A stochastic population model with hierarchic size-structure

TL;DR

This paper introduces a hierarchical population model combining deterministic delay equations and stochastic individual-based models, analyzing their relationship and showing the deterministic model approximates the stochastic one for large populations.

Contribution

It formulates a novel hierarchical population model linking deterministic and stochastic approaches, providing a formula to approximate the stochastic birth rate and establishing their large population relationship.

Findings

Deterministic model's stationary birth rate approximates stochastic quasi-stationary rate for large populations.

Derived a formula to estimate the stochastic birth rate.

Established the deterministic model as a large population limit of the stochastic model.

Abstract

We consider a hierarchically structured population in which the amount of resources an individual has access to is affected by individuals that are larger, and that the intake of resources by an individual only affects directly the growth rate of the individual. We formulate a deterministic model, which takes the form of a delay equation for the population birth rate. We also formulate an individual based stochastic model, and study the relationship between the two models. In particular the stationary birth rate of the deterministic model is compared to that of the quasi-stationary birth rate of the stochastic model. Since the quasi-stationary birth rate cannot be obtained explicitly, we derive a formula to approximate it. We show that the stationary birth rate of the deterministic model can be obtained as the large population limit of the quasi-stationary birth rate of the stochastic model. This relation suggests that the deterministic model is a good approximation of the stochastic model when the number of individuals is sufficiently large.