# Mean field approximation of a heterogeneous population of plants in competition

## TL;DR

This paper develops a mean-field approximation for modeling large, heterogeneous plant populations in competition, simplifying complex interactions into a manageable mathematical framework with potential applications in ecological inference.

## Contribution

It introduces a mean-field approach to analyze large heterogeneous populations with pairwise interactions, enabling simplified modeling and inference in ecological systems.

## Key findings

- Mean-field approximation effectively models large plant populations.
- Simulation using semi-Lagrangian scheme and Gaussian process regression.
- Asymptotic independence of individuals in large populations.

## Abstract

The processes of interplant competition within a field are still poorly understood. However, they explain a large part of the heterogeneity in a field and may have longer-term consequences, especially in mixed stands. Modeling can help to better understand these phenomena but requires simulating the interactions between different individuals. In the case of large populations, assessing the parameters of a heterogeneous population model from experimental data is intractable computationally. This paper investigates the mean-field approximation of large dynamical systems with random initial conditions and individual parameters, and with interaction being represented by pairwise potentials between individuals. Under this approximation, each individual is in interaction with an infinitely-crowded population, summarized by a probability measure, the mean-field limit distribution, being itself the weak solution of a non-linear hyperbolic partial differential equation. In particular, the phenomenon of chaos propagation implies that the individuals are independent asymptotically when the size of the population tends towards infinity. This result provides perspectives for a possible simplification of the inference problem. The simulation of the mean-field distribution, consisting in a semi-Lagrangian scheme with an interpolation step using Gaussian process regression, is illustrated for a heterogeneous population model representing plants in competition for light.