Derivation and quasi-invariant asymptotics of phenotype-structured integro-differential models
Emanuele Bernardi, Tommaso Lorenzi, Andrea Tosin
TL;DR
The paper develops a mass-varying kinetic framework for phenotype-structured populations, deriving an integro-differential equation (IDE) from a stochastic agent-based model and then a non-local Fokker–Planck–type PDE in the quasi-invariant limit of small, frequent phenotype changes. It provides rigorous a priori estimates, convergence results, and moment-matching properties, thereby linking micro-level dynamics to macro-level evolutionary behavior across three model levels. The authors demonstrate, both analytically and numerically, that the IDE accurately captures population dynamics across regimes and that the PDE emerges as the asymptotic limit as mutation steps become infinitesimal. This unitary approach enables finer representation of phenotype-change mechanisms via mutation kernels and lays groundwork for future spatial extensions and broader limiting analyses.
Abstract
Building upon kinetic theory approaches for multi-agent systems and generalising them to scenarios where the total mass of the system is not conserved, we develop a modelling framework for phenotype-structured populations that makes it possible to bridge individual-level mechanisms with population-scale evolutionary dynamics. We start by formulating a stochastic agent-based model, which describes the dynamics of single population members undergoing proliferation, death, and phenotype changes. Then, we formally derive the corresponding mesoscopic model, which consists of an integro-differential equation for the distribution of population members over the space of phenotypes, where phenotype changes are modelled via an integral kernel. Finally, considering a quasi-invariant regime of small but frequent phenotype changes, we rigorously derive a non-local Fokker-Planck-type equation counterpart of this model, wherein phenotype changes are taken into account by an advection-diffusion term. The theoretical results obtained are illustrated through a sample of results of numerical simulations.