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

Abstract

Predicting the adaptation of populations to a changing environment is crucial to assess the impact of human activities on biodiversity. Many theoretical studies have tackled this issue by modeling the evolution of quantitative traits subject to stabilizing selection around an optimum phenotype, whose value is shifted continuously through time. In this context, the population fate results from the equilibrium distribution of the trait, relative to the moving optimum. Such a distribution may vary with the shape of selection, the system of reproduction, the number of loci, the mutation kernel or their interactions. Here, we develop a methodology that provides quantitative measures of population maladaptation and potential of survival directly from the entire profile of the phenotypic distribution, without any a priori on its shape. We investigate two different models of reproduction (asexual and infinitesimal sexual models of inheritance), with general forms of selection. {In particular, we recover that fitness functions such that selection weakens away from the optimum lead to evolutionary tipping points, with an abrupt collapse of the population when the speed of environmental change is too high. Our unified framework furthermore allows highlighting the underlying mechanisms that lead to this phenomenon.} More generally, it allows discussing similarities and discrepancies between the two reproduction models, the latter being ultimately explained by different constraints on the evolution of the phenotypic variance. We demonstrate that the mean fitness in the population crucially depends on the shape of the selection function in the sexual infinitesimal model, in contrast with the asexual model. In the asexual model, we also investigate the effect of the mutation kernel and we show that kernels with higher kurtosis tend to reduce maladaptation and improve fitness, especially in fast changing environments.

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

“…Assuming the selection gradient is smooth and non-constant, a sufficient analytical condition for the existence of a local maximum of the selection gradient is the co-existence of multiple equilibria (located where the selection gradient cancels). This is in agreement with what occurs in a single-habitat framework under changing environment with non-quadratic selection functions for which maladaptation stabilizes away from the optimum ([Osmond and Klausmeier 2017], [Garnier et al 2022]). Indeed, in this case, the selection gradient under stable environment cancels in the optimum trait and converges to 0 in−∞.…”

“…In this model, it is assumed to be constant across families, time and space. Accordingly, at a time t , the number of individuals born with a trait z in the habitat i is given by the following formula (also used Turelli and Barton 1990; Mirrahimi and Raoul 2013; Calvez, Garnier, and Patout 2019; Patout 2020; Dekens and Lavigne 2021; Dekens 2022; Garnier et al 2022): We consider the dynamics of the local trait distributions given by: The environment changes through time at a constant speed c which is modelled by a linear increase of the local optima: θ 1 = − θ + ct and θ 2 = θ + ct .…”

“…Key features underlying the analytical results of these studies resides in the following assumptions: a Gaussian approximation of the trait distribution within the population and a quadratic selection function implying that maladaptation increases more steeply away from the optimal trait. This approach has next been extended to include the effects of plasticity (Chevin, Lande, and Mace 2010), multidimensional quantitative traits (Duputié et al 2012), age-structure (Cotto and Ronce 2014; Cotto, Sandell, et al 2019), or to examine the influence of the mode of reproduction (Waxman and Peck 1999, Garnier et al 2022).…”

“…This happens because, for these selection functions leading to a change of convexity in the selection gradient, the lag grows indefinitely past a certain threshold. This feature (among many others) was also characterized analytically in [Garnier et al 2022] that investigated more broadly the influence of the selection functions on the adaptation of sexually and asexually reproduction populations to a changing environment. However, while these evolutionary tipping points are linked to particular choices of selection functions in a single habitat framework, they have been reported to arise in more complex frameworks (Klausmeier et al 2020).…”

Uniform contractivity of the Fisher infinitesimal model with strongly convex selection

“…For any two given individuals with traits X and X ′ respectively, the associated parental traits (X 1 , X 2 ) and (X ′ 1 , X ′ 2 ) are closer to each other than (X, X ′ ) in some sense, see also [24,Appendix F.2] for a visual explanation. To make sense of this contraction, we shall work with the L ∞ Wasserstein distance, denoted by W ∞ (in contrast with the usual L 2 Wasserstein distance).…”