A demographic model for sex ratio evolution and the effects of sex‐biased offspring costs

The evolution of the primary sex ratio, the proportion of male births in an individual's offspring production strategy, is a frequency‐dependent process that selects against the more common sex. Because reproduction is shaped by the entire life cycle, sex ratio theory would benefit from explicitly two‐sex models that include some form of life cycle structure. We present a demographic approach to sex ratio evolution that combines adaptive dynamics with nonlinear matrix population models. We also determine the evolutionary and convergence stability of singular strategies using matrix calculus. These methods allow the incorporation of any population structure, including multiple sexes and stages, into evolutionary projections. Using this framework, we compare how four different interpretations of sex‐biased offspring costs affect sex ratio evolution. We find that demographic differences affect evolutionary outcomes and that, contrary to prior belief, sex‐biased mortality after parental investment can bias the primary sex ratio (but not the corresponding reproductive value ratio). These results differ qualitatively from the widely held conclusions of previous models that neglect demographic structure.     

Partial life cycle analysis: a model for pre-breeding census data

Matrix population models have become popular tools in research areas as diverse as population dynamics, life history theory, wildlife management, and conservation biology. Two classes of matrix models are commonly used for demographic analysis of age‐structured populations: age‐structured (Leslie) matrix models, which require age‐specific demographic data, and partial life cycle models, which can be parameterized with partial demographic data. Partial life cycle models are easier to parameterize because data needed to estimate parameters for these models are collected much more easily than those needed to estimate age‐specific demographic parameters. Partial life cycle models also allow evaluation of the sensitivity of population growth rate to changes in ages at first and last reproduction, which cannot be done with age‐structured models. Timing of censuses relative to the birth‐pulse is an important consideration in discrete‐time population models but most existing partial life cycle models do not address this issue, nor do they allow fractional values of variables such as ages at first and last reproduction. Here, we fully develop a partial life cycle model appropriate for situations in which demographic data are collected immediately before the birth‐pulse (pre‐breeding census). Our pre‐breeding census partial life cycle model can be fully parameterized with five variables (age at maturity, age at last reproduction, juvenile survival rate, adult survival rate, and fertility), and it has some important applications even when age‐specific demographic data are available (e.g., perturbation analysis involving ages at first and last reproduction). We have extended the model to allow non‐integer values of ages at first and last reproduction, derived formulae for sensitivity analyses, and presented methods for estimating parameters for our pre‐breeding census partial life cycle model. We applied the age‐structured Leslie matrix model and our pre‐breeding census partial life cycle model to demographic data for several species of mammals. Our results suggest that dynamical properties of the age‐structured model are generally retained in our partial life cycle model, and that our pre‐breeding census partial life cycle model is an excellent proxy for the age‐structured Leslie matrix model.     

Can possible evolutionary outcomes be determined directly from the population dynamics?

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria — one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.     

Can possible evolutionary outcomes be determined directly from the population dynamics?

Traditionally, to determine the possible evolutionary behaviour of an ecological system using adaptive dynamics, it is necessary to calculate the fitness and its derivatives at a singular point. We investigate the claim that the possible evolutionary behaviour can be predicted directly from the population dynamics, without the need for calculation, by applying three criteria — one based on the form of the density dependent rates and two on the role played by the evolving parameters. Taking a general continuous time model, with broad ecological range, we show that the claim is true. Initially, we assume that individuals enter in class 1 and move through population classes sequentially; later we relax these assumptions and find that the criteria still apply. However, when we consider models where the evolving parameters appear non-linearly in the dynamics, we find some aspects of the criteria fail; useful but weaker results on possible evolutionary behaviour now apply.     

