arXiv subjects

Matthieu Alfaro

Publications and source records attributed to Matthieu Alfaro.

At least 37 records · Page 2Linked to original sources

arXiv2021

Populations facing a nonlinear environmental gradient: steady states and pulsating fronts

We consider a population structured by a spacevariable and a phenotypical trait, submitted to dispersion,mutations, growth and nonlocal competition. This population is facing an {\it environmental gradient}: to survive at location $x$, an individual must have a trait close to some optimal trait $y_{opt}(x)$. Our main focus is to understand the effect of a {\it nonlinear} environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with $y_{opt}(x) = \varepsilonθ(x)$, $0<\vert \varepsilon \vert \ll 1$. We construct steady states solutions and, when $θ$ is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable $y$ we take advantage of a Hilbert basis of $L^{2}(\mathbb{R})$ made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable $x$ we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights.

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arXiv2020

Blow-up phenomena for positive solutions of semilinear diffusion equations in a half-space: the influence of the dispersion kernel

We consider the semilinear diffusion equation $\partial$ t u = Au + |u| $α$ u in the half-space R N + := R N --1 x (0, +$\infty$), where A is a linear diffusion operator, which may be the classical Laplace operator, or a fractional Laplace operator, or an appropriate non regularizing nonlocal operator. The equation is supplemented with an initial data u(0, x) = u 0 (x) which is nonnegative in the half-space R N + , and the Dirichlet boundary condition u(t, x ' , 0) = 0 for x ' $\in$ R N --1. We prove that if the symbol of the operator A is of order a|$ξ$| $β$ near the origin $ξ$ = 0, for some $β$ $\in$ (0, 2], then any positive solution of the semilinear diffusion equation blows up in finite time whenever 0 < $α$ $\le$ $β$/(N + 1). On the other hand, we prove existence of positive global solutions of the semilinear diffusion equation in a half-space when $α$ > $β$/(N + 1). Notice that in the case of the half-space, the exponent $β$/(N + 1) is smaller than the so-called Fujita exponent $β$/N in R N. As a consequence we can also solve the blow-up issue for solutions of the above mentioned semilinear diffusion equation in the whole of R N , which are odd in the x N direction (and thus sign changing).

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arXiv2019

Quantitative estimates of the threshold phenomena for propagation in reaction-diffusion equations

We focus on the (sharp) threshold phenomena arising in some reaction-diffusion equations supplemented with some compactly supported initial data. In the so-called ignition and bistable cases, we prove the first sharp quantitative estimate on the (sharp) threshold values. Furthermore, numerical explorations allow to conjecture some refined estimates. Last we provide related results in the case of a degenerate monostable nonlinearity "not enjoying the hair trigger effect". AMS Subject Classifications: 35K57 (Reaction-diffusion equations), 35K15 (Initial value problems for second-order parabolic equations), 35B40 (Asymptotic behavior of solutions).

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arXiv2019

Evolution equations involving nonlinear truncated Laplacian operators

We first study the so-called Heat equation with two families of elliptic operators whichare fully nonlinear, and depend on some eigenvalues of the Hessian matrix. The equationwith operators including the "large" eigenvalues has strong similarities with a Heatequation in lower dimension whereas, surprisingly, for operators including "small" eigenvalues it shares some properties with some transport equations. In particular, forthese operators, the Heat equation (which is nonlinear) not only does not have theproperty that "disturbances propagate with infinite speed" but may lead to quenchingin finite time. Last, based on our analysis of the Heat equations (for which we providea large variety of special solutions) for these operators, we inquire on the associated Fujita blow-up phenomena.

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arXiv2019

Density dependent replicator-mutator models in directed evolution

We analyze a replicator-mutator model arising in the context of directed evolution [23], where the selection term is modulated over time by the mean-fitness. We combine a Cumulant Generating Function approach [13] and a spatio-temporal rescaling related to the Avron-Herbst formula [1] to give of a complete picture of the Cauchy problem. Besides its well-posedness, we provide an implicit/explicit expression of the solution, and analyze its large time behaviour. As a by product, we also solve a replicator-mutator model where the mutation coefficient is socially determined, in the sense that it is modulated by the mean-fitness. The latter model reveals concentration or anti diffusion/diffusion phenomena.

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arXiv2018

Generation of fine transition layers and their dynamics for the stochastic Allen--Cahn equation

We study an $\ep$-dependent stochastic Allen--Cahn equation with a mild random noise on a bounded domain in $\mathbb{R}^n$, $n\geq 2$. Here $\ep$ is a small positive parameter that represents formally the thickness of the solution interface, while the mild noise $ξ^\ep(t)$ is a smooth random function of $t$ of order $\mathcal O(\ep^{-γ})$ with $0<γ<1/3$ that converges to white noise as $\ep\rightarrow 0^+$. We consider initial data that are independent of $\ep$ satisfying some non-degeneracy conditions, and prove that steep transition layers---or interfaces---develop within a very short time of order $\ep^2|\ln\ep|$, which we call the "generation of interface". Next we study the motion of those transition layers and derive a stochastic motion law for the sharp interface limit as $\ep\rightarrow 0^+$. Furthermore, we prove that the thickness of the interface for $\ep$ small is indeed of order $\mathcal O(\ep)$ and that the solution profile near the interface remains close to that of a (squeezed) travelling wave, this means that the presence of the noise does not destroy the solution profile near the interface as long as the noise is spatially uniform. Our results on the motion of interface improve the earlier results of Funaki (1999) and Weber (2010) by considerably weakening the requirements for the initial data and establishing the robustness of the solution profile near the interface that has not been known before.

