Frequency-dependent growth in class-structured populations: continuous dynamics in the limit of weak selection.

In this paper we consider class-structured populations in discrete time in the limit of weak selection and with the inverse of the intensity of selection as unit of time. The aim is to establish a continuous model that approximates the discrete model. More precisely, we study frequency-dependent growth in an infinite haploid population structured into a finite number of classes such that individuals in each class contribute to a given subset of classes from one time step to the next. These contributions take the form of generalized fecundity parameters with perturbations of order 1 / N that depends on the class frequencies of each type and the type frequencies. Moreover, they satisfy some mild conditions that ensure mixing in the long run. The dynamics in the limit as [Formula: see text] with N time steps as unit of time is considered first in the case of a single type, and second in the case of multiple types. The main result is that the type frequencies as [Formula: see text] obey the replicator equation with instantaneous growth rates for the different types that depend only on instantaneous equilibrium class frequencies and reproductive values. An application to evolutionary game theory complemented by simulation results is presented.

Climate Effects on Growth, Body Condition, and Survival Depend on the Genetic Characteristics of the Population.

Climatic change is expected to affect individual life histories and population dynamics, potentially increasing vulnerability to extinction. The importance of genetic diversity has been highlighted for adaptation and population persistence. However, whether responses of life-history traits to a given environmental condition depend on the genetic characteristics of a population remains elusive. Here we tested this hypothesis in the lizard Zootoca vivipara by simultaneously manipulating habitat humidity, a major climatic predictor of Zootoca's distribution, and adult male color morph frequency, a trait with genome-wide linkage. Interactive effects of humidity and morph frequency had immediate effects on growth and body condition of juveniles and yearlings, as well as on adult survival, and delayed effects on offspring size. In yearlings, higher humidity led to larger female body size and lower humidity led to higher male compared to female survival. In juveniles and yearlings, some treatment effects were compensated over time. The results show that individual responses to environmental conditions depend on the population's color morph frequency, age class, and sex and that these affect intra- and inter-age class competition. Moreover, humidity affected the competitive environment rather than imposing trait-based selection on specific color morphs. This indicates that species' responses to changing environments (e.g., to climate change) are highly complex and difficult to accurately reconstruct and predict without information on the genetic characteristics and demographic structure of populations.

Variable prey development time suppresses predator-prey cycles and enhances stability.

Although theoretical models have demonstrated that predator-prey population dynamics can depend critically on age (stage) structure and the duration and variability in development times of different life stages, experimental support for this theory is non-existent. We conducted an experiment with a host-parasitoid system to test the prediction that increased variability in the development time of the vulnerable host stage can promote interaction stability. Host-parasitoid microcosms were subjected to two treatments: Normal and High variance in the duration of the vulnerable host stage. In control and Normal-variance microcosms, hosts and parasitoids exhibited distinct population cycles. In contrast, insect abundances were 18-24% less variable in High- than Normal-variance microcosms. More significantly, periodicity in host-parasitoid population dynamics disappeared in the High-variance microcosms. Simulation models confirmed that stability in High-variance microcosms was sufficient to prevent extinction. We conclude that developmental variability is critical to predator-prey population dynamics and could be exploited in pest-management programs.

Ant Mutualism Increases Long-Term Growth and Survival of a Common Amazonian Tree.

How ecological context shapes mutualistic relationships remains poorly understood. We combined long-term tree census data with ant censuses in a permanent 25-ha Amazonian forest dynamics plot to evaluate the effect of the mutualistic ant Myrmelachista schumanni (Formicinae) on the growth and survival of the common Amazonian tree Duroia hirsuta (Rubiaceae), considering its interactions with tree growth, population structure, and habitat. We found that the mutualist ant more than doubled tree relative growth rates and increased odds of survival. However, host tree size and density of conspecific neighbors modified the effect of the ant. Smaller trees hosting the mutualist ant consistently grew faster when surrounded by higher densities of conspecifics, suggesting that the benefit to the tree outweighs any negative effects of high conspecific densities. Moreover, our findings suggest that the benefit afforded by the ant diminishes with plant age and also depends on the density of conspecific neighbors. We provide the first long-term large-scale evidence of how mutualism affects the population biology of an Amazonian tree species.

Definitions of fitness in age-structured populations: Comparison in the haploid case.

