Large population limit of interacting population dynamics via generalized gradient structures

TL;DR

This paper derives a nonlocal Fisher-KPP population model from microscopic stochastic processes using generalized gradient structures and Gamma-convergence, providing a variational framework and chaos propagation results.

Contribution

It introduces a novel derivation of the nonlocal Fisher-KPP model via a generalized gradient flow approach from microscopic stochastic dynamics.

Findings

Derivation of the nonlocal Fisher-KPP model from microscopic stochastic processes.

Establishment of a generalized gradient flow structure for the limit equation.

Proof of entropic propagation of chaos in the large population limit.

Abstract

This chapter focuses on the derivation of a doubly nonlocal Fisher-KPP model, which is a macroscopic nonlocal evolution equation describing population dynamics in the large population limit. The derivation starts from a microscopic individual-based model described as a stochastic process on the space of atomic measures with jump rates that satisfy detailed balance w.r.t. to a reference measure. We make use of the so-called `cosh' generalized gradient structure for the law of the process to pass to the large population limit using evolutionary Gamma-convergence. In addition to characterizing the large population limit as the solution of the nonlocal Fisher-KPP model, our variational approach further provides a generalized gradient flow structure for the limit equation as well as an entropic propagation of chaos result.