arXiv subjects

Jérôme Coville

Publications and source records attributed to Jérôme Coville.

At least 19 recordsLinked to original sources

arXiv2026

An abstract framework for a class of nonlocal structured population models: existence, uniqueness and stability of steady states

This paper is concerned with the study of a class of nonlinear nonlocal functional evolution problems defined in an abstract Banach algebra. We introduce an abstract functional setting that encompasses a wide range of structured population models appearing in biomathematical literature. Within this framework, we analyze the well-posedness of the Cauchy problem and the existence of stationary solutions in the positive cone of the Banach algebra. By reviewing a large number of approaches, we also derive conditions for the local and global stability of these stationary solutions. Additionally, we explore the limits of these conditions by exhibiting explicit counterexamples. In particular, for mutation--selection models with symmetric mutation operators, we uncover both sufficient conditions for existence, uniqueness and stability, and counterexamples to existence or stability.

math.AP↗

arXiv2026

Travelling waves in nonlocal reaction-dispersion equations with diffuse L{é}vy measures

In this paper, we focus on the existence of travelling fronts for nonlocal reaction-dispersion models. Our aim is to provide a unified existence proof in a very broad framework using simple real analysis tools. In particular, we review the existing literature and fill in the gaps. It appears that the central case is the bistable nonlinearity, which we then extend to other classical nonlinearities.

math.AP↗

arXiv2026

Propagation rates in integro-differential equations of ignition type

This paper provides sharp rates of invasion in nonlinear integro-differential equations of ignition type. The framework covers both integrable convolution kernels and singular fractional-type kernels. While the emergence of a threshold on the decay $1+2s$ of the jump kernel is expected from earlier works, far fewer quantitative results were available in such a general framework. In particular, in addition to the spreading rates, our study provides information on the shape of the invasion profiles. Importantly, we obtain the first sharp spreading estimate in the critical case $s=\frac12$, which separates linear-in-time and accelerated regimes.

math.AP↗

arXiv2025

Sharp exponent of acceleration in general nonlocal equations with a weak Allee effect

We study an acceleration phenomenon arising in monostable integro-differential equations with a weak Allee effect. Previous works have shown its occurrence and have given correct upper bounds on the rate of expansion in some particular cases, but precise lower bounds were still missing. In this paper, we provide a sharp lower bound for this acceleration rate, valid for a large class of dispersion operators. Our results manage to cover fractional Laplace operators and standard convolutions in a unified way, which is new in the literature. A first very important result of the paper is a general flattening estimate of independent interest: this phenomenon appears regularly in acceleration situations, but getting quantitative estimates is most of the time open. This estimate at hand, we construct a very subtle sub-solution that captures the expected dynamics of the accelerating solution (rates of expansion and flattening) and identifies several various regimes that appear in the dynamics depending on the parameters of the problem.

math.AP↗

arXiv2024

Acceleration or finite speed propagation in integro-differential equations with logarithmic Allee effect

This paper is devoted to studying propagation phenomena in integro-differential equations with a weakly degenerate non-linearity. The reaction term can be seen as an intermediate between the classical logistic (or Fisher-KPP) non-linearity and the standard weak Allee effect one. We study the effect of the tails of the dispersal kernel on the rate of expansion. When the tail of the kernel is sub-exponential, the exact separation between existence and non-existence of travelling waves is exhibited. This, in turn, provides the exact separation between finite speed propagation and acceleration in the Cauchy problem. Moreover, the exact rates of acceleration for dispersal kernels with sub-exponential and algebraic tails are provided. Our approach is generic and covers a large variety of dispersal kernels including those leading to convolution and fractional Laplace operators. Numerical simulations are provided to illustrate our results.

math.AP↗

arXiv2024

Group Dispersal Modelling Revisited

In this paper we revisit the notion of grouped dispersal that have been introduced by Soubeyrand and co-authors \cite{soubeyrand2011patchy} to model the simultaneous (and hence dependent) dispersal of several propagules from a single source in a homogeneous environment. We built a time continuous measure valued process that takes into account the main feature of a grouped dispersal and derive its infinitesimal generator. To cope with the mutligeneration aspect associated to the demography we introduce two types of propagules in the description of the population which is one of the main innovations here. We also provide a rigorous description of the process and its generator. We derive as well, some large population asymptotics of the process unveilling the degenerate ultra parabolic system of PDE satisfied by the density of population. Finally, we also show that such a PDE system has a non-trivial solution which is unique in a certain functional space.

