Matthieu Alfaro

Travelling Waves in a Nonlocal Reaction-Diffusion Equation as a Model for a Population Structured by a Space Variable and a Phenotypic Trait

We consider a nonlocal reaction-diffusion equation as a model for a population structured by a space variable and a phenotypic 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 ≥ c* and the nonexistence when 0 ≤ c < c*.

Explicit Solutions for Replicator-Mutator Equations: Extinction versus Acceleration

We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models.

Convergence of a mass conserving Allen–Cahn equation whose Lagrange multiplier is nonlocal and local

We consider the mass conserving Allen-Cahn equation proposed in \cite{Bra-Bre}: the Lagrange multiplier which ensures the conservation of the mass contains not only nonlocal but also local effects (in contrast with \cite{Che-Hil-Log}). As a parameter related to the thickness of a diffuse internal layer tends to zero, we perform formal asymptotic expansions of the solutions. Then, equipped with these approximate solutions, we rigorously prove the convergence to the volume preserving mean curvature flow, under the assumption that classical solutions of the latter exist. This requires a precise analysis of the error between the actual and the approximate Lagrange multipliers.

Motion by anisotropic mean curvature as sharp interface limit of an inhomogeneous and anisotropic Allen-Cahn equation

We consider the spatially inhomogeneous and anisotropic reaction–diffusion equation ut = m(x) −1 div[ m(x)ap(x,∇u) ] + e −2 f(u) , involving a small parameter e > 0 and a bistable nonlinear term whose stable equilibria are 0 and 1. We use a Finsler metric related to the anisotropic diffusion term and work in relative geometry. We prove a weak comparison principle and perform an analysis of both the generation and the motion of interfaces. More precisely, we show that, within the time-scale of order e2|ln e|, the unique weak solution u e develops a steep transition layer that separates the regions {u e ≈ 0} and {u e | 1}. Then, on a much slower time-scale, the layer starts to propagate. Consequently, as e → 0, the solution u e converges almost everywhere (a.e.) to 0 in Ω − t and 1 in Ω + t , where Ω − t and Ω + t are sub-domains of Ω separated by an interface Гt, whose motion is driven by its anisotropic mean curvature. We also prove that the thickness of the transition layer is of order e.

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.

On a nonlocal system for vegetation in drylands

Several mathematical models are proposed to understand spatial patchy vegetation patterns arising in drylands. In this paper, we consider the system with nonlocal dispersal of plants (through a redistribution kernel for seeds) proposed by Pueyo et al. (Oikos 117:1522–1532, 2008) as a model for vegetation in water-limited ecosystems. It consists in two reaction diffusion equations for surface water and soil water, combined with an integro-differential equation for plants. For this system, under suitable assumptions, we prove well-posedness using the Schauder fixed point theorem. In addition, we consider the stationary problem from the viewpoint of vegetated pattern formation, and show a transition of vegetation patterns when parameter values (rainfall, seed dispersal range, seed germination rate) in the system vary. The influence of the shape of the redistribution kernel is also discussed.

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 parameter for evolutionary branching to occur.

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

We consider the heat equation with a logarithmic nonlinearity, on the real line. For a suitable sign in front of the nonlinearity, we establish the existence and uniqueness of solutions of the Cauchy problem, for a well-adapted class of initial data. Explicit computations in the case of Gaussian data lead to various scenarii which are richer than the mere comparison with the ODE mechanism, involving (like in the ODE case) double exponential growth or decay for large time. Finally, we prove that such phenomena remain, in the case of compactly supported initial data.

Interface dynamics of the porous medium equation with a bistable reaction term

We consider a degenerate partial differential equation arising in population dynamics, namely the porous medium equation with a bistable reaction term. We study its asymptotic behavior as a small parameter, related to the thickness of a diffuse interface, tends to zero. We prove the rapid formation of transition layers which then propagate. We prove the convergence to a sharp interface limit whose normal velocity, at each point, is that of the underlying degenerate travelling wave.