Individual heterogeneity in life histories and eco‐evolutionary dynamics

Individual heterogeneity in life history shapes eco‐evolutionary processes, and unobserved heterogeneity can affect demographic outputs characterising life history and population dynamical properties. Demographic frameworks like matrix models or integral projection models represent powerful approaches to disentangle mechanisms linking individual life histories and population‐level processes. Recent developments have provided important steps towards their application to study eco‐evolutionary dynamics, but so far individual heterogeneity has largely been ignored. Here, we present a general demographic framework that incorporates individual heterogeneity in a flexible way, by separating static and dynamic traits (discrete or continuous). First, we apply the framework to derive the consequences of ignoring heterogeneity for a range of widely used demographic outputs. A general conclusion is that besides the long‐term growth rate lambda, all parameters can be affected. Second, we discuss how the framework can help advance current demographic models of eco‐evolutionary dynamics, by incorporating individual heterogeneity. For both applications numerical examples are provided, including an empirical example for pike. For instance, we demonstrate that predicted demographic responses to climate warming can be reversed by increased heritability. We discuss how applications of this demographic framework incorporating individual heterogeneity can help answer key biological questions that require a detailed understanding of eco‐evolutionary dynamics.     

Post‐reproductive life span and demographic stability

Recent field studies suggest that it is common in nature for animals to outlive their reproductive viability. Post‐reproductive life span has been observed in a broad range of vertebrate and invertebrate species. But post‐reproductive life span poses a paradox for traditional theories of life history evolution. The only commonly‐cited explanation is the ‘grandmother hypothesis’, which is limited to higher, social mammals. We propose that post‐reproductive life span evolves to stabilize population dynamics, avoiding local extinctions. Predator–prey and other ecosystem interactions tend to produce volatility that can create population crashes and local extinctions. Total fertility rates that exceed the ecosystem's recovery rate contribute to population overshoot, followed by collapse. These local extinctions may constitute a potent group selection mechanism, driving evolution toward controlled rates of population growth, even when there is a significant individual cost. In this paper, we consider the question: what life history characteristics support demographic homeostasis at the least cost to individual fitness? In individual‐based evolutionary simulations, we find that reduction in fertility is sufficient to avoid population instabilities leading to extinction, but that life histories that include senescence can accomplish the same thing at a lower cost to individual fitness. Furthermore, life histories that include the potential for a post‐reproductive period are yet more efficient at stabilizing population dynamics, while minimizing the impact on individual fitness.     

Resolving the life cycle alters expected impacts of climate change

Recent models predict contrasting impacts of climate change on tropical and temperate species, but these models ignore how environmental stress and organismal tolerance change during the life cycle. For example, geographical ranges and extinction risks have been inferred from thermal constraints on activity during the adult stage. Yet, most animals pass through a sessile embryonic stage before reaching adulthood, making them more susceptible to warming climates than current models would suggest. By projecting microclimates at high spatio-temporal resolution and measuring thermal tolerances of embryos, we developed a life cycle model of population dynamics for North American lizards. Our analyses show that previous models dramatically underestimate the demographic impacts of climate change. A predicted loss of fitness in 2% of the USA by 2100 became 35% when considering embryonic performance in response to hourly fluctuations in soil temperature. Most lethal events would have been overlooked if we had ignored thermal stress during embryonic development or had averaged temperatures over time. Therefore, accurate forecasts require detailed knowledge of environmental conditions and thermal tolerances throughout the life cycle.     

Natural selection, fitness entropy, and the dynamics of coevolution

The coevolutionary dynamics of interacting populations were studied by combining continuous time Lotka-Volterra models of population growth with single-locus genetic models of weak selection. The effects of natural selection on population growth were evaluated using Ginzburg's fitness entropy function as a measure of the deviation of a population's initial allele frequencies from their polymorphic equilibrium values. This entropy measure was used to relate the dynamics of a community composed of evolving populations to the dynamics of a "reference community" whose populations are initially in genetic equilibrium. Specifically, a quantity called the "selective difference area" was defined as the total difference between the population size trajectories of a reference and evolving population. The selective difference area represents the amount of extra life a species would realize if the entire community were at genetic equilibrium. It was shown that this selective difference area is a simple linear function of the initial fitness entropies of each species. This prediction is independent of the strength of selection and holds for any arbitrary set of initial population densities. Numerical examples were presented to illustrate the results. Under the assumption of weak selection, a generalization for arbitrary population growth models was outlined.     