math.AP↗

arXiv2023

Acceleration or finite speed propagation in weakly monostable reaction-diffusion equations

This paper focuses on propagation phenomena in reaction-diffusion equations with a weaklymonostable nonlinearity. The reaction term can be seen as an intermediate between the classicallogistic one (or Fisher-KPP) and the standard weak Allee effect one. We investigate the effect ofthe decay rate of the initial data on the propagation rate. When the right tail of the initial datais sub-exponential, finite speed propagation and acceleration may happen and we derive the exactseparation between the two situations. When the initial data is sub-exponentially unbounded, accel-eration unconditionally occurs. Estimates for the locations of the level sets are expressed in termsof the decay of the initial data. In addition, sharp exponents of acceleration for initial data withsub-exponential and algebraic tails are given. Numerical simulations are presented to illustrate theabove findings.

math.AP↗

arXiv2023

A simple flattening lower bound for solutions to some linear integrodifferential equations

Estimates on the asymptotic behaviour of solution to linear integro-differential equations are fundamental in understanding the dynamics occuring in many nonlocal evolution problems. They are usually derived by using precise decay estimates on the heat kernel of the considered diffusion process. In this note, we show that for some generic jump diffusion and particular initial data, one can derive a lower bound of the asymptotic behaviour of the solution using a simple PDE argument. This is viewed as an independant preliminary brick to study invasion phenomena in nonlinear reaction diffusion problems.

math.AP↗

arXiv2022

Mapping the intensity function of a non-stationary point process in unobserved areas

Seismic networks provide data that are used as basis both for public safety decisions and for scientific research. Their configuration affects the data completeness, which in turn, critically affects several seismological scientific targets (e.g., earthquake prediction, seismic hazard...). In this context, a key aspect is how to map earthquakes density in seismogenic areas from censored data or even in areas that are not covered by the network. We propose to predict the spatial distribution of earthquakes from the knowledge of presence locations and geological relationships, taking into account any interaction between records. Namely, in a more general setting, we aim to estimate the intensity function of a point process, conditional to its censored realization, as in geostatistics for continuous processes. We define a predictor as the best linear unbiased combination of the observed point pattern. We show that the weight function associated to the predictor is the solution of a Fredholm equation of second kind. Both the kernel and the source term of the Fredholm equation are related to the first-and second-order characteristics of the point process through the intensity and the pair correlation function. Results are presented and illustrated on simulated non-stationary point processes and real data for mapping Greek Hellenic seismicity in a region with unreliable and incomplete records.

stat.ME↗

arXiv2021

Acceleration in integro-differential combustion equations

We study acceleration phenomena in monostable integro-differential equations with ignition nonlinearity. Our results cover fractional Laplace operators and standard convolutions in a unified way, which is also a contribution of this paper. To achieve this, we construct a sub-solution that captures the expected dynamics of the accelerating solution, and this is here the main difficulty. This study involves the flattening effect occurring in accelerated propagation phenomena.

math.AP↗

arXiv2020

Can a population survive in a shifting environment using non-local dispersion

In this article, we analyse the non-local model : $\partial$ t U (t, x) = J $\star$ U (t, x) -- U (t, x) + f (x -- ct, U (t, x)) for t > 0, and x $\in$ R, where J is a positive continuous dispersal kernel and f (x, s) is a heterogeneous KPP type non-linearity describing the growth rate of the population. The ecological niche of the population is assumed to be bounded (i.e. outside a compact set, the environment is assumed to be lethal for the population) and shifted through time at a constant speed c. For compactly supported dispersal kernels J, assuming that for c = 0 the population survive, we prove that there exists a critical speeds c * ,$\pm$ and c * * ,$\pm$ such that for all --c * ,-- < c < c * ,+ then the population will survive and will perish when c $\ge$ c * * ,+ or c $\le$ --c * * ,--. To derive this results we first obtain an optimal persistence criteria depending of the speed c for non local problem with a drift term. Namely, we prove that for a positive speed c the population persists if and only if the generalized principal eigenvalue $λ$ p of the linear problem cD x [$Φ$] + J $\star$ $Φ$ -- $Φ$ + $\partial$ s f (x, 0)$Φ$ + $λ$ p $Φ$ = 0 in R, is negative. $λ$ p is a spectral quantity that we defined in the spirit of the generalized first eigenvalue of an elliptic operator. The speeds c * ,$\pm$ and c * * ,pm are then obtained through a fine analysis of the properties of $λ$ p with respect to c. In particular, we establish its continuity with respect to the speed c. In addition, for any continuous bounded non-negative initial data, we establish the long time behaviour of the solution U (t, x).

