Effective noisy dynamics within the phenotypic space of a growth-rate maximizing population

Microbial systems exhibit marked variability in metabolic phenotypes. A recently-proposed class of models explains this feature within a minimal mathematical setup which assumes that populations evolve towards maximum growth rate in a 'phenotypic space' subject to an intrinsic 'diffusive' stochasticity that causes small random changes in single-cell phenotypes. In such a framework, variability results from the exploration-exploitation balance between hardly accessible fast-growing phenotypes and easily accessible slow-growing ones. Here we extend the above scheme to include a degree of extrinsic noise, showing that the population dynamics over the phenotypic space is captured by an effective process that conflates both sources of randomness. This in turn leads to a simple approximation for the asymptotic distribution of the population over the phenotypic space, highlighting the connection between the strength of the noise that affects the dynamics and the degree of optimization. The theory thus obtained displays an excellent agreement with numerical simulations of low-dimensional systems.