Time delays in age-structured populations

A combination of analytical and computational techniques is employed to investigate age-structured populations in which the life cycle consists of two sequential demographic phases. Individuals within each phase have identical demographic rates that are functions of population size, but these rates may differ between phases. A model consisting of a system of delay ordinary differential equations is derived, and existence and stability of equilibria are discussed. Analysis reveals how equilibrium abundances depend on all demographic variables and, in particular, on the lengths of the demographic phases.     

Kin selection in age-structured populations

There have been several discussions in the literature as to how to weight interactions between individuals of different ages in models of kin selection. It has commonly been assumed that the reproductive value of a given age is the most appropriate weight, for the purpose of calculating its contribution to inclusive fitness. This paper analyses a model of kin selection in an age-structured population. It is shown that reproductive value is relevant to behavioural interactions involving effects on survival, although the reproductive value of a given age does not provide an exact weighting of its fitness contribution in either discrete- or continuous-time populations. Reproductive value is not relevant to interactions involving effects on fecundity. The results are discussed in relation to observations on behavioural asymmetries involving age differences.     

Measures of variability in age-structured populations

An expression for the entropy of a population was derived in Demetrius (1974) by using a variational principle argument. This entropy measure is precisely the information content of the distribution in the ages of reproducing individuals in a stationary population. This paper introduces another expression for the entropy by considering the variation in the ages at which offspring will be produced by newborn individuals.The relation between these two measures of entropy and their biological significance are discussed.     

Estimating fluctuating selection in age-structured populations

In age-structured populations, viability and fecundity selection of varying strength may occur in different age classes. On the basis of an original idea by Fisher of weighting individuals by their reproductive value, we show that the combined effect of selection on traits at different ages acts through the individual reproductive value defined as the stochastic contribution of an individual to the total reproductive value of the population the following year. The selection differential is a weighted sum of age-specific differentials that are the covariances between the phenotype and the age-specific relative fitness defined by the individual reproductive value. This enables estimation of weak selection on a multivariate quantitative character in populations with no density regulation by combinations of age-specific linear regressions of individual reproductive values on the traits. Demographic stochasticity produces random variation in fitness components in finite samples of individuals and affects the statistical inference of the temporal average directional selection as well as the magnitude of fluctuating selection. Uncertainties in parameter estimates and test power depend strongly on the demographic stochasticity. Large demographic variance results in large uncertainties in yearly estimates of selection that complicates detection of significant fluctuating selection. The method is illustrated by an analysis of age-specific selection in house sparrows on a fitness-related two-dimensional morphological trait, tarsus length and body mass of fledglings.     

Sex-linked genes in age-structured populations.

We study the progress towards equilibrium of the frequencies of sex-linked genes in elementary discrete time models of age-structured, overlapping generation populations. It is found that, if a finite upper age limit is assumed, the difference in the frequencies of an allele in males and females will oscillate as in the familiar non-overlapping generation models, although the oscillations may be irregular. Monotonic convergence of that difference, as found by Nagylaki (1975) in continuous-time overlapping generation models without age-structure, occurs in the models considered here only when there is no upper age limit and when there is "sufficient" overlap of generations.     

On two-sex models leading to stable populations

A mathematical, 2-sex, stable-population model that treats sex and age simultaneously was developed. The birth function is expressed in the form of an integral formula which when solved yields the intrinsic growth rate. Some of the concepts involved include the intrinsic sex ratio, the intrinsic age-specific birthrates, and the gross and net reproduction rates for both sexes. The new model demonstrates that both sexes can coexist in a stable population, whereas in the 1-sex model the intrinsic rates are internally inconsistent in regard to sex. 7 forms of the model are discussed and applied to data for the United States from 1963.     

The dynamics of hierarchical age-structured populations

An age-structured population is considered in which the birth and death rates of an individual of age a is a function of the density of individuals older and/or younger than a. An existence/uniqueness theorem is proved for the McKendrick equation that governs the dynamics of the age distribution function. This proof shows how a decoupled ordinary differential equation for the total population size can be derived. This result makes a study of the population's asymptotic dynamics (indeed, often its global asymptotic dynamics) mathematically tractable. Several applications to models for intra-specific competition and predation are given.     

