Invasion implies substitution in ecological communities with class-structured populations

Abstract

<jats:title>Abstract</jats:title><jats:p>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.</jats:p>

Metacommunities, fitness and gradual evolution

“…We find that the approximation (1)-(2) holds for small but non-zero δ where the perturbation of steady states caused by the invasion is taken into account (a result given without proof previously), and, for all times t ≥ t * where t * is some finite time that depends on δ. Moreover, and finally, we sharpen earlier representations of (1) for class structured populations given in terms of a reproductive value weighted mutant frequency (Roze and Rousset, 2004;Rousset and Ronce, 2004;Rousset, 2004;Van Cleve et al, 2010;Priklopil and Lehmann, 2020), by allowing to take any average for the mutant frequency with a special case of the arithmetic mean.…”

“…which defines the critical manifold of the system (e.g., Kuehn, 2015 for general considerations and Priklopil and Lehmann, 2020 in the context of the "invasion implies substitution"-principle). The critical manifold thus gives the steady states of the general solution of (76) and is a curve consisting of an infinite number of steady states (equilibria) parametrized by pv (as discussed in the above paragraph).…”

“…fluctuations in the quality of the local habitat). Second, we provide a detailed and step-by-step proof of (1)-(2) by singularly perturbing the limiting metacommunity model where δ = 0 (Fenichel, 1979;Hek, 2010;Kuehn, 2015;Priklopil and Lehmann, 2020) and pay particular attention to formulate the dynamics of relatedness. We find that the approximation (1)-(2) holds for small but non-zero δ where the perturbation of steady states caused by the invasion is taken into account (a result given without proof previously), and, for all times t ≥ t * where t * is some finite time that depends on δ.…”

Resident-invader dynamics of similar strategies in fluctuating environments

“…where F(x i , e t , θ t ) is a × matrix whose entries are transition rates between different states of individuals with strategy x i and depending on the current environmental condition (e t , θ t ). Model (3.1) is an extension of the deterministic models used in Durinx et al (2008) and Priklopil and Lehmann (2019), but now with an explicit formulation of environmental feedback and environmental stochasticity.…”

Hamilton’s rule, gradual evolution, and the optimal (feedback) control of phenotypically plastic traits

“…We have thus formalised an "invasion implies substitution"-principle (see Priklopil and Lehmann, 2020 for a review) for function-valued traits in the homogeneous island population and which takes the form of Hamilton's rule: the mutant spreads if r(u)b η (u) − c η (u) > 0. This novel result is a multidimensional generalisation of previous analogous results for scalar traits (Roze andRousset, 2003, 2004;Rousset, 2004;Lehmann and Rousset, 2014).…”