Mating, births, and transitions: a flexible two-sex matrix model for evolutionary demography

A sex skew in life-history research: the problem of missing males

Life-history strategies are diverse. While understanding this diversity is a fundamental aim of evolutionary biology and biodemography, life-history data for some traits-in particular, age-dependent reproductive investment-are biased towards females. While other authors have highlighted this sex skew, the general scale of this bias has not been quantified and its impact on our understanding of evolutionary ecology has not been discussed. This review summarizes why the sexes can evolve different life-history strategies. The scale of the sex skew is then discussed and its magnitude compared between taxonomic groups, laboratory and field studies, and through time. We discuss the consequences of this sex skew for evolutionary and ecological research. In particular, this sex bias means that we cannot test some core evolutionary theory. Additionally, this skew could obscure or drive trends in data and hinder our ability to develop effective conservation strategies. We finally highlight some ways through which this skew could be addressed to help us better understand broad patterns in life-history strategies.

Reformulation of Trivers-Willard hypothesis for parental investment

The Trivers-Willard hypothesis (TWH) plays a central role in understanding the optimal investment strategies to male and female offspring. Empirical studies of TWH, however, yielded conflicting results. Here, we present models to predict optimal comprehensive multi-element parental strategies composed of primary sex ratio, brood size, resource allocation among offspring, and the resultant secondary sex ratio. Our results reveal that the optimal strategy depends on sex differences in the shape of offspring fitness function rather than in fitness variance. Also, the slope of the tangent line (through the origin) to the offspring fitness function can be used to predict the preferred offspring sex. We also briefly discuss links between the model and the empirical research. This comprehensive reformulation of TWH will offer a thorough understanding of multi-element parental investment strategies beyond the classical TWH.

A two-sex renewal model and its population projection

Population projections rely on one-sex renewal models. Consequently, changing the projection of male mortality does not affect the projection of birth, contradicting commonsense. A two-sex renewal model is presented in this paper to provide a better description of reproduction and more reasonable population projections. This model is nonlinear and includes the one-sex renewal models as special cases. In this model, age-specific birth rates are defined for two sexes jointly; total fertility, net reproduction rate, and intrinsic growth rate are also derived for two sexes jointly; and age-specific populations approach or converge to stable status. Applying the two-sex renewal model to Australia, it indicates that one-sex models underestimated the intrinsic growth rate by 14 percent. Compared to the results of one-sex models, the two-sex model would provide higher growth rate for low-fertility countries, and lower growth rate for high-fertility countries. In other words, the one-sex models are commonly biased. If the two-sex model is applied to all the countries, it would project smaller populations for the world in the future.

Poaching of African elephants indirectly decreases population growth through lowered orphan survival

Prolonged maternal care is vital to the well-being of many long-lived mammals.¹ The premature loss of maternal care, i.e., orphaning, can reduce offspring survival even after weaning is complete.^2-5 However, ecologists have not explicitly assessed how orphaning impacts population growth. We examined the impact of orphaning on population growth in a free-ranging African elephant population, using 19 years of individual-based demographic monitoring data. We compared orphan and nonorphan survival, performed a sensitivity analysis to understand how population growth responds to the probability of being orphaned and orphan survival, and investigated how sensitivity to these orphan parameters changed with level of poaching. Orphans were found to have lower survival compared to nonorphaned age mates, and population growth rate was negatively correlated with orphaning probability and positively correlated with orphan survival. This demonstrates that, in addition to its direct effects, adult elephant death indirectly decreases population growth through orphaning. Population growth rate's sensitivity to orphan survival increased for the analysis parameterized using only data from years of more poaching, indicating orphan survival is more important for population growth as orphaning increases. We conclude that orphaning substantively decreases population growth for elephants and should not be overlooked when quantifying the impacts of poaching. Moreover, we conclude that population models characterizing systems with extensive parental care benefit from explicitly incorporating orphan stages and encourage research into quantifying effects of orphaning in other social mammals of conservation concern.

