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 other’s 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.

A model of coevolution and local adaptation between hosts and parasites in continuous space

“…The lifetime expected number of offspring (which, because mortality is equal for all individuals, we refer to as fitness) is determined by the trait of the parent along with the traits of other individuals the parent interacts with. This is similar to the starting points taken by Week et al (2021) in the derivation of a diffuse-coevolution model and by Week and Nuismer (2021) in the derivation of the offset-matching coevolution model, except neither of those models have a spatial component. To model fitness, we first consider the effects of abiotic selection 𝒜 S and biotic selection ℬ S separately for host and parasite species ( S = H, P ). We decompose the effects of biotic selection into sources due to intraspecific competition and interspecific parasitism so that .…”

“…In general, spatially heterogeneous population densities can affect the action of selection (Kirkpatrick and Barton 1997). However, because we assume population densities are spatially homogeneous, we follow Week et al (2021) to obtain expressions for the local dynamics of mean traits in response to selection. We use to denote the instantaneous rate of change of in response to selection.…”

“…The continuous time analog of this model is trait evolution following Brownian motion, which has been widely applied as a phenomenological model in the field of phylogenetic comparative methods (Felsenstein 1973; Manceau et al 2016). Mechanistically, this result has been formalized in continuous time by Week et al (2021). Denoting the instantaneous rate of change of the mean trait at location x in response to drift, Week et al (2021) found where G S is the additive genetic variance, N S is the local population density, and can be thought of informally as an infinitesimal amount of Gaussian noise drawn independently at each spatial location x ∈ ℝ 2 .…”

“…Mechanistically, this result has been formalized in continuous time by Week et al (2021). Denoting the instantaneous rate of change of the mean trait at location x in response to drift, Week et al (2021) found where G S is the additive genetic variance, N S is the local population density, and can be thought of informally as an infinitesimal amount of Gaussian noise drawn independently at each spatial location x ∈ ℝ 2 . Formally, is a space-time white noise process that, when integrated across regions of space and intervals of time, returns a normally distributed variable with mean zero and variance equal to the area of the space-time region integrated over (Walsh 1986).…”

“…Taking expectations provide where the approximation holds when s H , s P ≥ 1, which we assume from hereon. Then, assuming every parasite can potentially infect every host, the components of biotic selection due to interspecific interactions for each species are approximated by Using our weak biotic selection assumption s H , s P ≪ 1, it will be convenient to rewrite these expressions as To model mutation and spatial movement, we assume offspring trait values are normally distributed around their parental value (technically, this is done with breeding values, see Week et al 2021) and offspring locations are bivariate normal around their parental locations with i.i.d. displacements in the two spatial dimensions. To take a diffusion limit of this individual-based process, we follow Week et al (2021).…”