Disease regulation of age-structured host populations

A lethal, contagious disease can generate a density-dependent regulation of its host, provided the hosts' contact rate grows with population size. The condition for disease-induced population control is that the expected number of offspring of an infected newborn be less than one. In vertebrates that acquired immunity if they survive infection, the disease changes the age structure of its host population. The steady-state age structure of a disease-regulated host with age-dependent fecundity is computed. Local stability analysis indicates that the equilibrium age structure is always stable. However, when the usual exponentially distributed duration of the disease is replaced by a constant duration, the population can exhibit oscillations with a long period.     

Selection against demographic stochasticity in age-structured populations

It has been shown that differences in fecundity variance can influence the probability of invasion of a genotype in a population; i.e., a genotype with lower variance in offspring number can be favored in finite populations even if it has a somewhat lower mean fitness than a competitor. In this article, Gillespie's results are extended to population genetic systems with explicit age structure, where the demographic variance (variance in growth rate) calculated in the work of Engen and colleagues is used as a generalization of "variance in offspring number" to predict the interaction between deterministic and random forces driving change in allele frequency. By calculating the variance from the life-history parameters, it is shown that selection against variance in the growth rate will favor a genotypes with lower stochasticity in age-specific survival and fertility rates. A diffusion approximation for selection and drift in a population with two genotypes with different life-history matrices (and therefore different mean growth rates and demographic variances) is derived and shown to be consistent with individual-based simulations. It is also argued that for finite populations, perturbation analyses of both the mean and the variance in growth rate may be necessary to determine the sensitivity of fitness to changes in the life-history parameters.     

Optimal lineage principle for age-structured populations

We present a formulation of branching and aging processes that allows age distributions along lineages to be studied within populations, and provides a new interpretation of classical results in the theory of aging. We establish a variational principle for the stable age distribution along lineages. Using this optimal lineage principle, we show that the response of a population's growth rate to age-specific changes in mortality and fecundity--a key quantity that was first calculated by Hamilton--is given directly by the age distribution along lineages. We apply our method also to the Bellman-Harris process, in which both mother and progeny are rejuvenated at each reproduction event, and show that this process can be mapped to the classic aging process such that age statistics in the population and along lineages are identical. Our approach provides both a theoretical framework for understanding the statistics of aging in a population, and a new method of analytical calculations for populations with age structure. We discuss generalizations for populations with multiple phenotypes, and more complex aging processes. We also provide a first experimental test of our theory applied to bacterial populations growing in a microfluidics device.     

The effect of skewed distributions of vital statistics on growth of age-structured populations

Given that the variance of vital statistics can influence population projections, it seems reasonable that positive skew as observed for the distribution of larval survivorship of spruce budworms might also have a significant effect on stochastic projections of population growth. Simulations of population growth, using variable survivorship for a single age class, demonstrate that shape of the distribution of survivorship influences the outcome of stochastic population growth, and therefore is important for evolutionary and ecological theory. Unfortunately, empirical distributions of survivorships or fecundities for single life history stages are rare in the current literature.     

Optimal reproductive strategies in age-structured populations of zooplankton

SUMMARY. We describe a model of zooplankton population dynamics that accounts for differences in mortality and physiology among animals of different ages or sizes. The model follows changes in numbers of individuals and changes in individual and egg biomass through time and it expresses mortality and net assimilation as functions of animal size.
We investigated the effect of egg size, age at first reproduction, and size at first reproduction on the per capita growth rates of populations growing under different conditions. In the absence of predation or when exposed to vertebrate predators that prefer large prey, populations achieve maximum growth rates when animals hatch from small eggs and reach maturity quickly at small sizes. Populations exposed to invertebrate predators that concentrate on small animals may increase r in two different ways. One way is for animals to increase juvenile survivorship by hatching from large eggs and by shortening the juvenile period. An alternative strategy is for animals to hatch from small eggs and to postpone maturity until they grow beyond the range of sizes available to their predators. Certain life history strategies maximize r if animals continue to grow after they reach maturity. By growing larger, non-primiparous females are able to hatch larger clutches and thereby increase the overall rate of population growth.
The model analysis shows how to assess age-dependent mortality rates from field data. The net rate of population increase and the age distribution of eggs together provide specific, quantitative information about mortality.     

