From Individual-Based Models to General Stochastic Reaction Diffusion Equations
Abstract
In this article, we construct, for a broad class of one-dimensional stochastic reaction-diffusion equations, a sequence of spatially-structured individual-based models whose rescaled empirical processes converge to the solution of the equation. The purpose of this article is to show that, whenever the rescaled one-step mean and variance of a reasonable individual-based model converge to those of an SDE with drift b(x)x(1−x)+θ+(1−x)−θ-x and noise coefficient √(σ(x)x(1−x)), under certain moment conditions there also exists a corresponding multi-island model, composed of independent and identically distributed copies of the individual-based model on each island, with selection and mutation scaled to the diffusive time scale, coupled through migration, which converges to an SPDE with a Laplacian plus the same drift and noise coefficients, provided this SPDE is unique in law.
Related
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BibTeX
@misc{casanova2026from,
title = {{From Individual-Based Models to General Stochastic Reaction Diffusion Equations}},
author = {Adrián González Casanova and Johnny and Yang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02669},
eprint = {2610.02669},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02669},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - From Individual-Based Models to General Stochastic Reaction Diffusion Equations
AU - Adrián González Casanova
AU - Johnny
AU - Yang
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02669
UR - https://arxiv.org/abs/2610.02669
AB - In this article, we construct, for a broad class of one-dimensional stochastic reaction-diffusion equations, a sequence of spatially-structured individual-based models whose rescaled empirical processes converge to the solution of the equation. The purpose of this article is to show that, whenever the rescaled one-step mean and variance of a reasonable individual-based model converge to those of an SDE with drift b(x)x(1−x)+θ^+(1−x)−θ^-x and noise coefficient √(σ(x)x(1−x)), under certain moment conditions there also exists a corresponding multi-island model, composed of independent and identically distributed copies of the individual-based model on each island, with selection and mutation scaled to the diffusive time scale, coupled through migration, which converges to an SPDE with a Laplacian plus the same drift and noise coefficients, provided this SPDE is unique in law.
ER -