Analysis of the dynamics induced by a competition index in a heterogeneous population of plants: from an individual-based model to a macroscopic model

TL;DR

This paper investigates how a specific competition index influences the dynamics of a heterogeneous plant population, deriving conditions for the transition from individual-based models to macroscopic models and analyzing the population's behavior as it grows large.

Contribution

It provides mathematical conditions ensuring the validity of mean-field limits for a common competition index in heterogeneous populations, linking individual and population scales.

Findings

Conditions for well-defined population dynamics are established.

Convergence to mean-field distribution is proved as population size increases.

Numerical simulations illustrate the heterogeneous population dynamics.

Abstract

Competition indices are models frequently used in ecology to account for the impact of density and resource distribution on the growth of a plant population. They allow to define simple individual-based models, by integrating information relatively easy to collect at the population scale, which are generalized to a macroscopic scale by mean-field limit arguments. Nevertheless, up to our knowledge, few works have studied under which conditions on the competition index or on the initial configuration of the population the passage from the individual scale to the population scale is mathematically guaranteed. We consider in this paper a competition index commonly used in the literature, expressed as an average over the population of a pairwise potential depending on a measure of plants' sizes and their respective distances. In line with the literature on mixed-effect models, the population is assumed to be heterogeneous, with inter-individual variability of growth parameters. Sufficient conditions on the initial configuration are given so that the population dynamics, taking the form of a system of non-linear differential equations, is well defined. The mean-field distribution associated with an infinitely crowded population is then characterized by the characteristic flow, and the convergence towards this distribution for an increasing population size is also proved. The dynamics of the heterogeneous population is illustrated by numerical simulations, using a Lagrangian scheme to visualize the mean-field dynamics.