The effective strength of selection in random environment
Adrián González Casanova, Dario Spanò, Maite Wilke-Berenguer
TL;DR
This work develops a two-type Lambda-Fleming-Viot framework with random environmental fluctuations and skewed offspring reproduction to study how rare, environment-driven selection interacts with genetic drift. It builds a finite Wright-Fisher graph (DASG) with a time-varying environment, proves strong forward–backward dualities, and derives a robust scaling limit to a two-type Fleming-Viot process with both weak and rare selection (FVWRS), along with a branching-coalescing process in random environment (BCPRE) that is moment-dual to the forward process. A central result is the universal extinction/fixation threshold $\alpha_{\rm Eff}=\alpha_{\mathfrak s}\alpha^*+w= c\beta^*$, with $\alpha^*=\mathbb{E}[1/(1+V\mathbb{E}[K_{Y^{\mathfrak s}}-1|Y^{\mathfrak s}])]$, showing when strong enough selection eliminates the weaker allele regardless of other details. The paper also provides a Griffiths-type representation of the generator, establishes Feller and conservativeness properties for the dual process, and proves convergence of finite-population genealogies to the limiting BCPre, highlighting the influence of environment variability on long-term evolutionary outcomes. Overall, the results give rigorous criteria for extinction versus fixation in populations with random environmental selection and highly skewed offspring distributions, with broad implications for understanding cataclysmic selection events and their evolutionary consequences.
Abstract
We analyse a family of two-types Wright-Fisher models with selection in a random environment and skewed offspring distribution. We provide a calculable criterion to quantify the impact of different shapes of selection on the fate of the weakest allele, and thus compare them. The main mathematical tool is duality, which we prove to hold, also in presence of random environment (quenched and in some cases annealed), between the population's allele frequencies and genealogy, both in the case of finite population size and in the scaling limit for large size. Duality also yields new insight on properties of branching-coalescing processes in random environment, such as their long term behaviour.