Fitness versus longevity in age-structured population dynamics

 We examine the dynamics of an age-structured population model in which the life expectancy of an offspring may be mutated with respect to that of the parent. While the total population of the system always reaches a steady state, the fitness and age characteristics exhibit counter-intuitive behavior as a function of the mutational bias. By analytical and numerical study of the underlying rate equations, we show that if deleterious mutations are favored, the average fitness of the population reaches a steady state, while the average population age is a decreasing function of the average fitness. When advantageous mutations are favored, the average population fitness grows linearly with time t, while the average age is independent of the average fitness. For no mutational bias, the average fitness grows as $t^{2/3}$. Received: 21 December 1999 / Revised version: 31 October 2001 / Published online: 14 March 2002     

The effective population size of an age-structured population with a sex-linked locus

Let a population have the same age distribution and age-specific sex ratios at times 0, 1, 2,..., and let M, F, and L, respectively, be the numbers of males and females in the youngest age group and the generation interval. It can then be shown that if there is a sex-linked locus the fixation probabilities of a neutral allele are respectively 1/3LM or 1/3LF if the allele first appears in one newborn male or in one newborn female. The effective population size can then be derived. It is the same as for a population with discrete generations having the same means, variances, and covariances of male and female progeny during a lifetime and the same number of individuals entering the population per generation.     

Decomposing variation in population growth into contributions from environment and phenotypes in an age-structured population

Evaluating the relative importance of ecological drivers responsible for natural population fluctuations in size is challenging. Longitudinal studies where most individuals are monitored from birth to death and where environmental conditions are known provide a valuable resource to characterize complex ecological interactions. We used a recently developed approach to decompose the observed fluctuation in population growth of the red deer population on the Isle of Rum into contributions from climate, density and their interaction and to quantify their relative importance. We also quantified the contribution of individual covariates, including phenotypic and life-history traits, to population growth. Fluctuations in composition in age and sex classes ((st)age structure) of the population contributed substantially to the population dynamics. Density, climate, birth weight and reproductive status contributed less and approximately equally to the population growth. Our results support the contention that fluctuations in the population's (st)age structure have important consequences for population dynamics and underline the importance of including information on population composition to understand the effect of human-driven changes on population performance of long-lived species.     

Intense selection in an age-structured population

In a population with overlapping generations, intense selection can perturb the age distribution and thus affect the rate of increase of an advantageous allele. We found that the age-specific nature of intense selection, such as that generated by many diseases, can affect the outcome of selection on loci, such as those conferring disease resistance. We also found that the temporal dynamics of selection alter the speed of evolution, particularly when selection is intense, and even more so when it is age-specific. We relate our model and results to selection for disease resistance, although the results have broader implications for inferences about past selection pressures in general.     

Existence and stability of equilibria in age-structured population dynamics

The existence of positive equilibrium solutions of the McKendrick equations for the dynamics of an age-structured population is studied as a bifurcation phenomenon using the inherent net reproductive rate n as a bifurcation parameter. The local existence and uniqueness of a branch of positive equilibria which bifurcates from the trivial (identically zero) solution at the critical value n=1 are proved by implicit function techniques under very mild smoothness conditions on the death and fertility rates as functional of age and population density. This first requires the development of a suitable linear theory. The lowest order terms in the Liapunov-Schmidt expansions are also calculated. This local analysis supplements earlier global bifurcation results of the author. The stability of both the trivial and the positive branch equilibria is studied by means of the principle of linearized stability. It is shown that in general the trivial solution losses stability as n increases through one while the stability of the branch solution is stable if and only if the bifurcation is supercritical. Thus the McKendrick equations exhibit, in the latter case, a standard exchange of stability with regard to equilibrium states as they depend on the inherent net reproductive rate. The derived lower order terms in the Liapunov-Schmidt expansions yield formulas which explicitly relate the direction of bifurcation to properties of the age-specific death and fertility rates as functionals of population density. Analytical and numerical results for some examples are given which illustrate these results.     

Evolution of cannibalism in an age-structured population

One of Bobisud's models for the evolution of cannibalism is reanalyzed by applying the method of finding evolutionarily stable strategies (or ESS's). It is demonstrated that ‘no cannibalism’ never will be an ESS if the initial rate of cannibalism is too large. It is further demonstrated that individual selection may even result in the evolution of cannibalism during food abundance. Some empirical case studies are briefly discussed in relation to this model.     

