[Paper Review] Rare mutations limit of a steady state dispersion trait model

Benoı̂t Perthame, Panagiotis E. Souganidis|arXiv (Cornell University)|May 13, 2015

Evolution and Genetic Dynamics29 references3 citations

TL;DR

This paper studies the evolution of dispersal traits in a population structured by space and a genetic trait influencing dispersal rate, using a rare mutation limit approach. As mutation rates vanish, the population concentrates on a single trait with the lowest dispersal rate, governed by a constrained Hamilton-Jacobi equation derived via WKB and viscosity solution methods, with the evolutionary speed determined by an effective fitness function.

ABSTRACT

The evolution of dispersal is a classical question in evolutionary ecology, which has been widely studied with several mathematical models. The main question is to define the fittest dispersal rate for a population in a bounded domain, and, more recently, for traveling waves in the full space. In the present study, we reformulate the problem in the context of adaptive evolution. We consider a population structured by space and a genetic trait acting directly on the dispersal (diffusion) rate under the effect of rare mutations on the genetic trait. We show that, as in simpler models, in the limit of vanishing mutations, the population concentrates on a single trait associated to the lowest dispersal rate. We also explain how to compute the evolution speed towards this evolutionary stable distribution. The mathematical interest stems from the asymptotic analysis which requires a completely different treatment of the different variables. For the space variable, the ellipticity leads to the use the maximum principle and Sobolev-type regularity results. For the trait variable, the concentration to a Dirac mass requires a different treatment. This is based on the WKB method and viscosity solutions leading to an effective Hamiltonian (effective fitness of the population) and a constrained Hamilton-Jacobi equation.

Motivation & Objective

• To understand how dispersal traits evolve in a spatially heterogeneous environment under rare mutations.
• To determine the evolutionary stable distribution in a population structured by space and a continuous dispersal trait.
• To derive the long-term evolutionary dynamics of the fittest trait in the limit of vanishing mutations.
• To establish a mathematical framework linking the steady-state dispersal model to evolutionary dynamics through effective fitness and Hamilton-Jacobi equations.

Proposed method

• Models a population with spatial and trait structure, where dispersal rate depends on a genetic trait θ, using a nonlocal, nonlinear PDE with diffusion in both space and trait variables.
• Applies the rare mutation limit ε→0, treating space and trait variables differently: maximum principle and Sobolev regularity for space, WKB and viscosity solutions for trait concentration.
• Uses a constrained Hamilton-Jacobi equation to describe the evolution of the fittest trait, with the effective fitness derived from a principal eigenvalue problem.
• Introduces a time-dependent parabolic version of the model to study dynamic evolution, leading to a canonical equation for the trait's time evolution.
• Derives the evolutionary speed of the fittest trait using the Hessian and gradient of the effective fitness at the optimal trait.
• Employs periodic boundary conditions in θ and Neumann conditions in x to simplify analysis and ensure a priori estimates.

Experimental results

Research questions

• RQ1 What trait does a population evolve toward under rare mutations when dispersal rate is a heritable trait?
• RQ2 How does the effective fitness of a population depend on the dispersal trait in a spatially heterogeneous environment?
• RQ3 What is the mathematical structure of the evolutionary dynamics in the rare mutation limit for a spatially structured population?
• RQ4 How can the speed of evolutionary change in the dispersal trait be quantified in the limit of vanishing mutations?
• RQ5 What role does the principal eigenvalue of a nonlocal elliptic operator play in determining evolutionary stability?

Key findings

• In the rare mutation limit ε→0, the population density concentrates on a single trait, corresponding to the lowest dispersal rate, forming a Dirac mass in the trait variable.
• The optimal trait θ̄ is determined by maximizing the effective fitness, which is defined as the principal eigenvalue of a nonlocal elliptic operator depending on the population distribution.
• The evolutionary dynamics of the fittest trait is governed by a constrained Hamilton-Jacobi equation involving the effective fitness and the gradient of the fitness function.
• The evolutionary speed of the trait is given by a canonical equation that depends on the Hessian of the value function and the gradient of the effective fitness at the optimal trait.
• Numerical simulations confirm that the fittest trait evolves toward lower dispersal rates over time, with the population distribution concentrating around θ=0 in the example case.
• The effective fitness function H(θ, N) is defined through the principal eigenvalue of the operator −D(θ)Δx + K(x) − N, with N being the total population density.