arXiv subjects

Gaël Raoul

Publications and source records attributed to Gaël Raoul.

14 recordsLinked to original sources

arXiv2026

A model for a population of trees structured by phenological traits

In the context of global warming, tree populations rely on two primary mechanisms of adaptation: phenotypic plasticity, which enables individuals to adjust their behavior in response to environmental stress, and genetic evolution, driven by natural selection and genetic diversity within the population. Understanding the interplay between these mechanisms is crucial for assessing the impacts of climate change on forest ecosystems and for informing sustainable management strategies. In this manuscript, we focus on a specific phenological adaptation: the ability of trees to enter summer dormancy once a critical temperature threshold is exceeded. Individuals are characterized by this threshold temperature and by their seed production capacity. We first establish a detailed mathematical model describing the population dynamics under these traits, and progressively reduce it to a system of two coupled ordinary differential equations. This simpler macroscopic model is then analyzed numerically, to investigate how the population reacts to a shift in its environment: an temperature increase, a drop in precipitation levels, or a combination of the two. Our results highlight contrasting effects of water stress and temperature stress on population dynamics, as well as the ambivalent effect of the plasticity.

q-bio.PE↗

arXiv2025

On measure-valued solutions for a structured population model with transfers

We consider a transfer operator where two interacting cells carrying non-negative traits transfer a random fraction of their trait to each other. These transfers can lead to population having singular distributions in trait. We extend the definition of the transfer operator to non-negative measures with a finite second moment, and we discuss the regularity of the fixed distributions of that transfer operator. Finally, we consider a dynamic transfer model where an initial population distribution is affected by a transfer operator: we prove the existence and uniqueness of mild measure-valued solutions for that Cauchy problem.

math.AP↗

arXiv2025

Macroscopic limit from a structured population model to the Kirkpatrick-Barton model

We consider an ecology model in which the population is structured by a spatial variable and a phenotypic trait. The model combines a parabolic operator on the spatial variable with a kinetic operator on the trait variable. We prove the existence of solutions to that model, and show that these solutions are unique. The kinetic operator present in the model, that represents the effect of sexual reproductions, satisfies a Tanaka-type inequality: it implies a contraction of the Wasserstein distance in the space of phenotypic traits. We combine this contraction argument with parabolic estimates controlling the spatial regularity of solutions to prove the convergence of the population size and the mean phenotypic trait to solutions of the Kirkpatrick-Barton model, which is a well-established model in evolutionary ecology. Specifically, at high reproductive rates, we provide explicit convergence estimates for the moments of solutions of the kinetic model.

math.AP↗

arXiv2024

Spreading speed of locally regulated population models in macroscopically heterogeneous environments

We consider a certain lattice branching random walk with on-site competition and in an environment which is heterogeneous at a macroscopic scale $1/\varepsilon$ in space and time. This can be seen as a model for the spatial dynamics of a biological population in a habitat which is heterogeneous at a large scale (mountains, temperature or precipitation gradient\ldots). The model incorporates another parameter, $K$, which is a measure of the local population density. We study the model in the limit when first $\varepsilon\to 0$ and then $K\to\infty$. In this asymptotic regime, we show that the rescaled position of the front as a function of time converges to the solution of an explicit ODE. We further discuss the relation with another popular model of population dynamics, the Fisher-KPP equation, which arises in the limit $K\to\infty$. Combined with known results on the Fisher-KPP equation, our results show in particular that the limits $\varepsilon\to0$ and $K\to\infty$ do not commute in general. We conjecture that an interpolating regime appears when $\log K$ and $1/\varepsilon$ are of the same order.

math.PR↗

arXiv2021

Exponential convergence to a steady-state for a population genetics model with sexual reproduction and selection

We are interested in the dynamics of a population structured by a phenotypic trait. Individuals reproduce sexually, which is represented by a non-linear integral operator. This operator is combined to a multiplicative operator representing selection. When the strength of selection is small, we show that the dynamics of the population is governed by a simple macroscopic differential equation, and that solutions converge exponentially to steady-states that are locally unique. The analysis is based on Wasserstein distance inequalities using a uniform lower bound on distributions. These inequalities are coupled to tail estimates to show the stability of the steady-states.

math.AP↗

arXiv2020

Interactions between different predator-prey states. A method for the derivation of the functional and numerical response

In this paper we introduce a formal method for the derivation of a predator's functional response from a system of fast state transitions of the prey or predator on a time scale during which the total prey and predator densities remain constant. Such derivation permits an explicit interpretation of the structure and parameters of the functional response in terms of individual behaviour. The same method is also used here to derive the corresponding numerical response of the predator as well as of the prey.

q-bio.PE↗

arXiv2015

Existence of self-accelerating fronts for a non-local reaction-diffusion equations