Optimal harvesting of an age-structured population

Here we investigate the optimal harvesting of an age-structured population. We use the McKendrick model of population dynamics, and optimize a discounted yield on an infinite time horizon. The harvesting function is allowed to depend arbitrarily on age and time and its magnitude is unconstrained. We obtain, in addition to existence, the qualitative result that an optimal harvesting policy consists of harvesting at no more than three distinct ages.     

On the stability of separable solutions of a sexual age-structured population dynamics model

The Sharpe-Lotka-McKendrick-von Foerster equations for non-dispersing age-sex-structured populations with a harmonic mean type mating law are considered and their separable solutions are analysed. For certain forms of the demographic rates the underlying evolution equations are reduced to systems of ODEs, the long time behavior of their solutions is studied, and the stability of separable solutions is discussed. It is found that for the constant death rates and constant sex ratio of newborns with stationary birth rates this model admits one one-parameter class of separable solutions, two such classes (repeated or different) or no such ones. In the case of special forms of age-dependent birth rates, solutions of one of these two different classes corresponding to the greater root of the characteristic equation are locally stable, solutions of the other one corresponding to the smaller root are unstable, and the population dies out if the model does not admit separable solutions or if initial densities of newborns are small enough in the case of the existence of separable solutions. In the case of constant vital rates, the model has no separable solutions or admits only one class of such ones that are globally stable.     

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.     

Dynamics of a harvested moose population in a variable environment

1. Population size, calves per female, female mean age and adult sex ratio of a moose ( Alces alces ) population in Vefsn, northern Norway were reconstructed from 1967 to 1993 using cohort analysis and catch-at-age data from 96% (6752) of all individuals harvested.
2. The dynamics of the population were influenced mainly by density-dependent harvesting, stochastic variation in climate and intrinsic variation in the age-structure of the female segment of the population.
3. A time delay in the assignment of hunting permits in relation to population size increased fluctuations in population size.
4. Selective harvesting of calves and yearlings increased the mean age of adult females in the population, and, because fecundity in moose is strongly age-specific, the number of calves per female concordantly increased. However, after years with high recruitment, the adult mean age decreased as large cohorts entered the adult age-groups. This age-structure effect generated cycles in the rate of recruitment to the population and fluctuations introduced time-lags in the population dynamics.
5. An inverse relationship between recruitment rate and population density, mediated by a density-dependent decrease in female body condition, could potentially have constituted a regulatory mechanism in the dynamics of the population, but this effect was counteracted by a density-dependent increase in the mean age of adult females.
6. Stochastic variation in winter snow depth and summer temperature had delayed effects on recruitment rate and in turn population growth rate, apparently through effects on female body condition before conception.     

Kin selection and natal dispersal in an age-structured population

We examine the effect of iteroparity on the evolution of dispersal for a species living in a stable but fragmented habitat. We use a kin selection model that incorporates the effects of demographic stochasticity on the local age structure and age-specific genetic identities. We consider two cases: when the juvenile dispersal rate is allowed to change with maternal age and when it is not. In the latter case, we find that the unconditional evolutionarily stable dispersal rate increases when the adult survival rate increases. Two antagonistic forces act upon the evolution of age-specific dispersal rates. First, when the local age structure varies between patches of habitat, the intensity of competition between adults and juveniles in the natal patch is, on average, lower for offspring born to older senescent mothers. This selects for decreasing dispersal with maternal age. Second, offspring born to older parents are on average more related to other juveniles in the same patch and they experience a higher intensity of kin competition, which selects for increasing dispersal with maternal age. We show that the evolutionary outcome results from a balance between these two opposing forces, which depends on the amount of variance in age structure among sub-populations.     

Senescence and antibiotic resistance in an age-structured population model

Different theories have been proposed to understand the growing problem of antibiotic resistance of microbial populations. Here we investigate a model that is based on the hypothesis that senescence is a possible explanation for the existence of so-called persister cells which are resistant to antibiotic treatment. We study a chemostat model with a microbial population which is age-structured and show that if the growth rates of cells in different age classes are sufficiently close to a scalar multiple of a common growth rate, then the population will globally stabilize at a coexistence steady state. This steady state persists under an antibiotic treatment if the level of antibiotics is below a certain threshold; if the level exceeds this threshold, the washout state becomes a globally attracting equilibrium.     