math.AP↗

arXiv2020

Propagation phenomena with nonlocal diffusion in presence of an obstacle

We consider a nonlocal semi-linear parabolic equation on a connected exterior domain of the form $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a compact "obstacle". The model we study is motivated by applications in biology and takes into account long range dispersal events that may be anisotropic depending on how a given population perceives the environment. To formulate this in a meaningful manner, we introduce a new theoretical framework which is of both mathematical and biological interest. The main goal of this paper is to construct an entire solution that behaves like a planar travelling wave as $t\to-\infty$ and to study how this solution propagates depending on the shape of the obstacle. We show that whether the solution recovers the shape of a planar front in the large time limit is equivalent to whether a certain Liouville type property is satisfied. We study the validity of this Liouville type property and we extend some previous results of Hamel, Valdinoci and the authors. Lastly, we show that the entire solution is a generalised transition front.

math.AP↗

arXiv2020

Propagation acceleration in reaction diffusion equations with anomalous diffusions

In this paper, we are interested in the properties of solution of the nonlocal equation $$\begin{cases}u_t+(-Δ)^su=f(u),\quad t>0, \ x\in\mathbb{R}\\ u(0,x)=u_0(x),\quad x\in\mathbb{R}\end{cases}$$ where $0\le u_0<1$ is a Heaviside type function, $Δ^s$ stands for the fractional Laplacian with $s\in (0,1)$, and $f\in C([0,1],\mathbb{R}^+)$ is a non negative nonlinearity such that $f(0)=f(1)=0$ and $f'(1)<0$. In this context, it is known that the solution $u(t,s)$ converges locally uniformly to 1 and our aim here is to understand how fast this invasion process occur. When $f$ is a Fisher-KPP type nonlinearity and $s \in (0,1)$, it is known that the level set of the solution $u(t,x)$ moves at an exponential speed whereas when $f$ is of ignition type and $s\in \left(\frac{1}{2},1\right)$ then the level set of the solution moves at a constant speed. In this article, for general monostable nonlinearities $f$ and any $s\in (0,1)$ we derive generic estimates on the position of the level sets of the solution $u(t,x)$ which then enable us to describe more precisely the behaviour of this invasion process. In particular, we obtain a algebraic generic upper bound on the "speed" of level set highlighting the delicate interplay of $s$ and $f$ in the existence of an exponential acceleration process. When $s\in\left (0,\frac{1}{2}\right]$ and $f$ is of ignition type, we also complete the known description of the behaviour of $u$ and give a precise asymptotic of the speed of the level set in this context. Notably, we prove that the level sets accelerate when $s\in\left(0,\frac{1}{2}\right)$ and that in the critical case $s=\frac{1}{2}$ although no travelling front can exist, the level sets still move asymptotically at a constant speed. These new results are in sharp contrast with the bistable situation where no such acceleration may occur, highlighting therefore the qualitative difference between the two type of nonlinearities.

math.AP↗

arXiv2019

Optimising reactive disease management using spatially explicit models at the landscape scale

Increasing rates of global trade and travel, as well as changing climatic patterns, have led to more frequent outbreaks of plant disease epidemics worldwide. Mathematical modelling is a key tool in predicting where and how these new threats will spread, as well as in assessing how damaging they might be. Models can also be used to inform disease management, providing a rational methodology for comparing the performance of possible control strategies against one another. For emerging epidemics, in which new pathogens or pathogen strains are actively spreading into new regions, the spatial component of spread becomes particularly important, both to make predictions and to optimise disease control. In this chapter we illustrate how the spatial spread of emerging plant diseases can be modelled at the landscape scale via spatially explicit compartmental models. Our particular focus is on the crucial role of the dispersal kernel-which parameterises the probability of pathogen spread from an infected host to susceptible hosts at any given distance-in determining outcomes of epidemics. We add disease management to our model by testing performance of a simple ''one off'' form of reactive disease control, in which sites within a particular distance of locations detected to contain infection are removed in a single round of disease management. We use this simplified model to show how ostensibly arcane decisions made by the modeller-most notably whether or not the underpinning disease model allows for stochasticity (i.e. randomness)-can greatly impact on disease management recommendations. Our chapter is accompanied by example code in the programming language R available via an online repository, allowing the reader to run the models we present for him/herself.

