Invasion implies substitution in ecological communities with class-structured populations

Abstract

Long-term evolution of quantitative traits is classically and usefully described as the directional change in phenotype due to the recurrent fixation of new mutations. A formal justification for such continual evolution ultimately relies on the ''invasion implies substitution''-principle. Here, whenever a mutant allele causing a small phenotypic change can successfully invade a population, the ancestral (or wild-type) allele will be replaced, whereby fostering gradual phenotypic change if the process is repeated. It has been argued that this principle holds in a broad range of situations, including spatially and demographically structured populations experiencing frequency-and density-dependent selection under demographic and environmental fluctuations. However, prior studies have not been able to account for all aspects of population structure, leaving unsettled the conditions under which the ''invasion implies substitution''-principle really holds. In this paper, we start by laying out a program to explore and clarify the generality of the ''invasion implies substitution''-principle. Particular focus is given on finding an explicit and functionally constant representation of the selection gradient on a quantitative trait. Using geometric singular perturbation methods, we then show that the ''invasion implies substitution''-principle generalizes to well-mixed and scalar-valued polymorphic multispecies ecological communities that are structured into finitely many demographic (or physiological) classes. The selection gradient is shown to be constant over the evolutionary timescale and that it depends only on the resident phenotype, individual growth-rates, population steady states and reproductive values, all of which are calculated from the resident dynamics. Our work contributes to the theoretical foundations of evolutionary ecology.

Evolution of class-structured populations in periodic 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).…”

“…This leads to a separation of time scales, with the class frequencies and individual reproductive values being fast variables and the weighted mutant frequency being a slow variable. Importantly, the unweighted average, f m , is not a slow variable because its dynamics (3) depends on O (1) terms (see also Priklopil & Lehmann (2020)).…”

Multi-morph eco-evolutionary dynamics in structured populations

“…A drawback of this increased realism is that it inflates the number of ecological and genetic variables we need to track. We therefore apply recent theory on reproductive values (Lion, 2018a,b; Priklopil & Lehmann, 2020) to simplify the oligomorphic analysis and obtain a compact description of how the morph-level trait distributions change over time when there are demographic transitions between classes. The key idea is to define a weighted trait distribution that gives us a way of examining how selection acts on a particular morph across all the classes (Fisher, 1930; Taylor & Frank, 1996; Frank, 1998; Rousset, 2004; Lehmann & Rousset, 2014; Lion, 2018a).…”

“…On this fast time scale, we can derive the following quasi-equilibrium approximation from equations (10) and (13): so that the vectors of class frequencies and reproductive values are respectively the right and left eigenvectors of the matrix associated with eigenvalue 0, where it follows from equation (7) that has elements . This is a multi-morph extension of a standard result from monomorphic theory (Taylor, 1990; Rousset, 2004; Lehmann & Rousset, 2014; Lion, 2018a,b; Priklopil & Lehmann, 2020).…”

“…The effect of selection on the morph mean is thus scaled by the morph variance, V i , and by the selection gradient , which depends on (1) the marginal effect of a change in the trait on the between-class transition rates r kj ( z ), evaluated at the morph mean, (2) the relative quality of class k for morph- i individuals, measured by the reproductive value , and (3) the relative quantity of class- k individuals among morph- i individuals. This is the multi-morph extension of the classical expression for the class-structured selection gradient (Taylor, 1990; Rousset, 1999, 2004; Lehmann & Rousset, 2014; Lion, 2018a,b; Priklopil & Lehmann, 2020), which can be recovered by noting that, in the single-morph case, , so that u i = f and v i = v .…”