[Paper Review] Population invasion with bistable dynamics and adaptive evolution: the evolutionary rescue

Matthieu Alfaro, Arnaud Ducrot|arXiv (Cornell University)|Jan 22, 2018

Mathematical and Theoretical Epidemiology and Ecology Models12 references4 citations

TL;DR

This paper mathematically proves that adaptive evolution can rescue a population from extinction in a reaction-diffusion model with bistable dynamics and an Allee effect. By coupling ecological spread with evolutionary adaptation through a trait-dependent Allee threshold, the authors show that even a small, initially doomed population can persist and spread globally due to evolutionary rescue driven by genetic variance and selection.

ABSTRACT

We consider the system of reaction-diffusion equations proposed in [8] as a population dynamics model. The first equation stands for the population density and models the ecological effects, namely dispersion and growth with a Allee effect (bistable nonlinearity). The second one stands for the Allee threshold, seen as a trait mean, and accounts for evolutionary effects. Precisely, the Allee threshold is submitted to three main effects: dispersion (mirroring ecology), asymmetrical gene flow and selection. The strength of the latter depends on the population density and is thus coupling ecology and evolution. Our main result is to mathematically prove evolutionary rescue: any small initial population, that would become extinct in the sole ecological context, will persist and spread thanks to evolutionary factors.

Motivation & Objective

• To rigorously establish the phenomenon of evolutionary rescue in a population model with bistable dynamics and adaptive evolution.
• To analyze how genetic variance and selection pressure enable population persistence when ecological dynamics alone would lead to extinction.
• To demonstrate that a small, localized population with low initial density can invade and spread globally due to evolutionary adaptation of the Allee threshold.
• To extend the classical bistable reaction-diffusion framework by incorporating a dynamic, evolving trait (Allee threshold) that couples ecology and evolution.

Proposed method

• Formulates a coupled system of reaction-diffusion equations: one for population density $u(t,x)$ with a bistable nonlinearity involving $a^2$, and one for the evolving Allee threshold $a(t,x)$.
• Models the Allee threshold $a$ as a trait subject to diffusion, asymmetric gene flow (via $\partial_x \ln u$), and selection proportional to $\varepsilon(1-u)a$, linking ecological density to evolutionary change.
• Applies the energy method and comparison principles to analyze long-time behavior and establish uniform bounds on $u$ and $a$.
• Uses parabolic regularity and compactness arguments to extract a limit solution from rescaled sequences, proving convergence to the uniform state $u \equiv 1$.
• Employs a contradiction argument to show that $\limsup_{t\to\infty} \sup_x u(t,x) = 1$, implying global persistence.
• Applies the parabolic comparison principle to compare the solution with a reference problem having a fixed, lower Allee threshold, establishing convergence to 1 locally uniformly in space.

Experimental results

Research questions

• RQ1 Can adaptive evolution rescue a population from extinction when ecological dynamics alone lead to extinction due to a strong Allee effect?
• RQ2 How does the coupling of ecological spread and evolutionary adaptation through a dynamic Allee threshold affect long-term population persistence?
• RQ3 What is the role of genetic variance $\varepsilon > 0$ in enabling invasion when $\varepsilon = 0$ leads to population extinction?
• RQ4 Under what conditions does the population density $u(t,x)$ converge to 1 globally in space, despite a small initial population and a high Allee threshold?
• RQ5 Can the evolutionary dynamics of the Allee threshold $a(t,x)$ prevent the population from being trapped in a low-density, extinction-prone state?

Key findings

• Any small, localized initial population with $u_0 \leq a_0^2$ and $u_0 \not\equiv 0$ will persist and spread globally when $\varepsilon > 0$, despite extinction under $\varepsilon = 0$.
• The solution $u(t,x)$ satisfies $\limsup_{t\to\infty} \sup_x u(t,x) = 1$, indicating that the population reaches near-maximal density everywhere in space over time.
• The population density $u(t,x)$ converges to 1 locally uniformly in $x$ as $t \to \infty$, even when the initial population is below the Allee threshold.
• The Allee threshold $a(t,x)$ decays over time due to selection, reducing the threshold for population growth and enabling persistence.
• The evolutionary rescue mechanism is driven by the coupling of ecological dynamics (diffusion and growth) with evolutionary dynamics (gene flow and selection), where $\varepsilon > 0$ is essential for rescue.
• The proof relies on comparison principles and the construction of a limiting entire solution that must be identically 1, forcing the population to reach full density.