Evolutionary dynamics of complex traits in sexual populations in a heterogeneous environment: how normal?

Abstract

When studying the dynamics of trait distribution of populations in a heterogeneous environment, classical models from quantitative genetics choose to look at its system of moments, specifically the first two ones. Additionally, in order to close the resulting system of equations, they often assume that the trait distribution is Gaussian (see for instance Ronce and Kirkpatrick 2001). The aim of this paper is to introduce a mathematical framework that follows the whole trait distribution (without prior assumption) to study evolutionary dynamics of sexually reproducing populations. Specifically, it focuses on complex traits, whose inheritance can be encoded by the infinitesimal model of segregation (Fisher 1919). We show that it allows us to derive a regime in which our model gives the same dynamics as when assuming a Gaussian trait distribution. To support that, we compare the stationary problems of the system of moments derived from our model with the one given in Ronce and Kirkpatrick 2001 and show that they are equivalent under this regime and do not need to be otherwise. Moreover, under this regime of equivalence, we show that a separation bewteen ecological and evolutionary time scales arises. A fast relaxation toward monomorphism allows us to reduce the complexity of the system of moments, using a slow-fast analysis. This reduction leads us to complete, still in this regime, the analytical description of the bistable asymmetrical equilibria numerically found in Ronce and Kirkpatrick 2001. More globally, we provide explicit modelling hypotheses that allow for such local adaptation patterns to occur.

Adaptation of a quantitative trait to a changing environment: new analytical insights on the asexual and infinitesimal sexual models

“…To understand this relationship, the approximation of the phenotype distribution appeared necessary. This approach is indeed robust, as shown by several studies following the same methodology in spatial structured population models: discrete patches ([Mirrahimi, 2017] in asexual model and [Dekens, 2020] in the infinitesimal sexual model); dispersal evolution ([Perthame and Souganidis, 2016, Lam and Lou, 2017, Lam, 2017, W Hao, 2021, Calvez et al, 2018] in the asexual case and [Dekens and Lavigne, 2021] in the infinitesimal sexual case). Moreover, this methodology is expected to be efficient to investigate other structured population models.…”

Sharp habitat shifts, evolutionary tipping points and rescue: quantifying the perilous path of a specialist species toward a refugium in a changing environment

“…The model follows a classical setting of quantitative genetic models for heterogeneous environments (Ronce and Kirkpatrick 2001; Hendry, Day, and Taylor 2001; Débarre, Ronce, and Gandon 2013; Mirrahimi and Gandon 2020; Dekens 2022), which is summarized in Fig. 1.…”

“…However, these methods are not suited to study the patterns of dispersal in fragmented and patchy environments, where the demographic dynamics and the trait dynamics are quite difficult to disentangle, even under stable environment (see Ronce and Kirkpatrick 2001; Hendry, Day, and Taylor 2001; Dekens 2022 for sexual reproduction and Débarre, Ronce, and Gandon 2013; Mirrahimi 2017; Mirrahimi and Gandon 2020 for asexual ones). Therefore, most models studying adaptation to a changing and fragmented environment rely mostly on numerical simulations to explore complex metacommunities dynamics (see Cotto, Wessely, et al 2017 for such a model with multiple traits, species and an age structure, and Walters and Berger 2019 on the contribution of genetic variance on the time to extinction in a migration-mutation-selection-drift framework), or to assess the interplay between dispersal and local competition under a warming climate (Thompson and Fronhofer 2019,McManus et al 2021).…”

“…If the environmental speed is too strong, will they be strained to adapt to the refugium, and if so, how successfully? To answer these questions, I will, as a starting point, leverage the results of an analogous model under stable environment [Dekens 2022] that includes the analytical derivation of the source-sink dynamics characteristic of a specialist species that I use here as the initial state of the system.…”

“…My aim is to predict analytically the dynamics of niche evolution of this specialist population as a function of the speed of environmental change. To this aim, in Section 2, I show that, in a regime of small within-family variance allowing to separate ecological and evolutionary time scales, all the features of the dynamics of adaptation can be deduced from a simple modification of the phase-line analysis of the selection gradient derived in a previous study of a two-patch model under stable environment (Dekens 2022). Th results are presented in Section 3.…”