Comparison between effective and individual growth rates in a heterogeneous population

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, which can represent genetic polymorphism, 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 mutations. The fitness (or Malthus parameter, or effective fitness) of such heterogeneous populations 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, both cases yield similar expressions. By comparing the fitness and effective growth rates of populations with different degrees of heterogeneity or different heredity/mutation laws 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.