[Paper Review] Invasion fronts with variable motility: phenotype selection, spatial sorting and wave acceleration

Émeric Bouin, Vincent Calvez+5 more|arXiv (Cornell University)|Jul 10, 2012

Evolution and Genetic Dynamics13 references4 citations

TL;DR

This paper analyzes a reaction-diffusion model with variable motility to explain wave acceleration and phenotype selection in biological invasion fronts. It derives a Burgers-type PDE with a source term for the evolution of the fittest trait at the front edge, showing that unbounded motility leads to $t^{3/2}$ front scaling and spatial sorting favors higher motility, even without reproductive advantage.

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.

Motivation & Objective

• To understand how variable motility influences invasion front dynamics in ecological systems.
• To explain the observed acceleration of invasion fronts when motility is unbounded.
• To derive a canonical equation for the evolution of the fittest trait at the front under rare mutation conditions.
• To establish the role of spatial sorting in selecting more motile phenotypes, even without fitness trade-offs.
• To connect the WKB asymptotic analysis with adaptive dynamics and front propagation in reaction-diffusion systems.

Proposed method

• Uses a reaction-diffusion equation with a diffusion term in phenotype space ($\alpha \partial_\theta^2 n$) to model mutations affecting motility ($\theta$).
• Applies the WKB asymptotic method to derive the leading-order front dynamics via the Hamilton-Jacobi equation and characteristic curves.
• Derives a dispersion relation linking wave speed to spatial decay, enabling analysis of traveling wave solutions.
• Constructs a formal canonical equation for the locally selected trait $\overline{\theta}(t,x)$ using first-order optimality and differentiation of the action function $u^0$.
• Identifies the front edge as the locus where $u^0(t,x,\theta) = 0$, and computes its position via the solution to a cubic equation in $Z$.
• Derives a PDE for $\overline{\theta}(t,x)$ that takes the form of a viscous Burgers equation with a source term, capturing evolutionary drift toward higher motility.

Experimental results

Research questions

• RQ1 What is the scaling law for the invasion front position when motility is unbounded?
• RQ2 How does spatial sorting select for higher motility in the absence of reproductive trade-offs?
• RQ3 What is the canonical equation governing the evolution of the fittest trait at the front under rare mutations?
• RQ4 How does the front's wave speed relate to the dispersion relation in the traveling wave regime?
• RQ5 Can the WKB approximation and Hamilton-Jacobi framework be used to derive a PDE-based canonical equation in this context?

Key findings

• When motility is unbounded, the front position scales as $X_{\text{edge}}(t) = \frac{4}{3}(\alpha^{1/4}r^{3/4})t^{3/2}$, confirming $t^{3/2}$ acceleration.
• For bounded motility ($\theta \in (0,\Theta)$), a minimal wave speed $c^*$ exists, and the most motile phenotypes are selected due to spatial sorting.
• The canonical equation for the locally selected trait $\overline{\theta}(t,x)$ is a Burgers-type PDE with a source term: $\partial_t \overline{\theta} - 2\overline{\theta} \partial_x u^0 \partial_x \overline{\theta} = \frac{|\partial_x u^0|^2}{-\partial_{\theta\theta}^2 u^0}$.
• The transport speed in the canonical equation matches the local minimal wave speed, linking front dynamics to trait evolution.
• The source term in the canonical equation drives evolutionary drift toward higher motility, explaining the observed increase in dispersal ability in invasive populations.
• The front edge is determined by the nullset of the action function $u^0(t,x,\theta) = 0$, which is computed via a cubic equation in $Z$.