Estimating density dependence in time-series of age-structured populations

For a life history with age at maturity alpha, and stochasticity and density dependence in adult recruitment and mortality, we derive a linearized autoregressive equation with time-lags of from 1 to alpha years. Contrary to current interpretations, the coefficients for different time-lags in the autoregressive dynamics do not simply measure delayed density dependence, but also depend on life-history parameters. We define a new measure of total density dependence in a life history, D, as the negative elasticity of population growth rate per generation with respect to change in population size, D = - partial differential lnlambda(T)/partial differential lnN, where lambda is the asymptotic multiplicative growth rate per year, T is the generation time and N is adult population size. We show that D can be estimated from the sum of the autoregression coefficients. We estimated D in populations of six avian species for which life-history data and unusually long time-series of complete population censuses were available. Estimates of D were in the order of 1 or higher, indicating strong, statistically significant density dependence in four of the six species.     

The optimal sex ratio for age-structured populations

A new nonlinear age-dependent model for age-structured sexual populations is introduced, based on two assumptions: (1) the birth function depends on the ages of the two parents; and (2) the death functions of the two sexes are composed of two types of additive terms depending on age and sex and on time evolution of population densities, respectively. Formal arguments are given that suggest that time-persistent age profiles may exist and that the intrinsic rate of growth for the two sexes is the same. If the ratio between the number of newborn females and the number of newborn males is equal to the square root of the ratio of the corresponding per capita birth rates, then the intrinsic rate of growth has an optimal value. The optimal sex ratio for the whole population is equal to the reciprocal value of the sex ratio at birth.     

Competition between juveniles and adults in age-structured populations

The effect of competition between juveniles and adults is examined in a generalized, two-age-class, discrete-time model. Adult fecundity and juvenile survival are functions of both age-class densities. Possible configurations of the zero growth isoclines are examined, giving special attention to the isocline shapes, the number of equilibria, and the manner in which the population approaches these equilibria. It is found that small increases in the density of one age class may have either a positive or a negative effect on recruitment into the other class, depending upon the degree of density dependence in fecundity and survival. Closely allied to this, an increase in the resources for a given age class may result in either an increase or a decrease in its equilibrium density. Strong juvenile-adult competition generally has destabilizing effects on the population's equilibrium, with the system being more sensitive to juveniles competing with adults than to the reverse.     

Stability effects of a juvenile period in age-structured populations

Prior theoretical studies have shown that the juvenile period's length is an important determinant of local stability in age-structured population dynamics. For example, both short and long periods produce stability, but intermediate lengths can cause instability. Short juvenile periods significantly increase stability (compared to no juvenile period) if fecundity is independent of adult age. Here I re-examine these and other patterns, using a model which includes a variable juvenile period, juvenile mortality, density-dependent fecundity and adult mortality, and age-dependence is adult fecundity. Among other things, the results confirm the stable-unstable-stable pattern with increasing juvenile period length, but show that the stabilizing effect of short periods disappears when fecundity varies with adult age. Broadly speaking, the results suggest that age-dependence in adult fecundity has important dynamical consequences, and that models assuming that fecundity is independent of adult age may be unreliable guides to the dynamics of populations for which this assumption is not reasonably accurate.     

Random deaths in a computational model for age-structured populations

Summary The concept of random deaths in a computational model for population dynamics is critically examined. We claim that it is just an artifact, albeit useful, of computational models to limit the size of the populations through the use of the socalled Verhulst factor and has no biological foundation. Alternative implementations of random deaths strategies are discussed and compared.     

On the optimal harvesting of persistent age-structured populations

Summary This paper discusses optimal harvesting policies for age-structured populations harvested with effort independent of age.