Resident-invader dynamics of similar strategies in fluctuating environments

Abstract

We study resident-invader dynamics in fluctuating environments when the invader and the resident have close but distinct strategies. First we focus on a class of continuous-time models of unstructured populations of multi-dimensional strategies, which incorporates environmental feedback and environmental stochasticity. Then we generalize our results to a class of structured population models. We classify the generic population dynamical outcomes of an invasion event when the resident population in a given environment is non-growing on the long-run and stochastically persistent. Our approach is based on the series expansion of a model with respect to the small strategy difference, and on the analysis of a stochastic fast-slow system induced by time-scale separation. Theoretical and numerical analyses show that the total size of the resident and invader population varies stochastically and dramatically in time, while the relative size of the invader population changes slowly and asymptotically in time. Thereby the classification is based on the asymptotic behavior of the relative population size, and which is shown to be fully determined by invasion criteria (i.e., without having to study the full generic dynamical system). Our results extend and generalize previous results for a stable resident equilibrium (particularly, Geritz in J Math Biol 50(1):67–82, 2005; Dercole and Geritz in J Theor Biol 394:231-254, 2016) to non-equilibrium resident population dynamics as well as resident dynamics with stochastic (or deterministic) drivers.

Evolution of class-structured populations in periodic environments

“…An interesting avenue for future work would be to extend our approach to stochastic stationary ecological dynamics. Since the “invasion implies fixation” principle has also been proven for stochastic fluctuations in the environment (Cai & Geritz, 2020), we think this extension is feasible and would help link our method with classical theory on the influence of environmental stochasticity on life-history evolution (Frank & Slatkin, 1990; Sasaki & Ellner, 1995; Lande et al, 2017). However, this would require a careful definition of the concept of time-dependent reproductive value in stochastic environments.…”

“…The only difference is that the quantity and quality of each class are allowed to fluctuate periodically over time. Second, the direction of selection can be obtained by computing the average, over one period of the resident attractor, of the instantaneous selection gradient, and we recover a periodic extension of the “invasion implies fixation” principle (Geritz, 2005; Cai & Geritz, 2020; Priklopil & Lehmann, 2020).…”

“…However, under weak selection, these fast fluctuations can be ironed out. This is known as the averaging principle (Cai & Geritz, 2020), which allows us to approximate the dynamics of the mutant frequency on the slow time scales by the solution of the so-called averaged system where is the selection gradient averaged over one period of the resident attractor (i.e. for any given τ on the periodic attractor).…”

Metacommunities, fitness and gradual evolution

“…Our model also applies to situations where reproduction, dispersal, survival and physiological development depends on locally fluctuating external factors, defined as any factors that are not contained in the environmental feedback, and can thus generate the effect of environmental stochasticity on a local (group) scale and whenever the external factors can be expressed as discrete variables. Extending our model to capture globally fluctuating external factors would require to introduce an additional "environmental" state space E whose elements would affect all quantities used in this paper but not change their functional form, that is, all rates used in this paper would also be functions of E (in the context of this present paper but for well-mixed populations see a recent approach in Cai and Geritz, 2020). Yet, we expect that by taking an appropriate average over the ergodic path of environmental states including such environmental change is unlikely to change the generic shape of the selection coefficient, a point illustrated by Lenormand et al (2009), proven more generally by Lion (2018b, eq.…”

“…In this paper we provide sufficient conditions for selective sweeps when evolution occurs in an asexual focal species that resides in a wider metacommunity (Wilson, 1992;Leibold et al, 2004), and where socially interacting individuals are structured into discrete classes that determine their physiological state, such as age and size, as well as their local biotic and abiotic environment. We provide these conditions in terms of an "invasion implies substitution"-principle, which is a formal characterization of the invariance of selection with respect to a population-wide frequency of a mutant allele that affects quantitative traits in the context of density-and frequency-dependent selection (Hamilton, 1964;Rousset, 2004;Meszéna et al, 2005;Dercole and Rinaldi, 2008;Lehmann and Rousset, 2014;Dercole, 2016;Cantrell et al, 2017;Ito et al, 2020;Cai and Geritz, 2020;Priklopil and Lehmann, 2020). The principle consists of two propositions (Priklopil and Lehmann, 2020).…”