for these classes and also study some extensions. In particular, we describe the behavior of the system following an initial surge of activity. We show that the model can give rise to many diverse qualitative dynamics. We also provide a variety of numerical simulations to illustrate our results and to reveal further properties and open questions.

math.AP↗

arXiv2020

Propagation of epidemics along lines with fast diffusion

It has long been known that epidemics can travel along communication lines, such as roads. In the current COVID-19 epidemic, it has been observed that major roads have enhanced its propagation in Italy. We propose a new simple model of propagation of epidemics which exhibits this effect and allows for a quantitative analysis. The model consists of a classical $SIR$ model with diffusion, to which an additional compartment is added, formed by the infected individuals travelling on a line of fast diffusion. Exchanges between individuals on the line and in the rest of the domain are taken into account. A classical transformation allows us to reduce the proposed model to a system analogous to one we had previously introduced [5] to describe the enhancement of biological invasions by lines of fast diffusion. We establish the existence of a minimal spreading speed and we show that it may be quite large, even when the basic reproduction number $R_0$ is close to $1$. More subtle qualitative features of the final state, showing the important influence of the line, are also proved here.

math.AP↗

arXiv2020

Reaction-diffusion equations in the half-space

We study reaction-diffusion equations of various types in the half-space. For bistable reactions with Dirichlet boundary conditions, we prove conditional uniqueness: there is a unique nonzero bounded steady state which exceeds the bistable threshold on large balls. Moreover, solutions starting from sufficiently large initial data converge to this steady state as $t \to \infty$. For compactly supported initial data, the asymptotic speed of this propagation agrees with the unique speed $c_*$ of the one-dimensional traveling wave. We furthermore construct a traveling wave in the half-plane of speed $c_*$. In parallel, we show analogous results for ignition reactions under both Dirichlet and Robin boundary conditions. Using our ignition construction, we obtain stronger results for monostable reactions with the same boundary conditions. For such reactions, we show in general that there is a unique nonzero bounded steady state. Furthermore, monostable reactions exhibit the hair-trigger effect: every solution with nontrivial initial data converges to this steady state as $t \to \infty$. Given compactly supported initial data, this disturbance propagates at a speed $c_*$ equal to the minimal speed of one-dimensional traveling waves. We also construct monostable traveling waves in the Dirichlet or Robin half-plane with any speed $c \geq c_*$.

math.AP↗

arXiv2020

A parsimonious model for spatial transmission and heterogeneity in the COVID-19 propagation

Raw data on the cumulative number of deaths at a country level generally indicate a spatially variable distribution of the incidence of COVID-19 disease. An important issue is to determine whether this spatial pattern is a consequence of environmental heterogeneities, such as the climatic conditions, during the course of the outbreak. Another fundamental issue is to understand the spatial spreading of COVID-19. To address these questions, we consider four candidate epidemiological models with varying complexity in terms of initial conditions, contact rates and non-local transmissions, and we fit them to French mortality data with a mixed probabilistic-ODE approach. Using standard statistical criteria, we select the model with non-local transmission corresponding to a diffusion on the graph of counties that depends on the geographic proximity, with time-dependent contact rate and spatially constant parameters. This original spatially parsimonious model suggests that in a geographically middle size centralized country such as France, once the epidemic is established, the effect of global processes such as restriction policies, sanitary measures and social distancing overwhelms the effect of local factors. Additionally, this modeling approach reveals the latent epidemiological dynamics including the local level of immunity, and allows us to evaluate the role of non-local interactions on the future spread of the disease. In view of its theoretical and numerical simplicity and its ability to accurately track the COVID-19 epidemic curves, the framework we develop here, in particular the non-local model and the associated estimation procedure, is of general interest in studying spatial dynamics of epidemics.

q-bio.PE↗

arXiv2019

Influence of a road on a population in an ecological niche facing climate change

We introduce a model designed to account for the influence of a line with fast diffusion-such as a road or another transport network-on the dynamics of a population in an ecological niche. This model consists of a system of coupled reaction-diffusion equations set on domains with different dimensions (line / plane). We first show that the presence of the line is always deleterious and can even lead the population to extinction. Next, we consider the case where the niche is subject to a displacement, representing the effect of a climate change or of seasonal variation of resources. We find that in such case the presence of the line with fast diffusion can help the population to persist. We also study several qualitative properties of this system. The analysis is based on a notion of generalized principal eigenvalue developed by the authors in [5].

math.AP↗

arXiv2019

The existence and stability of spike solutions for a chemotaxis system modeling crime pattern formation

This paper is a continuation of the paper Berestycki, Wei and Winter \cite{Berestycki2014}. In \cite{Berestycki2014}, the existence of multiple symmetric and asymmetric spike solutions of a chemotaxis system modeling crime pattern formation, suggested by Short, Bertozzi, and Brantingham \cite{Short2010}, has been proved in the one-dimensional case. The problem of stability of these spike solutions has been left open. In this paper, we establish the existence of a single radial symmetric spike solution for the system in the one and two-dimensional cases. The main difficulty is to deal with quasilinear elliptic problems whose diffusion coefficients vary largely near the core. We also study the linear stability of the spike solutions in both one-dimensional and two-dimensional cases which show complete different behaviors. In the one-dimensional case, we show that when the reaction time ratio $τ>0$ is small enough, or large enough, the spike solution is linearly stable. In the two-dimensional case, when $τ$ is small enough, the spike solution is linearly stable; while when $τ$ is large enough, the spike solution is linearly unstable and Hopf bifurcation occurs from the spike solution at some $τ=τ_h$.

math.AP↗

arXiv2019

Predator-prey models with competition, Part II: uniform regularity estimates

We study a system of elliptic equations with strong competition and an arbitrary large number of components. The system is related to a model of predators and prey, with a single and where several predators compete with each other. In this paper we derive regularity estimates of the solutions that are independent of the number of components (i.e., groups of predators) and the strength of competition between the components.

math.AP↗

arXiv2019

Coupled reaction-diffusion equations on adjacent domains

We consider a reaction-diffusion system for two densities lying in adjacent domains of $\mathbb{R}^N$. We treat two configurations: either a cylinder and its complement, or two half-spaces. Diffusion and reaction heterogeneities for the two densities are considered, and an exchange occurs through the separating boundary. We study the long-time behavior of the solution, and, when it converges to a positive steady state, we prove the existence of an asymptotic speed of propagation in some specific directions. Moreover, we determine how such a speed qualitatively depends with respect to several parameters appearing in the model. In the case $N=2$, we compare such properties to those studied in [6-9] for a model with a line representing a road of fast diffusion at the boundary of a half-plane, which can be seen as a singular limit of the problem studied here.

math.AP↗