Understanding evolutionary and ecological dynamics using a continuum limit

Limits on the evolutionary rates of biological traits

This paper focuses on the maximum speed at which biological evolution can occur. I derive inequalities that limit the rate of evolutionary processes driven by natural selection, mutations, or genetic drift. These rate limits link the variability in a population to evolutionary rates. In particular, high variances in the fitness of a population and of a quantitative trait allow for fast changes in the trait's average. In contrast, low variability makes a trait less susceptible to random changes due to genetic drift. The results in this article generalize Fisher's fundamental theorem of natural selection to dynamics that allow for mutations and genetic drift, via trade-off relations that constrain the evolutionary rates of arbitrary traits. The rate limits can be used to probe questions in various evolutionary biology and ecology settings. They apply, for instance, to trait dynamics within or across species or to the evolution of bacteria strains. They apply to any quantitative trait, e.g., from species' weights to the lengths of DNA strands.

Evolutionary rescue via niche construction: Infrequent construction can prevent post-invasion extinction

A population experiencing habitat loss can avoid extinction by undergoing genetic adaptation-a process known as evolutionary rescue. Here we analytically approximate the probability of evolutionary rescue via a niche-constructing mutation that allows carriers to convert a novel, unfavorable reproductive habitat to a favorable state at a cost to their fecundity. We analyze competition between mutants and non-niche-constructing wild types, who ultimately require the constructed habitats to reproduce. We find that over-exploitation of the constructed habitats by wild types can generate damped oscillations in population size shortly after mutant invasion, thereby decreasing the probability of rescue. Such post-invasion extinction is less probable when construction is infrequent, habitat loss is common, the reproductive environment is large, or the population's carrying capacity is small. Under these conditions, wild types are less likely to encounter the constructed habitats and, consequently, mutants are more likely to fix. These results suggest that, without a mechanism that deters wild type inheritance of the constructed habitats, a population undergoing rescue via niche construction may remain prone to short-timescale extinction despite successful mutant invasion.

A stochastic analysis of the interplay between antibiotic dose, mode of action, and bacterial competition in the evolution of antibiotic resistance

The use of an antibiotic may lead to the emergence and spread of bacterial strains resistant to this antibiotic. Experimental and theoretical studies have investigated the drug dose that minimizes the risk of resistance evolution over the course of treatment of an individual, showing that the optimal dose will either be the highest or the lowest drug concentration possible to administer; however, no analytical results exist that help decide between these two extremes. To address this gap, we develop a stochastic mathematical model of bacterial dynamics under antibiotic treatment. We explore various scenarios of density regulation (bacterial density affects cell birth or death rates), and antibiotic modes of action (biostatic or biocidal). We derive analytical results for the survival probability of the resistant subpopulation until the end of treatment, the size of the resistant subpopulation at the end of treatment, the carriage time of the resistant subpopulation until it is replaced by a sensitive one after treatment, and we verify these results with stochastic simulations. We find that the scenario of density regulation and the drug mode of action are important determinants of the survival of a resistant subpopulation. Resistant cells survive best when bacterial competition reduces cell birth and under biocidal antibiotics. Compared to an analogous deterministic model, the population size reached by the resistant type is larger and carriage time is slightly reduced by stochastic loss of resistant cells. Moreover, we obtain an analytical prediction of the antibiotic concentration that maximizes the survival of resistant cells, which may help to decide which drug dosage (not) to administer. Our results are amenable to experimental tests and help link the within and between host scales in epidemiological models.

Revisiting the number of self-incompatibility alleles in finite populations: From old models to new results

Under gametophytic self-incompatibility (GSI), plants are heterozygous at the self-incompatibility locus (S-locus) and can only be fertilized by pollen with a different allele at that locus. The last century has seen a heated debate about the correct way of modelling the allele diversity in a GSI population that was never formally resolved. Starting from an individual-based model, we derive the deterministic dynamics as proposed by Fisher (The genetical theory of natural selection - A complete, Variorum edition, Oxford University Press, 1958) and compute the stationary S-allele frequency distribution. We find that the stationary distribution proposed by Wright (Evolution, 18, 609, 1964) is close to our theoretical prediction, in line with earlier numerical confirmation. Additionally, we approximate the invasion probability of a new S-allele, which scales inversely with the number of resident S-alleles. Lastly, we use the stationary allele frequency distribution to estimate the population size of a plant population from an empirically obtained allele frequency spectrum, which complements the existing estimator of the number of S-alleles. Our expression of the stationary distribution resolves the long-standing debate about the correct approximation of the number of S-alleles and paves the way for new statistical developments for the estimation of the plant population size based on S-allele frequencies.

Analyzing dynamic species abundance distributions using generalized linear mixed models

Understanding the mechanisms of ecological community dynamics and how they could be affected by environmental changes is important. Population dynamic models have well known ecological parameters that describe key characteristics of species such as the effect of environmental noise and demographic variance on the dynamics, the long‐term growth rate, and strength of density regulation. These parameters are also central for detecting and understanding changes in communities of species; however, incorporating such vital parameters into models of community dynamics is challenging. In this paper, we demonstrate how generalized linear mixed models specified as intercept‐only models with different random effects can be used to fit dynamic species abundance distributions. Each random effect has an ecologically meaningful interpretation either describing general and species‐specific responses to environmental stochasticity in time or space, or variation in growth rate and carrying capacity among species. We use simulations to show that the accuracy of the estimation depends on the strength of density regulation in discrete population dynamics. The estimation of different covariance and population dynamic parameters, with corresponding statistical uncertainties, is demonstrated for case studies of fish and bat communities. We find that species heterogeneity is the main factor of spatial and temporal community similarity for both case studies.

Weak Selection and the Separation of Eco-evo Time Scales using Perturbation Analysis

We show that under the assumption of weak frequency-dependent selection a wide class of population dynamical models can be analysed using perturbation theory. The inner solution corresponds to the ecological dynamics, where to zeroth order, the genotype frequencies remain constant. The outer solution provides the evolutionary dynamics and corresponds, to zeroth order, to a generalisation of the replicator equation. We apply this method to a model of public goods dynamics and construct, using matched asymptotic expansions, a composite solution valid for all times. We also analyse a Lotka-Volterra model of predator competition and show that to zeroth order the fraction of wild-type predators follows a replicator equation with a constant selection coefficient given by the predator death rate. For both models, we investigate how the error between approximate solutions and the solution to the full model depend on the order of the approximation and show using numerical comparison, for [Formula: see text] and 2, that the error scales according to [Formula: see text], where [Formula: see text] is the strength of selection and k is the order of the approximation.