A game-theoretic mechanism for aggregation and dispersal of interacting populations

TL;DR

This paper introduces a PDE model using a fitness function from evolutionary game theory to explain how populations aggregate or disperse spatially, capturing behaviors like prey aggregation and spatial pattern formation.

Contribution

It adapts a fitness gradient-based dispersal mechanism into PDEs, providing new insights into spatial population structures and steady states, including weak solutions and instabilities.

Findings

Populations can reach spatially structured steady states with constant fitness.

Prey aggregation aligns with Hamilton's selfish herd hypothesis.

Fitness gradient flux can induce instabilities leading to pattern formation.

Abstract

We adapt a fitness function from evolutionary game theory as a mechanism for aggregation and dispersal in a partial differential equation (PDE) model of two interacting populations, described by density functions $u$ and $v$. We consider a spatial model where individuals migrate up local fitness gradients, seeking out locations where their given traits are more advantageous. The resulting system of fitness gradient equations is a degenerate system having spatially structured, smooth, steady state solutions characterized by constant fitness throughout the domain. When populations are viewed as predator and prey, our model captures prey aggregation behavior consistent with Hamilton's selfish herd hypothesis. We also present weak steady state solutions in 1d that are continuous but in general not smooth everywhere, with an associated fitness that is discontinuous, piecewise constant. We give numerical examples of solutions that evolve toward such weak steady states. We also give an example of a spatial Lotka--Volterra model, where a fitness gradient flux creates instabilities that lead to spatially structured steady states. Our results also suggest that when fitness has some dependence on local interactions, a fitness-based dispersal mechanism may act to create spatial variation across a habitat.