Emeric Bouin

Travelling waves for the cane toads equation with bounded traits.

In this paper, we study propagation in a non-local reaction–diffusion–mutation model describing the invasion of cane toads in Australia (Phillips et al 2006 Nature 439 803). The population of toads is structured by a space variable and a phenotypical trait and the space diffusivity depends on the trait. We use a Schauder topological degree argument for the construction of some travelling wave solutions of the model. The speed c* of the wave is obtained after solving a suitable spectral problem in the trait variable. An eigenvector arising from this eigenvalue problem gives the flavour of the profile at the edge of the front. The major difficulty is to obtain uniform L∞ bounds despite the combination of non-local terms and a heterogeneous diffusivity.

Population Formulation of Adaptative Meso-evolution: Theory and Numerics

The population formalism of “adaptive evolution” has been developed in the last twenty years along ideas presented in other chapters in this volume. This mathematical formalism addresses the question of explaining how selection of a favorable phenotypical trait in a population occurs. In the language of Metz’s Chapter, it refers to meso-evolution. It uses models based, usually, on integro-differential equations for the population structured by a phenotypical trait. A self-contained mathematical formulation of adaptive evolution also contains the description of mutations and leads to partial differential equations. Then the complete evolution picture follows from the model ingredients mostly driven by the changing adaptive landscape.

Hyperbolic traveling waves driven by growth

We perform the analysis of a hyperbolic model which is the analog of the Fisher-KPP equation. This model accounts for particles that move at maximal speed

Thin front limit of an integro–differential Fisher–KPP equation with fat–tailed kernels

\epsilon^{-1}

Super-linear spreading in local bistable cane toads equations

(

Exponential Decay to Equilibrium for a Fiber Lay-Down Process on a Moving Conveyor Belt

\epsilon>0

Invasion fronts with variable motility: Phenotype selection, spatial sorting and wave acceleration

), and proliferate according to a reaction term of monostable type. We study the existence and stability of traveling fronts. We exhibit a transition depending on the parameter

Propagation in a Kinetic Reaction-Transport Equation: Travelling Waves And Accelerating Fronts

\epsilon

A Hamilton-Jacobi approach for a model of population structured by space and trait

: for small

A kinetic eikonal equation

\epsilon