Fisher's (1930) Fundamental Theorem of Natural Selection (FTNS), and in particular the development of an explicit age-structured version of the theorem, is of everlasting interest. In a recent paper, Grafen (2015a) argues that Fisher regarded his theorem as justifying individual rather than population fitness maximization. The argument relies on a new definition of fitness in age-structured populations in terms of individual birth and death rates and age-specific reproductive values in agreement with a principle of neutrality. The latter are frequency-dependent and defined without reference to genetic variation. In the same paper, it is shown that the rate of increase in the mean of the breeding values of fitness weighted by the reproductive values, but keeping the breeding values constant as in Price (1972) is equal to the additive genetic variance in fitness. Therefore, this partial change is obtained by keeping constant not only the genotypic birth and death rates but also the mean age-specific birth and death rates from which the age-specific reproductive values are defined. In this paper we reaffirm that the Malthusian parameter which measures the relative rate of increase or decrease in reproductive value of each genotype in a continuous-time age-structured population is the definition of fitness used in Fisher's (1930) FTNS. This is shown by considering an age-structured asexual haploid population with constant age-specific birth and death (or survival) parameters for each type. Although the original statement of the FTNS is for a diploid population, this simplified haploid model allows us to address the definition of fitness meant in this theorem without the complexities and effects of a changing genic environment. In this simplified framework, the rate of change in mean fitness in continuous time is expected to be exactly equal to the genetic variance in fitness (or to the genetic variance in fitness divided by the mean fitness in discrete time), which can be seen as a generalized growth-rate theorem. This theorem is shown to hold with the Malthusian parameter used as the definition of fitness. Moreover, in the same framework, it is shown that a discrete-time version of Grafen's definition may lead to a decrease in mean fitness. In the limit of weak selection with the unit of time proportional to the inverse of the intensity of selection, however, this definition predicts the right population dynamics in agreement with the growth-rate theorem. This clarifies the domain of application of the new definition, at least as far as population dynamics is concerned, and reconciles the new definition with the original one.

The concept of critical age group for density dependence: bridging the gap between demographers, evolutionary biologists and behavioural ecologists.

Density dependence plays an important role in population regulation in the wild. It involves a decrease in population growth rate when the population size increases. Fifty years ago, Charlesworth introduced the concept of 'critical age group', denoting the age classes in which variation in the number of individuals most strongly contributes to density regulation. Since this pioneering work, this concept has rarely been used. In light of Charlesworth's concept, we discuss the need to develop work between behavioural ecology, demography and evolutionary biology to better understand the mechanisms acting in density-regulated age-structured populations. We highlight demographic studies that explored age-specific contributions to density dependence and discuss the underlying evolutionary processes. Understanding competitive interactions among individuals is pivotal to identify the ages contributing most strongly to density regulation, highlighting the need to move towards behavioural ecology to decipher mechanisms acting in density-regulated age-structured populations. Because individual characteristics other than age can be linked to competitive abilities, expanding the concept of critical age to other structures (e.g. sex, dominance rank) offers interesting perspectives. Linking research fields based on the concept of the critical age group is key to move from a pattern-oriented view of density regulation to a process-oriented approach.This article is part of the discussion meeting issue 'Understanding age and society using natural populations'.

Mathematical modeling of contact tracing and stability analysis to inform its impact on disease outbreaks; an application to COVID-19.

We develop a mathematical model to investigate the effect of contact tracing on containing epidemic outbreaks and slowing down the spread of transmissible diseases. We propose a discrete-time epidemic model structured by disease-age which includes general features of contact tracing. The model is fitted to data reported for the early spread of COVID-19 in South Korea, Brazil, and Venezuela. The calibrated values for the contact tracing parameters reflect the order pattern observed in its performance intensity within the three countries. Using the fitted values, we estimate the effective reproduction number Re and investigate its responses to varied control scenarios of contact tracing. Alongside the positivity of solutions, and a stability analysis of the disease-free equilibrium are provided.

The impact of maturation time distributions on the structure and growth of cellular populations.

Here we study how the structure and growth of a cellular population vary with the distribution of maturation times from each stage. We consider two cell cycle stages. The first represents early G1. The second includes late G1, S, G2, and mitosis. Passage between the two reflects passage of an important cell cycle checkpoint known as the restriction point. We model the population as a system of partial differential equations. After establishing the existence of solutions, we characterize the maturation rates and derive the steady-state age and stage distributions as well as the asymptotic growth rates for models with exponential and inverse Gaussian maturation time distributions. We find that the stable age and stage distributions, transient dynamics, and asymptotic growth rates are substantially different for these two maturation models. We conclude that researchers modeling cellular populations should take care when choosing a maturation time distribution, as the population growth rate and stage structure can be heavily impacted by this choice. Furthermore, differences in the models' transient dynamics constitute testable predictions that can help further our understanding of the fundamental process of cellular proliferation. We hope that our numerical methods and programs will provide a scaffold for future research on cellular proliferation.

Equivalences between age structured models and state dependent distributed delay differential equations.

We use the McKendrick equation with variable ageing rate and randomly distributed mat-uration time to derive a state dependent distributed delay differential equation. We show that the resulting delay differential equation preserves non-negativity of initial conditions and we characterise local stability of equilibria. By specifying the distribution of maturation age, we recover state depen-dent discrete, uniform and gamma distributed delay differential equations. We show how to reduce the uniform case to a system of state dependent discrete delay equations and the gamma distributed case to a system of ordinary differential equations. To illustrate the benefits of these reductions, we convert previously published transit compartment models into equivalent distributed delay differential equations.