q-bio.PE↗

arXiv2019

A note on Liouville type results for a fractional obstacle problem

This note is a synthesis of my reflexions on some questions that have emerged during the MATRIX event "Recent Trends on Nonlinear PDEs of Elliptic and Parabolic Type" concerning the qualitative properties of solutions to some non local reaction-diffusion equations of the form L[u](x) + f (u(x)) = 0, for x $\in$ R n \ K, where K $\subset$ R N is a bounded smooth compact "obstacle", L is non local operator and f is a bistable nonlinearity. When K is convex and the nonlocal operator L is a continuous operator of convolution type then some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity have been recently established by Brasseur, Coville, Hamel and Valdinoci [4]. Here, we show that for a bounded smooth convex obstacle K, similar Liouville type results hold true when the operator L is the regional s-fractional Laplacian.

math.AP↗

arXiv2018

On generalized principal eigenvalues of nonlocal operators with a drift *

This article is concerned with the following spectral problem: to find a positive function $Φ$ $\in$ C 1 ($Ω$) and $λ$ $\in$ R such that q(x)$Φ$ (x) + ^ $Ω$ J(x, y)$Φ$(y) dy + a(x)$Φ$(x) + $λ$$Φ$(x) = 0 for x $\in$ $Ω$, where $Ω$ $\subset$ R is a non-empty domain (open interval), possibly unbounded, J is a positive continuous kernel, and a and q are continuous coefficients. Such a spectral problem naturally arises in the study of nonlocal population dynamics models defined in a space-time varying environment encoding the influence of a climate change through a spatial shift of the coefficient. In such models, working directly in a moving frame that matches the spatial shift leads to consider a problem where the dispersal of the population is modeled by a nonlocal operator with a drift term. Assuming that the drift q is a positive function, for rather general assumptions on J and a, we prove the existence of a principal eigenpair ($λ$ p , $Φ$ p) and derive some of its main properties. In particular, we prove that $λ$ p ($Ω$) = lim R$\rightarrow$+$\infty$ $λ$ p ($Ω$ R), where $Ω$ R = $Ω$ $\cap$ (--R, R) and $λ$ p ($Ω$ R) corresponds to the principal eigenvalue of the truncation operator defined in $Ω$ R. The proofs especially rely on the derivation of a new Harnack type inequality for positive solutions of such problems.

math.AP↗

arXiv2018

A counterexample to the Liouville property of some nonlocal problems

In this paper, we construct a counterexample to the Liouville property of some nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,( u(y)-u(x) )\mathrm{d}y+f(u(x))=0, \quad x\in\R^N\setminus K,$$where $K\subset\mathbb{R}^N$ is a bounded compact set, called an "obstacle", and $f$ is a bistable nonlinearity. When $K$ is convex, it is known that solutions ranging in $[0,1]$ and satisfying $u(x)\to1$ as $|x|\to\infty$ must be identically $1$ in the whole space. We construct a nontrivial family of simply connected (non-starshaped) obstacles as well as data $f$ and $J$ for which this property fails.

math.AP↗

arXiv2017

Liouville type results for a nonlocal obstacle problem

This paper is concerned with qualitative properties of solutions to nonlocal reaction-diffusion equations of the form$$ \int\_{\mathbb{R}^N\setminus K} J(x-y)\,\big( u(y)-u(x) \big)\,\D y+f(u(x))=0, \quad x\in\R^N\setminus K,$$set in a perforated open set $\mathbb{R}^N\setminus K$, where $K\subset\mathbb{R}^N$ is a bounded compact "obstacle" and $f$ is a bistable nonlinearity. When $K$ is convex, we prove some Liouville-type results for solutions satisfying some asymptotic limiting conditions at infinity. We also establish a robustness result, assuming slightly relaxed conditions on $K$.

math.AP↗