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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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arXiv2018

Evolutionary branching via replicator-mutator equations

We consider a class of non-local reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. For a confining fitness function, we prove well-posedness and write the solution explicitly, via some underlying Schrödinger spectral elements (for which we provide new and non-standard estimates). As a consequence, the long time behaviour is determined by the principal eigenfunction or ground state. Based on this, we discuss (rigorously and via numerical explorations) the conditions on the fitness function and the mutation rate for evolutionary branching to occur.

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arXiv2018

Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue

We consider the system of reaction-diffusion equations proposed in [8] as a population dynamics model. The first equation stands for the population density and models the ecological effects, namely dispersion and growth with a Allee effect (bistable nonlinearity). The second one stands for the Allee threshold, seen as a trait mean, and accounts for evolutionary effects. Precisely, the Allee threshold is submitted to three main effects: dispersion (mirroring ecology), asymmetrical gene flow and selection. The strength of the latter depends on the population density and is thus coupling ecology and evolution. Our main result is to mathematically prove evolutionary rescue: any small initial population, that would become extinct in the sole ecological context, will persist and spread thanks to evolutionary factors.

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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

Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data.

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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

Replicator-mutator equations with quadratic fitness

This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is non-positive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is non-negative (inverted harmonic potential), solutions always become extinct in finite time.

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arXiv2016

Propagation phenomena in monostable integro-differential equations: acceleration or not?

We consider the homogeneous integro-differential equation$\partial \_t u=J*u-u+f(u)$ with a monostable nonlinearity $f$. Our interest is twofold: we investigate the existence/non existence of travelling waves, and the propagation properties of the Cauchy problem.When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Weinberger1982}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009}. %When the dispersion kernel $J$ is exponentially bounded, travelling waves are known to exist when $f$ belongs to one of the three main class of non-linearities (bistable, ignition or monostable), and solutions of the Cauchy problem typically propagate at a constant speed \cite{Schumacher1980}, \cite{Wei-82},\cite{Bates1997},\cite{Chen1997}, \cite{Carr2004}, \cite{Coville2007a}, \cite{Coville2008a}, \cite{Yagisita2009,Yagisita2009a}. On the other hand, when the dispersion kernel $J$ has heavy tails and the non-linearity $f$ is non degenerate, i.e $f'(0)\textgreater{}0$, travelling waves do not exist and solutions of the Cauchy problem propagate by accelerating \cite{Medlock2003}, \cite{Yagisita2009}, \cite{Garnier2011}. For a general monostable non-linearity, a dichotomy between these two types of propagation behaviour is still not known. The originality of our work is to provide such dichotomy by studying the interplay between the tails of the dispersion kernel and the Allee effect induced by the degeneracy of $f$, i.e. $f'(0)=0$. First, for algebraic decaying kernels, we prove the exact separation between existence and non existence of travelling waves. This in turn provides the exact separation between non acceleration and acceleration in the Cauchy problem. In the latter case, we provide a first estimate of the position of the level sets of the solution.

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arXiv2016

Pulsating fronts for Fisher-KPP systems with mutations as models in evolutionary epidemiology

We consider a periodic reaction diffusion system which, because of competition between $u$ and $v$, does not enjoy the comparison principle. It also takes into account mutations, allowing $u$ to switch to $v$ and vice versa. Such a system serves as a model in evolutionary epidemiology where two types of pathogens compete in a heterogeneous environment while mutations can occur, thus allowing coexistence.We first discuss the existence of nontrivial positive steady states, using some bifurcation technics. Then, to sustain the possibility of invasion when nontrivial steady states exist, we construct pulsating fronts. As far as we know, this is the first such construction in a situation where comparison arguments are not available.

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arXiv2016

Fujita blow up phenomena and hair trigger effect: the role of dispersal tails

We consider the nonlocal diffusion equation $\partial \_t u=J*u-u+u^{1+p}$ in the whole of $\R ^N$. We prove that the Fujita exponent dramatically depends on the behavior of the Fourier transform of the kernel $J$ near the origin, which is linked to the tails of $J$. In particular, for compactly supported or exponentially bounded kernels, the Fujita exponent is the same as that of the nonlinear Heat equation $\partial \_tu=Δu+u^{1+p}$. On the other hand, for kernels with algebraic tails, the Fujita exponent is either of the Heat type or of some related Fractional type, depending on the finiteness of the second moment of $J$. As an application of the result in population dynamics models, we discuss the hair trigger effect for $\partial \_t u=J*u-u+u^{1+p}(1-u)$

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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.

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arXiv2015

Slowing Allee effect vs. accelerating heavy tails in monostable reaction diffusion equations

We focus on the spreading properties of solutions of monostable reaction-diffusion equations. Initial data are assumed to have heavy tails, which tends to accelerate the invasion phenomenon. On the other hand, the nonlinearity involves a weak Allee effect, which tends to slow down the process. We study the balance between the two effects. For algebraic tails, we prove the exact separation between "no acceleration and acceleration". This implies in particular that, for tails exponentially unbounded but lighter than algebraic , acceleration never occurs in presence of an Allee effect. This is in sharp contrast with the KPP situation [19]. When algebraic tails lead to acceleration despite the Allee effect, we also give an accurate estimate of the position of the level sets.

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