We describe the accelerated propagation wave arising from a non-local reaction-diffusion equation. This equation originates from an ecological problem, where accelerated biological invasions have been documented. The analysis is based on the comparison of this model with a related local equation, and on the analysis of the dynamics of the solutions of this second model thanks to probabilistic methods.

math.AP↗

arXiv2015

Asymptotic profile in selection-mutation equations: Gauss versus Cauchy distributions

In this paper, we study the asymptotic (large time) behavior of a selection-mutation-competition model for a population structured with respect to a phenotypic trait, when the rate of mutation is very small. We assume that the reproduction is asexual, and that the mutations can be described by a linear integral operator. We are interested in the interplay between the time variable $t$ and the rate $\varepsilon$ of mutations. We show that depending on $α> 0$, the limit $\varepsilon \to 0$ with $t = \varepsilon^{-α}$ can lead to population number densities which are either Gaussian-like (when $α$ is small) or Cauchy-like (when $α$ is large).

math.AP↗

arXiv2015

The effect of climate shift on a species submitted to dispersion, evolution, growth and nonlocal competition

We consider a population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition. We introduce theclimate shift due to {\it Global Warming} and discuss the dynamicsof the population by studying the long time behavior of thesolution of the Cauchy problem. We consider three sets ofassumptions on the growth function. In the so-called {\it confinedcase} we determine a critical climate change speed for theextinction or survival of the population, the latter case taking place by "strictly following the climate shift". In the so-called {\itenvironmental gradient case}, or {\it unconfined case}, we additionally determine the propagation speedof the population when it survives: thanks to a combination of migration and evolution, it can here be different from the speed of the climate shift. Finally, we consider {\it mixed scenarios}, that are complex situations, where thegrowth function satisfies the conditions of the confined case on the right, and the conditions of the unconfined case on the left.The main difficulty comes from the nonlocal competition term that prevents the use of classical methods based on comparison arguments. This difficulty is overcome thanks to estimates on the tails of the solution, and a careful application of the parabolic Harnack inequality.

math.AP↗

arXiv2015

Dynamics of a kinetic model describing protein exchanges in a cell population

We consider a cell population structured by a positive real number, describing the number of P-glycoproteins carried by the cell. We are interested in the effect of those proteins on the growth of the population: those proteins are indeed involve in the resistance of cancer cells to chemotherapy drugs. To describe this dynamics, we introduce a kinetic model. We then introduce a rigorous hydrodynamic limit, showing that if the exchanges are frequent, then the dynamics of the model can be described by a system of two coupled differential equations. Finally, we also show that the kinetic model converges to a unique limit in large times. The main idea of this analysis is to use Wasserstein distance estimates to describe the effect of the kinetic operator, combined to more classical estimates on the macroscopic quantities.

math.AP↗

arXiv2014

Existence and qualitative properties of travelling waves for an epidemiological model with mutations

In this article, we are interested in a non-monotone system of logistic reaction-diffusion equations. This system of equations models an epidemics where two types of pathogens are competing, and a mutation can change one type into the other with a certain rate. We show the existence of minimal speed travelling waves, that are usually non monotonic. We then provide a description of the shape of those constructed travelling waves, and relate them to some Fisher-KPP fronts with non-minimal speed.

math.AP↗

arXiv2014

Local stability of perfect alignment for a spatially homogeneous kinetic model

We prove the nonlinear local stability of Dirac masses for a kinetic model of alignment of particles on the unit sphere, each point of the unit sphere representing a direction. A population concentrated in a Dirac mass then corresponds to the global alignment of all individuals. The main difficulty of this model is the lack of conserved quantities and the absence of an energy that would decrease for any initial condition. We overcome this difficulty thanks to a functional which is decreasing in time in a neighborhood of any Dirac mass (in the sense of the Wasserstein distance). The results are then extended to the case where the unit sphere is replaced by a general Riemannian manifold.

math.AP↗

arXiv2014

Confinement by biased velocity jumps: aggregation of Escherichia coli

We investigate a linear kinetic equation derived from a velocity jump process modelling bacterial chemotaxis in the presence of an external chemical signal centered at the origin. We prove the existence of a positive equilibrium distribution with an exponential decay at infinity. We deduce a hypocoercivity result, namely: the solution of the Cauchy problem converges exponentially fast towards the stationary state. The strategy follows [J. Dolbeault, C. Mouhot, and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Trans. AMS 2014]. The novelty here is that the equilibrium does not belong to the null spaces of the collision operator and of the transport operator. From a modelling viewpoint it is related to the observation that exponential confinement is generated by a spatially inhomogeneous bias in the velocity jump process.

math.AP↗

arXiv2012

Travelling waves in a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypical trait. To sustain the possibility of invasion in the case where an underlying principal eigenvalue is negative, we investigate the existence of travelling wave solutions. We identify a minimal speed $c^*>0$, and prove the existence of waves when $c\geq c^*$ and the non existence when $0\leq c

math.AP↗