arXiv subjects

Léo Girardin

Publications and source records attributed to Léo Girardin.

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

Stochastic dynamics at the back of a gene drive eradication wave

Gene drive alleles bias their own inheritance to offspring. They can fix in a wild-type population in spite of a fitness cost, and even lead to the eradication of the target population if the fitness cost is high. However, this outcome may be prevented or delayed if areas previously cleared by the drive are recolonised by wild-type individuals. Here, we investigate the conditions under which these stochastic wild-type recolonisation events are likely and when they are unlikely to occur in one spatial dimension. More precisely, we examine the conditions ensuring that the last individual carrying a wild-type allele is surrounded by a large enough number of drive homozygous individuals, resulting in a very low chance of wild-type recolonisation. To do so, we make a deterministic approximation of the distribution of drive alleles within the wave, and we split the distribution of wild-type alleles into a deterministic part and a stochastic part. Our analytical and numerical results suggest that the probability of wild-type recolonisation events increases with lower fitness of drive individuals and with smaller local carrying capacity. Numerical simulations show that these results extend to two spatial dimensions. The role of the migration rate however, is less clear but has a lower impact. We further demonstrate that, in the event of wild-type recolonization, the probability of subsequent drive reinvasion decreases with smaller values of the intrinsic growth rate of the population. Overall, our study paves the way for further analysis of wild-type recolonisation at the back of eradication traveling waves.

q-bio.PE↗

arXiv2026

Planar pulsating traveling wave solutions of non-cooperative Fisher--KPP systems in space-time periodic media

Non-cooperative Fisher-KPP systems with space-time periodic coefficients are motivated for instance by models for structured populations evolving in periodic environments. This paper is concerned with entire solutions describing the invasion of open space by a persistent population at constant speed. These solutions are important in the understanding of long-time behaviors for the Cauchy problem. Adapting methods developed for scalar equations satisfying the comparison principle as well as methods developed for systems with homogeneous coefficients, we prove, in each spatial direction, the existence of a critical speed such that: there exists no almost planar generalized transition waves with a smaller speed; if the direction is rational, each rational speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time and transverse space periodicity; if the coefficients are homogeneous in space, each speed not smaller than the critical speed is the speed of a planar pulsating traveling wave with time periodicity.

math.AP↗

arXiv2025

Spreading properties of the Fisher--KPP equation when the intrinsic growth rate is maximal in a moving patch of bounded size

This paper is concerned with spreading properties of space-time heterogeneous Fisher--KPP equations in one space dimension. We focus on the case of everywhere favorable environment with three different zones, a left half-line with slow or intermediate growth, a central patch with fast growth and a right half-line with slow or intermediate growth. The central patch moves at various speeds. The behavior of the front changes drastically depending on the speed of the central patch. Among other things, intriguing phenomena such as nonlocal pulling and locking may occur, which would make the behavior of the front further complicated. The problem we discuss here is closely related to questions in biomathematical modelling. By considering several special cases, we illustrate the remarkable diversity of possible behaviors. In particular, when the central patch has constant size and constant speed, we provide a complete set of explicit formulas for the spreading speed.

math.AP↗

arXiv2025

Persistence, extinction and spreading properties of non-cooperative Fisher--KPP systems in space-time periodic media

This paper is concerned with asymptotic persistence, extinction and spreading properties for non-cooperative Fisher-KPP systems with space-time periodic coefficients. Results are formulated in terms of a family of generalized principal eigenvalues associated with the linearized problem. When the maximal generalized principal eigenvalue is negative, all solutions to the Cauchy problem become locally uniformly positive in long-time, at least one space-time periodic uniformly positive entire solution exists, and solutions with compactly supported initial condition asymptotically spread in space at a speed given by a Freidlin-G{ä}rtner-type formula. When another, possibly smaller, generalized principal eigenvalue is nonnegative, then on the contrary all solutions to the Cauchy problem vanish uniformly and the zero solution is the unique space-time periodic nonnegative entire solution. When the twogeneralized principal eigenvalues differ and zero is in between, the long-time behavior depends on the decay at infinity of the initial condition. The proofs rely upon double-sided controls by solutions of cooperative systems. The control from below is new for such systems and makes it possible to shorten the proofs and extend the generality of the system simultaneously.

math.AP↗

arXiv2024

Generalized principal eigenvalues of space-time periodic, weakly coupled, cooperative, parabolic systems

This paper is concerned with generalizations of the notion of principal eigenvalue in the context of space-time periodic cooperative systems. When the spatial domain is the whole space, the Krein-Rutman theorem cannot be applied and this leads to more sophisticated constructions and to the notion of generalized principal eigenvalues. These are not unique in general and we focus on a one-parameter family corresponding to principal eigenfunctions that are space-time periodic multiplicative perturbations of exponentials of the space variable. Besides existence and uniqueness properties of such principal eigenpairs, we also prove various dependence and optimization results illustrating how known results in the scalar setting can, or cannot, be extended to the vector setting. We especially prove an optimization property on minimizers and maximizers among mutation operators valued in the set of bistochastic matrices that is, to the best of our knowledge, new.