A life‐history perspective on the demographic drivers of structured population dynamics in changing environments

Current understanding of life‐history evolution and how demographic parameters contribute to population dynamics across species is largely based on assumptions of either constant environments or stationary environmental variation. Meanwhile, species are faced with non‐stationary environmental conditions (changing mean, variance, or both) created by climate and landscape change. To close the gap between contemporary reality and demographic theory, we develop a set of transient life table response experiments (LTREs) for decomposing realised population growth rates into contributions from specific vital rates and components of population structure. Using transient LTREs in a theoretical framework, we reveal that established concepts in population biology will require revision because of reliance on approaches that do not address the influence of unstable population structure on population growth and mean fitness. Going forward, transient LTREs will enhance understanding of demography and improve the explanatory power of models used to understand ecological and evolutionary dynamics.     

Physiological dynamics,reproduction‐maintenance allocations,and life history evolution

Allocation of resources to competing processes of growth, maintenance, or reproduction is arguably a key process driving the physiology of life history trade‐offs and has been shown to affect immune defenses, the evolution of aging, and the evolutionary ecology of offspring quality. Here, we develop a framework to investigate the evolutionary consequences of physiological dynamics by developing theory linking reproductive cell dynamics and components of fitness associated with costly resource allocation decisions to broader life history consequences. We scale these reproductive cell allocation decisions to population‐level survival and fecundity using a life history approach and explore the effects of investment in reproduction or tissue‐specific repair (somatic or reproductive) on the force of selection, reproductive effort, and resource allocation decisions. At the cellular level, we show that investment in protecting reproductive cells increases fitness when reproductive cell maturation rate is high or reproductive cell death is high. At the population level, life history fitness measures show that cellular protection increases reproductive value by differential investment in somatic or reproductive cells and the optimal allocation of resources to reproduction is moulded by this level of investment. Our model provides a framework to understand the evolutionary consequences of physiological processes underlying trade‐offs and highlights the insights to be gained from considering fitness at multiple levels, from cell dynamics through to population growth.     

Feedback between Population and Evolutionary Dynamics Determines the Fate of Social Microbial Populations

The evolutionary spread of cheater strategies can destabilize populations engaging in social cooperative behaviors, thus demonstrating that evolutionary changes can have profound implications for population dynamics. At the same time, the relative fitness of cooperative traits often depends upon population density, thus leading to the potential for bi-directional coupling between population density and the evolution of a cooperative trait. Despite the potential importance of these eco-evolutionary feedback loops in social species, they have not yet been demonstrated experimentally and their ecological implications are poorly understood. Here, we demonstrate the presence of a strong feedback loop between population dynamics and the evolutionary dynamics of a social microbial gene, SUC2, in laboratory yeast populations whose cooperative growth is mediated by the SUC2 gene. We directly visualize eco-evolutionary trajectories of hundreds of populations over 50–100 generations, allowing us to characterize the phase space describing the interplay of evolution and ecology in this system. Small populations collapse despite continual evolution towards increased cooperative allele frequencies; large populations with a sufficient number of cooperators “spiral” to a stable state of coexistence between cooperator and cheater strategies. The presence of cheaters does not significantly affect the equilibrium population density, but it does reduce the resilience of the population as well as its ability to adapt to a rapidly deteriorating environment. Our results demonstrate the potential ecological importance of coupling between evolutionary dynamics and the population dynamics of cooperatively growing organisms, particularly in microbes. Our study suggests that this interaction may need to be considered in order to explain intraspecific variability in cooperative behaviors, and also that this feedback between evolution and ecology can critically affect the demographic fate of those species that rely on cooperation for their survival.     

