When the Allee threshold is an evolutionary trait: Persistence vs. extinction

Abstract

We consider a nonlocal parabolic equation describing the dynamics of a population structured by a spatial position and a phenotypic trait, submitted to dispersion, mutations and growth. The growth term may be of the Fisher-KPP type but may also be subject to an Allee effect which can be weak (non-KPP monostable nonlinearity, possibly degenerate) or strong (bistable nonlinearity). The type of growth depends on the value of a variable θ : the Allee threshold, which is considered here as an evolutionary trait. After proving the well-posedness of the Cauchy problem, we study the long time behavior of the solutions. Due to the richness of the model and the interplay between the various phenomena and the nonlocality of the growth term, the outcomes (extinction vs. persistence) are various and in sharp contrast with earlier results of the existing literature on local reaction-diffusion equations.

When expansion stalls: an extension to the concept of range pinning in ecology

“…Variations in the carrying capacity of the environment – which impacts pinning stability – can be mitigated by the rather short timespan of biological invasions. Evolution however, can occur during the course of the invasion, notably with regard to Allee effects (Erm and Phillips 2020, Alfaro et al 2021) or to the relationship between density and dispersal (Kun and Scheuring 2006, Travis et al 2009, Dahirel et al 2021). Therefore, pinning is likely to be observed in real range expansions, but the time scales on which it is stable may be limited by variations in the conditions required for its emergence.…”

Dynamics of traveling waves for predator-prey systems with Allee effect and time delay

“…The Allee effect was originally proposed by Warder Clyde Allee [2] to characterize the correlation between the population density and per capita growth rate of the population at low densities [27]. In recent decades, it has received considerable attention and numerous related studies have been conducted [1,19,23,29,31,34]. To investigate the impact of the reproductive Allee effect in prey growth, Dey et al [7] considered the predator-prey system with Beddington-DeAngelis functional response and reproductive Allee effect,…”

Spatial heterogeneity alters the trade-off between growth and dispersal during a range expansion

“…The objective of these works was mainly to explain the acceleration of the range expansion of can toads since their introduction in Australia, and the corresponding model is often referred to as the “cane toad equation”. Recently, this framework has been applied to other traits such as the Allee threshold (Alfaro et al, 2021). In all of these reaction-diffusion based models, the additional phenotypic variable only affects a single biological parameter, either directly when this variable is the trait itself such as the diffusion term D or the Allee threshold, or indirectly when the growth rate R ( x, y ) depends on an abstract trait y .…”