Effects of size- and sex-selective harvesting: An integral projection model approach

Harvesting is often size-selective, and in species with sexual size dimorphism, it may also be sex-selective. A powerful approach to investigate potential consequences of size- and/or sex-selective harvesting is to simulate it in a demographic population model. We developed a population-based integral projection model for a size- and sex-structured species, the commonly exploited pike (Esox lucius). The model allows reproductive success to be proportional to body size and potentially limited by both sexes. We ran all harvest simulations with both lower size limits and slot limits, and to quantify the effects of selective harvesting, we calculated sex ratios and the long-term population growth rate (λ). In addition, we quantified to what degree purely size-selective harvesting was sex-selective, and determined when λ shifted from being female to male limited under size- and sex-selective harvesting. We found that purely size-selective harvest can be sex-selective, and that it depends on the harvest limits and the size distributions of the sexes. For the size- and sex-selective harvest simulations, λ increased with harvest intensity up to a threshold as females limited reproduction. Beyond this threshold, males became the limiting sex, and λ decreased as more males were harvested. The peak in λ, and the corresponding sex ratio in harvest, varied with both the selectivity and the intensity of the harvest simulation. Our model represents a useful extension of size-structured population models as it includes both sexes, relaxes the assumption of female dominance, and accounts for size-dependent fecundity. The consequences of selective harvesting presented here are especially relevant for size- and sex-structured exploited species, such as commercial fisheries. Thus, our model provides a useful contribution toward the development of more sustainable harvesting regimes.

Is there a Trivers-Willard effect for parental investment? Modelling evolutionarily stable strategies using a matrix population model with nonlinear mating

The Trivers-Willard hypothesis (TWH) states that parents in good condition preferentially produce the sex with a higher variation in reproductive success, whereas parents in bad condition favour the opposite sex. Theorists distinguish two variants of the TWH: (a) a biased sex-ratio at birth and (b) biased parental investment after birth. It has been argued before that the conditions stated by Trivers and Willard (good condition is inherited and affects reproductive success more strongly for one of the sexes) are sufficient for the sex-ratio version but insufficient for the investment version of the TWH. However, it has not yet been investigated how these conditions affect parental investment in high and low quality parents, depending on the life-cycle of a species. The present study aims to fill this gap by introducing a multi-stage matrix population model with nonlinear mating to describe the effects of parental investment after birth on the reproductive values of male and female individuals. Using methods from adaptive dynamics and evolutionary invasion analysis, evolutionary trajectories and evolutionarily stable strategies are derived for different parameterizations of the model. Simulation results demonstrate that the conditions given by Trivers and Willard produce a general bias of parental investment towards the sex with higher variance in reproductive value. This bias is stronger for low-quality parents than for high-quality parents and matches the expected marginal offspring reproductive values for parental investment.

Selection in two-sex stage-structured populations: Genetics, demography, and polymorphism

The outcome of natural selection depends on the demographic processes of birth, death, and development. Here, we derive conditions for protected polymorphism in a population characterized by age- or stage-dependent demography with two sexes. We do so using a novel two-sex matrix population model including basic Mendelian genetics (one locus, two alleles, random mating). Selection may operate on survival, growth, or fertility, any or all of which may differ between the sexes. The model can therefore incorporate genes with arbitrary pleiotropic and sex-specific effects. Conditions for protected polymorphism are expressed in terms of the eigenvalues of the linearization of the model at the homozygote boundary equilibria. We show that in the absence of sexual dimorphism, polymorphism requires heterozygote superiority in the genotypic population growth rate. In the presence of sexual dimorphism, however, heterozygote superiority is not required; an inferior heterozygote may invade, reducing the population growth rate and even leading to extinction (so-called evolutionary suicide). Our model makes no assumptions about separation of time scales between ecological and evolutionary processes, and can thus be used to project sex×stage×genotype dynamics of eco-evolutionary processes. Empirical evidence that sexual dimorphism affects extinction risk is growing, yet sex differences are often ignored in evolutionary demography and in eco-evolutionary models. Our analysis highlights the importance of sexual dimorphism and suggests mechanisms by which an allele can be favored by selection, yet drive a population to extinction, as a result of the structure and interdependence of sex- and stage-specific processes.