Allele dynamics plots for the study of evolutionary dynamics in viral populations

Phylodynamic techniques combine epidemiological and genetic information to analyze the evolutionary and spatiotemporal dynamics of rapidly evolving pathogens, such as influenza A or human immunodeficiency viruses. We introduce 'allele dynamics plots' (AD plots) as a method for visualizing the evolutionary dynamics of a gene in a population. Using AD plots, we propose how to identify the alleles that are likely to be subject to directional selection. We analyze the method's merits with a detailed study of the evolutionary dynamics of seasonal influenza A viruses. AD plots for the major surface protein of seasonal influenza A (H3N2) and the 2009 swine-origin influenza A (H1N1) viruses show the succession of substitutions that became fixed in the evolution of the two viral populations. They also allow the early identification of those viral strains that later rise to predominance, which is important for the problem of vaccine strain selection. In summary, we describe a technique that reveals the evolutionary dynamics of a rapidly evolving population and allows us to identify alleles and associated genetic changes that might be under directional selection. The method can be applied for the study of influenza A viruses and other rapidly evolving species or viruses.     

Adaptive dynamics for physiologically structured population models

We develop a systematic toolbox for analyzing the adaptive dynamics of multidimensional traits in physiologically structured population models with point equilibria (sensu Dieckmann et al. in Theor. Popul. Biol. 63:309–338, 2003). Firstly, we show how the canonical equation of adaptive dynamics (Dieckmann and Law in J. Math. Biol. 34:579–612, 1996), an approximation for the rate of evolutionary change in characters under directional selection, can be extended so as to apply to general physiologically structured population models with multiple birth states. Secondly, we show that the invasion fitness function (up to and including second order terms, in the distances of the trait vectors to the singularity) for a community of N coexisting types near an evolutionarily singular point has a rational form, which is model-independent in the following sense: the form depends on the strategies of the residents and the invader, and on the second order partial derivatives of the one-resident fitness function at the singular point. This normal form holds for Lotka–Volterra models as well as for physiologically structured population models with multiple birth states, in discrete as well as continuous time and can thus be considered universal for the evolutionary dynamics in the neighbourhood of singular points. Only in the case of one-dimensional trait spaces or when N = 1 can the normal form be reduced to a Taylor polynomial. Lastly we show, in the form of a stylized recipe, how these results can be combined into a systematic approach for the analysis of the (large) class of evolutionary models that satisfy the above restrictions.      

Inference for nonlinear epidemiological models using genealogies and time series

Phylodynamics - the field aiming to quantitatively integrate the ecological and evolutionary dynamics of rapidly evolving populations like those of RNA viruses - increasingly relies upon coalescent approaches to infer past population dynamics from reconstructed genealogies. As sequence data have become more abundant, these approaches are beginning to be used on populations undergoing rapid and rather complex dynamics. In such cases, the simple demographic models that current phylodynamic methods employ can be limiting. First, these models are not ideal for yielding biological insight into the processes that drive the dynamics of the populations of interest. Second, these models differ in form from mechanistic and often stochastic population dynamic models that are currently widely used when fitting models to time series data. As such, their use does not allow for both genealogical data and time series data to be considered in tandem when conducting inference. Here, we present a flexible statistical framework for phylodynamic inference that goes beyond these current limitations. The framework we present employs a recently developed method known as particle MCMC to fit stochastic, nonlinear mechanistic models for complex population dynamics to gene genealogies and time series data in a Bayesian framework. We demonstrate our approach using a nonlinear Susceptible-Infected-Recovered (SIR) model for the transmission dynamics of an infectious disease and show through simulations that it provides accurate estimates of past disease dynamics and key epidemiological parameters from genealogies with or without accompanying time series data.     

Predicting trait values and measuring selection in complex life histories: reproductive allocation decisions in Soay sheep

Accurate prediction of life history phenomena and characterisation of selection in free-living animal populations are fundamental goals in evolutionary ecology. In density regulated, structured populations, where individual state influences fate, simple and widely used approaches based on individual lifetime measures of fitness are difficult to justify. We combine recently developed structured population modelling tools with ideas from modern evolutionary game theory (adaptive dynamics) to understand selection on allocation of female reproductive effort to singletons or twins in a size-structured population of feral sheep. In marked contrast to the classical selection analyses, our model-based approach predicts that the female allocation strategy is under negligible directional selection. These differences arise because classical selection analysis ignores components of offspring fitness and fails to consider selection over the complete life cycle.     

