A White Noise Approach to Evolutionary Ecology

Abstract

Although the evolutionary response to random genetic drift is classically modelled as a sampling process for populations with fixed abundance, the abundances of populations in the wild fluctuate over time. Furthermore, since wild populations exhibit demographic stochasticity, it is reasonable to consider the evolutionary response to demographic stochasticity and its relation to random genetic drift. Here we close this gap in the context of quantitative genetics by deriving the dynamics of the distribution of a quantitative character and the abundance of a biological population from a stochastic partial differential equation driven by space-time white noise. In the process we develop a useful set of heuristics to operationalize the powerful, but abstract theory of white noise and measure-valued stochastic processes. This approach allows us to compute the full implications of demographic stochasticity on phenotypic distributions and abundances of populations. We demonstrate the utility of our approach by deriving a quantitative genetic model of diffuse coevolution mediated by exploitative competition for a continuum of resources. In addition to trait and abundance distributions, this model predicts interaction networks defined by rates of interactions, competition coefficients, or selection gradients. Analyzing the relationship between selection gradients and competition coefficients reveals independence between linear selection gradients and competition coefficients. In contrast, absolute values of linear selection gradients and quadratic selection gradients tend to be positively correlated with competition coefficients. That is, competing species that strongly affect each others abundance tend to also impose selection on one another, but the directionality is not predicted. This approach contributes to the development of a synthetic theory of evolutionary ecology by formalizing first principle derivations of stochastic models that underlie rigorous investigations of the relationship between feedbacks of biological processes and the patterns of diversity they produce.