The evolutionary limit for models of populations interacting competitively with many resources

TL;DR

This paper analyzes a nonlinear integro-differential model of population evolution with trait structure, showing that under strong selection and small mutations, the population concentrates into Dirac masses governed by a Hamilton-Jacobi equation.

Contribution

It provides a rigorous characterization of the concentration phenomenon in population models with resource-dependent interactions, linking it to Hamilton-Jacobi equations.

Findings

Population density converges to Dirac masses in the limit

Resource concentrations are fully characterized by the Hamilton-Jacobi solution

The model extends understanding of trait evolution under resource competition

Abstract

We consider a integro-differential nonlinear model that describes the evolution of a population structured by a quantitative trait. The interactions between traits occur from competition for resources whose concentrations depend on the current state of the population. Following the formalism of\cite{DJMP}, we study a concentration phenomenon arising in the limit of strong selection and small mutations. We prove that the population density converges to a sum of Dirac masses characterized by the solution $\phi$ of a Hamilton-Jacobi equation which depends on resource concentrations that we fully characterize in terms of the function $\phi$.