math.AP↗

arXiv2023

Pulled, pushed or failed: the demographic impact of a gene drive can change the nature of its spatial spread

Understanding the temporal spread of gene drive alleles -- alleles that bias their own transmission -- through modeling is essential before any field experiments. In this paper, we present a deterministic reaction-diffusion model describing the interplay between demographic and allelic dynamics, in a one-dimensional spatial context. We focused on the traveling wave solutions, and more specifically, on the speed of gene drive invasion (if successful). We considered various timings of gene conversion (in the zygote or in the germline) and different probabilities of gene conversion (instead of assuming 100$\%$ conversion as done in a previous work). We compared the types of propagation when the intrinsic growth rate of the population takes extreme values, either very large or very low. When it is infinitely large, the wave can be either successful or not, and, if successful, it can be either pulled or pushed, in agreement with previous studies (extended here to the case of partial conversion). In contrast, it cannot be pushed when the intrinsic growth rate is vanishing. In this case, analytical results are obtained through an insightful connection with an epidemiological SI model. We conducted extensive numerical simulations to bridge the gap between the two regimes of large and low growth rate. We conjecture that, if it is pulled in the two extreme regimes, then the wave is always pulled, and the wave speed is independent of the growth rate. This occurs for instance when the fitness cost is small enough, or when there is stable coexistence of the drive and the wild-type in the population after successful drive invasion. Our model helps delineate the conditions under which demographic dynamics can affect the spread of a gene drive.

math.AP↗

arXiv2022

Agro-ecological control of a pest-host system: preventing spreading

We consider an agro-ecologically motivated coupling between a prey-predator system and a vector-borne epidemic system. The coupled system contains one ODE, two reaction-diffusion PDEs and one reaction-diffusionadvection PDE; it has no complete variational or monotonic structure and coefficients are spatially heterogeneous. We study the long-time behavior of solutions of the Cauchy-Neumann problem, which turns out to be largely decided by a linear stability criterion but still involves nonlinear subtleties. Then we consider an optimal control problem. Although it remains elusive analytically, we show numerically the variety of outcomes and the strong impact of the initial conditions.

math.AP↗

arXiv2021

Spatial segregation limit of traveling wave solutions for a fully nonlinear strongly coupled competitive system

The paper is concerned with a singular limit for the bistable traveling wave problem in a very large class of two-species fully nonlinear parabolic systems with competitive reaction terms. Assuming existence of traveling waves and enough compactness, we derive and characterize the limiting problem. The assumptions and results are discussed in detail. The free boundary problem obtained at the limit is specified for important applications.

math.AP↗

arXiv2021

Demographic feedbacks can hamper the spatial spread of a gene drive

This paper is concerned with a reaction--diffusion system modeling the fixation and the invasion in a population of a gene drive (an allele biasing inheritance, increasing its own transmission to offspring). In our model, the gene drive has a negative effect on the fitness of individuals carrying it, and is therefore susceptible of decreasing the total carrying capacity of the population locally in space. This tends to generate an opposing demographic advection that the gene drive has to overcome in order to invade. While previous reaction--diffusion models neglected this aspect, here we focus on it and try to predict the sign of the traveling wave speed. It turns out to be an analytical challenge, only partial results being within reach, and we complete our theoretical analysis by numerical simulations. Our results indicate that taking into account the interplay between population dynamics and population genetics might actually be crucial, as it can effectively reverse the direction of the invasion and lead to failure. Our findings can be extended to other bistable systems, such as the spread of cytoplasmic incompatibilities caused by Wolbachia.

math.AP↗

arXiv2021

A note on "Existence and uniqueness of coexistence states for an elliptic system coupled in the linear part", by Hei Li-Jun, Nonlinear Anal. Real World Appl. 5, 2004

In this short paper I report on a paper published in Nonlinear Analysis: Real World Applications in 2004. There is a major mistake early in that paper which makes most of its claims false. The class of reaction-diffusion systems considered in the paper has been the object of a renewed investigation in the past few years, by myself and others, and recent discoveries provide explicit counterexamples .

math.AP↗

arXiv2021

When the Allee threshold is an evolutionary trait: persistence vs. extinction

We consider a nonlocal parabolic equation describing the dynamics of a population structured by a spatial position and a phenotypic trait, submitted to dispersion , mutations and growth. The growth term may be of the Fisher-KPP type but may also be subject to an Allee effect which can be weak (non-KPP monostable nonlinearity, possibly degenerate) or strong (bistable nonlinearity). The type of growth depends on the value of a variable $θ$ : the Allee threshold, which is considered here as an evolutionary trait. After proving the well-posedness of the Cauchy problem, we study the long time behavior of the solutions. Due to the richness of the model and the interplay between the various phenomena and the nonlocality of the growth term, the outcomes (extinction vs. persistence) are various and in sharp contrast with earlier results of the existing literature on local reaction-diffusion equations.

