Existence of two thresholds in a bistable equation with nonlocal competition

TL;DR

The paper analyzes a nonlocal bistable reaction-diffusion model for phenotypically structured populations with mutation, trait-dependent fitness, and nonlocal competition, augmented by a pseudo-Allee term. It proves well-posedness and then reveals rich long-time dynamics, including extinction for both very small and very large initial data and persistence for intermediate data, implying the existence of at least two thresholds. Through analytical arguments and numerical experiments across different fitness landscapes (bounded below and quadratic), it shows that nonlocal competition fundamentally alters threshold phenomena compared to local bistable equations. The work highlights how stationary states and the Allee-like term interact to create complex invasion/persistence scenarios with potential implications for adaptive invasion and spillover dynamics in ecology and epidemiology.

Abstract

We consider a nonlocal bistable reaction-diffusion equation, which serves as a model for a population structured by a phenotypic trait, subject to mutation, trait-dependent fitness, and nonlocal competition. Within this replicator-mutator framework, we further incorporate a ''pseudo-Allee effect'' so that the long time behavior (extinction vs. survival) depends on the size of the initial data. After proving the well-posedness of the associated Cauchy problem, we investigate its long-time behavior. We first show that small initial data lead to extinction. More surprisingly, we then prove that that extinction may also occur for too large initial data, in particular when selection is not strong enough. Finally, we exhibit situations where intermediate initial data lead to persistence, thereby revealing the existence of (at least) two thresholds. These results stand in sharp contrast with the behavior observed in local bistable equations.

Figures (3)

• Figure 1: Schematic representation of a bump function $f$ inducing an Allee effect.
• Figure 2: Extinction--persistence diagrams obtained from the final value $u(T,\theta=0)$ as a function of the initial width $L$ and height $H$ of the initial condition $u_0^{L,H}$. Panels: (a) constant selection; (b) bounded selection profile; (c) weak quadratic decrease; (d) strong quadratic decrease. In all cases, $f(u) = 15u\!\left(1-\tfrac{u}{2\varepsilon}\right)^{2}\mathbf{1}_{\{u<2\varepsilon\}}$ with $\varepsilon=0.1$ and final time $T=5$.
• Figure 3: Final-time solution profiles $u(T,\theta)$ for four different selection functions $r$, at $T=5$. Numerically, all the initial conditions that lead to persistence converge to these profiles. A logarithmic scale in $\theta$ is adopted here to highlight the differences between the profiles obtained for the different choices of $r(\theta)$.

Theorems & Definitions (42)

• Theorem 2.1: Well-posedness
• Proposition 2.1: Extinction for small data
• Theorem 2.2: Extinction for large data
• Remark 2.1
• Theorem 2.3: Survival may occur
• Theorem 2.4: Survival for large selection
• Theorem 2.5: Existence of two stationary states
• Lemma 4.1
• proof
• Proposition 4.1