Interplay of nonlinear diffusion, initial tails and Allee effect on the speed of invasions

Abstract

We focus on the spreading properties of solutions of monostable equations with nonlinear 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 (nonlinear 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.

When fast diffusion and reactive growth both induce accelerating invasions

“…In particular, and roughly speaking, we show that the leading term of the position of the level sets is of the monomial type t β−m 2(β−1) , which is independent on α, thus on the tail of the initial data. This is in contrast with the other regimes fully described in [3].…”

“…See also [8], [5], [9], [2]. The nonlinear diffusion cases (m > 1 or 0 < m < 1) were recently solved in [3], revealing similar results, except in the fast diffusion range 0 < m < 1 − 2 α which yields a slightly stronger acceleration. • In presence of an Allee effect (β > 1): the linear diffusion (m = 1) case was studied in [1].…”

“…In [3], we proved that, in this regime, propagation occurs by accelerating but a precise estimate of the position of the level sets of u(t, ·) as t → +∞ was missing. In the present paper, we fill this gap by constructing very refined sub and supersolutions, which rely on self-similar solutions of…”

“…In this paper, companion of [3], we are concerned with the spreading properties of u(t, x) the solution of the nonlinear monostable reaction-diffusion equation…”

“…We refer to the introduction in [3] for references, comments and relevance in population dynamics models on the three main effects inserted in the Cauchy problem (1): nonlinear diffusion [15], [17], [4], Allee effect [6] (vs KPP nonlinearities [7], [16]), tail of the initial data. Let us briefly recall the available results on the propagation of solutions to (1), with a front-like initial data (whose behaviour at +∞ is of crucial importance on the speed of invasion).…”