math.AP↗

arXiv2020

A Liouville-type result for non-cooperative Fisher--KPP systems and nonlocal equations in cylinders

We address the uniqueness of the nonzero stationary state for a reaction-diffusion system of Fisher-KPP type that does not satisfy the comparison principle. Although the uniqueness is false in general, it turns out to be true under biologically natural assumptions on the parameters. This Liouville-type result is then used to characterize the wake of traveling waves. All results are extended to an analogous nonlocal reaction-diffusion equation that contains as a particular case the cane toads equation with bounded traits.

math.AP↗

arXiv2020

The effect of random dispersal on competitive exclusion -- A review

Does a high dispersal rate provide a competitive advantage when risking competitive exclusion? To this day, the theoretical literature cannot answer this question in full generality. The present paper focuses on the simplest mathematical model with two populations differing only in dispersal ability and whose one-dimensional territories are spatially segregated. Although the motion of the border between the two territories remains elusive in general, all cases investigated in the literature concur: either the border does not move at all because of some environmental heterogeneity or the fast diffuser chases the slow diffuser. Counterintuitively, it is better to randomly explore the hostile enemy territory, even if it means highly probable death of some individuals, than to "stay united". This directly contradicts a celebrated result on the intermediate competition case, emphasizing the importance of the competition intensity. Overall, the larger picture remains unclear and the optimal strategy regarding dispersal remains ambiguous. Several open problems worthy of a special attention are raised.

math.AP↗

arXiv2019

Catch me if you can: a spatial model for a brake-driven gene drive reversal

Population management using artificial gene drives (alleles biasing inheritance, increasing their own transmission to offspring) is becoming a realistic possibility with the development of CRISPR-Cas genetic engineering. A gene drive may however have to be stopped. "Antidotes" (brakes) have been suggested, but have been so far only studied in well-mixed populations. Here, we consider a reaction--diffusion system modeling the release of a gene drive (of fitness $1-a$) and a brake (fitness $1-b$, $b\leq a$) in a wild-type population (fitness $1$). We prove that, whenever the drive fitness is at most $1/2$ while the brake fitness is close to $1$, coextinction of the brake and the drive occurs in the long run. On the contrary, if the drive fitness is greater than $1/2$, then coextinction is impossible: the drive and the brake keep spreading spatially, leaving in the invasion wake a complicated spatio-temporally heterogeneous genetic pattern. Based on numerical experiments, we argue in favor of a global coextinction conjecture provided the drive fitness is at most $1/2$, irrespective of the brake fitness. The proof relies upon the study of a related predator--prey system with strong Allee effect on the prey. Our results indicate that some drives may be unstoppable, and that, if gene drives are ever deployed in nature, threshold drives, that only spread if introduced in high enough frequencies, should be preferred.

math.AP↗

arXiv2019

Two components is too simple: an example of oscillatory Fisher--KPP system with three components

In a recent paper by Cantrell, Cosner and Yu, two-component KPP systems with competition of Lotka--Volterra type were analyzed and their long-time behavior largely settled. In particular, the authors established that any constant positive steady state, if unique, is necessarily globally attractive. In the present paper, we give an explicit and biologically very natural example of oscillatory three-component system. Using elementary techniques or pre-established theorems, we show that it has a unique constant positive steady state with two-dimensional unstable manifold, a stable limit cycle, a predator--prey structure near the steady state, periodic wave trains and point-to-periodic rapid traveling waves. Numerically, we also show the existence of pulsating fronts and propagating terraces.

math.AP↗

arXiv2019

Invasion of open space by two competitors: spreading properties of monostable two-species competition--diffusion systems

This paper is concerned with some spreading properties of monostable Lotka--Volterra two-species competition--diffusion systems when the initial values are null or exponentially decaying in a right half-line. Thanks to a careful construction of super-solutions and sub-solutions, we improve previously known results and settle open questions. In particular, we show that if the weaker competitor is also the faster one, then it is able to evade the stronger and slower competitor by invading first into unoccupied territories. The pair of speeds depends on the initial values. If these are null in a right half-line, then the first speed is the KPP speed of the fastest competitor and the second speed is given by an exact formula describing the possibility of non-local pulling. Furthermore, the unbounded set of pairs of speeds achievable with exponentially decaying initial values is characterized, up to a negligible set.

math.AP↗

arXiv2018

Competition in periodic media: III -- Existence \& stability of segregated periodic coexistence states

In this paper we consider a system of parabolic reaction-diffusion equations with strong competition and two related scalar reaction-diffusion equations. We are mainly concerned with the case of periodic coefficients and periodic solutions. We show that, for sufficiently large periods, these models have stationary, non-constant, fully non-trivial and stable solutions. We compare our results with already known results about the existence and non-existence of such solutions. Finally, we provide ecological interpretations for these results in terms of resistance against an invasion.

math.AP↗