Modelling the Evolution of Traits in a Two-Sex Population,with an Application to Grandmothering

We present a mathematical simplification for the evolutionary dynamics of a heritable trait within a two-sex population. This trait is assumed to control the timing of sex-specific life-history events, such as the age of sexual maturity and end of female fertility, and each sex has a distinct fitness trade-off associated with the trait. We provide a formula for the fitness landscape of the population and show a natural extension of the result to an arbitrary number of heritable traits. Our method can be viewed as a dynamical systems generalisation of the Price equation to include two sexes, age structure and multiple traits. We use this formula to examine the effect of grandmothering, whereby post-fertile females subsidise their daughter’s fertility by provisioning grandchildren. Grandmothering can drive a shift towards increasingly male-biased mating sex ratios due to a post-fertile life stage in females, while male fertility continues to older ages. Our fitness landscapes show a net increase in fitness for both males and females at longer lifespans, and as a result, we find that grandmothering alone provides an evolutionary trajectory to higher longevities.     

Evolution of haploid–diploid life cycles when haploid and diploid fitnesses are not equal

Many organisms spend a significant portion of their life cycle as haploids and as diploids (a haploid–diploid life cycle). However, the evolutionary processes that could maintain this sort of life cycle are unclear. Most previous models of ploidy evolution have assumed that the fitness effects of new mutations are equal in haploids and homozygous diploids, however, this equivalency is not supported by empirical data. With different mutational effects, the overall (intrinsic) fitness of a haploid would not be equal to that of a diploid after a series of substitution events. Intrinsic fitness differences between haploids and diploids can also arise directly, for example because diploids tend to have larger cell sizes than haploids. Here, we incorporate intrinsic fitness differences into genetic models for the evolution of time spent in the haploid versus diploid phases, in which ploidy affects whether new mutations are masked. Life‐cycle evolution can be affected by intrinsic fitness differences between phases, the masking of mutations, or a combination of both. We find parameter ranges where these two selective forces act and show that the balance between them can favor convergence on a haploid–diploid life cycle, which is not observed in the absence of intrinsic fitness differences.     

The force of selection on the human life cycle

In this article, I present evidence for a robust and quite general force of selection on the human life cycle. The force of selection acts in remarkably invariant ways on human life histories, despite a great abundance of demographic diversity. Human life histories are highly structured, with mortality and fertility changing substantially through the life cycle. This structure necessitates the use of structured population models to understand human life history evolution. Using such structured models, I find that the vital rates to which fitness is most sensitive are prereproductive survival probabilities, particularly the survival of children ages 0 to 4 years. The fact that the preponderance of selection falls on transitions related to recruitment combined with the late age at first reproduction characteristic of the human life cycle creates a fitness bottleneck out of recruitment. Because of this, antagonistic pleiotropy with any trait that detracts from the constituent transitions to recruitment is expected. I explore the predictors of variation in the force of selection on early survival. High fertility increases the selective premium placed on early survivorship, whereas high life expectancy at birth decreases it.     

Evolutionary dynamics as a component of stage-structured matrix models: an example using Trillium grandiflorum

Evolution by natural selection improves fitness and may therefore influence population trajectories. Demographic matrix models are often employed in conservation studies to project population dynamics, but such analyses have not incorporated evolutionary dynamics. We project evolutionarily informed population trajectories for a population of the perennial plant Trillium grandiflorum, which is declining due to high levels of herbivory by white-tailed deer. Individuals with later flowering times are less often consumed, so there is selection on this trait. We first incorporated selection analyses into a deterministic matrix model in three ways (corresponding to different methods that have been used for analyzing evolution in structured populations). Because it is not clear which of these methods works best for stage-structured models, we compared each with a more realistic, individual-based model. Deterministic models using fitness averaged over the phenotypic distribution gave trajectories that were similar to those of the individual-based model, whereas the deterministic model using fitness at the mean phenotype gave a much faster rate of evolution than that which was observed. This illustrates that subtle differences in the way in which one splices evolution into demographic models can have a large effect on expected outcomes. This study demonstrates that, by combining demographic and selection analyses, one can gauge the potential relevance of evolution to population dynamics and persistence.