Comparison Between Effective and Individual Growth Rates in a Heterogeneous Population

TL;DR

This paper analyzes how heterogeneity in growth rates affects the long-term fitness of size-structured populations, deriving formulas to compare homogeneous and heterogeneous populations under different growth and division scenarios.

Contribution

It provides analytical formulas linking effective and individual growth rates in specific growth models, clarifying the impact of heterogeneity and heredity on population fitness.

Findings

Heterogeneity can influence long-term fitness depending on heredity.

Derived formulas show similar results for constant and linear growth models.

Heterogeneity benefits are more pronounced with strong heredity.

Abstract

Is there an advantage to heterogeneity in a population where individuals grow and divide by fission? This is a broad question, to which there is no easy universal answer. This article aims to provide a quantitative answer in the specific context of growth rate heterogeneity by comparing the fitness of homogeneous versus heterogeneous populations. We focus on size-structured populations, where the growth rate of each individual is set at birth by heredity and/or random mutations. The fitness (or Malthus parameter, or effective fitness) of such heterogeneous population is defined by its long-term behaviour, and we introduce the effective growth rate as the individual growth rate in the homogeneous population with the same fitness. We derive analytical formulae linking effective and individual growth rates in two paradigmatic cases: first, constant growth and division rates, second, linear growth rates and uniform fragmentation. Surprisingly, these two cases yield similar expressions. Then, by comparing the fitness and the effective growth rates of populations with different degrees of heterogeneity or different laws of heredity/mutation to those of average homogeneous populations, we quantitatively investigate the combined influence of heredity and heterogeneity, and revisit previous results stating that heterogeneity is beneficial in the case of strong heredity.