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

Abstract

Invasion fronts in ecology are well studied but very few mathematical results concern the case with variable motility (possibly due to mutations). Based on an apparently simple reaction-diffusion equation, we explain the observed phenomena of front acceleration (when the motility is unbounded) as well as other quantitative results, such as the selection of the most motile individuals (when the motility is bounded). The key argument for the construction and analysis of traveling fronts is the derivation of the dispersion relation linking the speed of the wave and the spatial decay. When the motility is unbounded we show that the position of the front scales as t 3/2 . When the mutation rate is low we show that the canonical equation for the dynamics of the fittest trait should be stated as a PDE in our context. It turns out to be a type of Burgers equation with source term. RésuméFronts d'invasion avec motilité variable : répartition des phénotypes et accélération de l'onde. Les fronts d'invasion enécologie ontété largementétudiés. Cependant peu de résultats mathématiques existent pour le cas d'un coefficient de motilité variable (à cause des mutations). A partir d'un modèle minimal de réactiondiffusion, nous expliquons le phénomène observé d'accélération du front (lorsque la motilité n'est pas bornée), et nous démontrons l'existence d'ondes progressives ainsi que la sélection des individus les plus motiles (lorsque la motilité est bornée). Le point clé pour la construction des fronts est la relation de dispersion qui relie la vitesse de l'onde avec la décroissance en espace. Lorsque la motilité n'est pas bornée nous montrons que la position du front suit une loi d'échelle en t 3/2 . Lorsque le taux de mutation est faible, nous montrons que, dans notre contexte, l'équation canonique pour la dynamique du meilleur trait est une EDP. C'est uneéquation de type Burgers avec terme source.

Super-linear spreading in local and non-local cane toads equations

“…In Section 2 we recall some facts from [8] on the Hamilton-Jacobi framework. Then we derive in Section 3 the upper bound common to Theorems 1.1 and 1.2 using an explicit super-solution that arises from the work in Section 2.…”

“…The cane toads equation has similarly attracted recent interest, mostly when the motility set Θ is a finite interval. An Hamilton-Jacobi framework has been formally applied to the non-local model in [8], and rigorously justified in [38]. In these works, the authors obtain the speed of propagation and the expected repartition of traits at the edge of the front by solving a spectral problem in the trait variable.…”

“…As far as unbounded traits are concerned, a formal argument in [8] using a Hamilton-Jacobi framework predicted front acceleration and spreading rate of O(t 3/2 ). In this paper, we give a rigorous proof of this spreading rate both in the local and non-local models.…”

Evolutionary dynamics of phenotype-structured populations: from individual-level mechanisms to population-level consequences

“…The analysis of nonlinear reaction-diffusion systems by maximal exploitation of relations to the ordinary diffusion equation has parallels in recent work of Alfaro and Carles [42] on non-local reaction-diffusion equations, while the notion of interacting, evolving phenotype distributions (in a genetic, rather than epigentic context, and with a somewhat different mathematical formulation) was pursued by May and Nowak two decades ago [43][44][45]. The inclusion of variation across physical space as well as across phenotype space or the incorporation of non-local couplings with more structure than in our simple coupling via the total population could produce a richer class of behaviours [46][47][48][49][50][51][52][53][54][55].…”

Spatial sorting as the spatial analogue of natural selection

“…Although these models may be more limited in their range of possible behaviors, they have the potential to generate cleaner notions of how certain biological factors modulate the dynamics of spatial sorting. In particular, an important generalisation of Fisher's (Fisher 1937) reaction-diffusion model has encapsulated the process of spatial sorting very simply into a description of how that model's "diffusion coefficient" evolves as a function of spatial processes (Benichou et al 2012;Bouin et al 2012;Bouin and Calvez 2014). This new class of models is currently the focus of intense and productive theoretical work, much of which necessarily involves rather advanced mathematics and is beyond the grasp of many biologists.…”