Effective size of a fluctuating age-structured population

Previous theories on the effective size of age-structured populations assumed a constant environment and, usually, a constant population size and age structure. We derive formulas for the variance effective size of populations subject to fluctuations in age structure and total population size produced by a combination of demographic and environmental stochasticity. Haploid and monoecious or dioecious diploid populations are analyzed. Recent results from stochastic demography are employed to derive a two-dimensional diffusion approximation for the joint dynamics of the total population size, N, and the frequency of a selectively neutral allele, p. The infinitesimal variance for p, multiplied by the generation time, yields an expression for the effective population size per generation. This depends on the current value of N, the generation time, demographic stochasticity, and genetic stochasticity due to Mendelian segregation, but is independent of environmental stochasticity. A formula for the effective population size over longer time intervals incorporates deterministic growth and environmental stochasticity to account for changes in N.     

Genetic variability at neutral markers,quantitative trait land trait in a subdivided population under selection

Genetic variability in a subdivided population under stabilizing and diversifying selection was investigated at three levels: neutral markers, QTL coding for a trait, and the trait itself. A quantitative model with additive effects was used to link genotypes to phenotypes. No physical linkage was introduced. Using an analytical approach, we compared the diversity within deme (H(S)) and the differentiation (F(ST)) at the QTL with the genetic variance within deme (V(W)) and the differentiation (Q(ST)) for the trait. The difference between F(ST) and Q(ST) was shown to depend on the relative amounts of covariance between QTL within and between demes. Simulations were used to study the effect of selection intensity, variance of optima among demes, and migration rate for an allogamous and predominantly selfing species. Contrasting dynamics of the genetic variability at markers, QTL, and trait were observed as a function of the level of gene flow and diversifying selection. The highest discrepancy among the three levels occurred under highly diversifying selection and high gene flow. Furthermore, diversifying selection might cause substantial heterogeneity among QTL, only a few of them showing allelic differentiation, while the others behave as neutral markers.     

Stochastic search variable selection for identifying multiple quantitative trait loci

In this article, we utilize stochastic search variable selection methodology to develop a Bayesian method for identifying multiple quantitative trait loci (QTL) for complex traits in experimental designs. The proposed procedure entails embedding multiple regression in a hierarchical normal mixture model, where latent indicators for all markers are used to identify the multiple markers. The markers with significant effects can be identified as those with higher posterior probability included in the model. A simple and easy-to-use Gibbs sampler is employed to generate samples from the joint posterior distribution of all unknowns including the latent indicators, genetic effects for all markers, and other model parameters. The proposed method was evaluated using simulated data and illustrated using a real data set. The results demonstrate that the proposed method works well under typical situations of most QTL studies in terms of number of markers and marker density.     

Segregation of a population in an environment

A mathematical model for spatial patterns and the segregation of a population is presented. Individuals in a population are assumed to move at random under the influence of a given environment potential V(x). The notion of kinetic excitation K(x) and intensity excitation Q(x) of a population is introduced. Then equilibrium states of a population are defined through a macroscopic relation K(x) + Q(x) + V(x) = constant. The problem of finding out equilibrium distributions is reduced to an eigenvalue problem. It is shown that a population is segregated by the nodal surfaces of the eigenfunctions, if it is excited. Some applications of the model to biological and ecological problems are indicated.     

Dynamics of an age-structured population with Allee effects and harvesting

We propose a discrete-time, age-structured population model to study the impact of Allee effects and harvesting. It is assumed that survival probabilities from one age class to the next are constants and fertility rate is a function of weighted total population size. Global extinction is certain if the maximal growth rate of the population is less than one. The model can have multiple attractors and the asymptotic dynamics of the population depend on its initial distribution if the maximal growth rate is larger than one. An Allee threshold depending on the components of the unstable interior equilibrium is derived when only the last age class can reproduce. The population becomes extinct if its initial population distribution is below the threshold. Harvesting on any particular age class can decrease the magnitude of the possible stable interior equilibrium and increase the magnitude of the unstable interior equilibrium simultaneously.     

Dynamics of an age-structured population with Allee effects and harvesting

We propose a discrete-time, age-structured population model to study the impact of Allee effects and harvesting. It is assumed that survival probabilities from one age class to the next are constants and fertility rate is a function of weighted total population size. Global extinction is certain if the maximal growth rate of the population is less than one. The model can have multiple attractors and the asymptotic dynamics of the population depend on its initial distribution if the maximal growth rate is larger than one. An Allee threshold depending on the components of the unstable interior equilibrium is derived when only the last age class can reproduce. The population becomes extinct if its initial population distribution is below the threshold. Harvesting on any particular age class can decrease the magnitude of the possible stable interior equilibrium and increase the magnitude of the unstable interior equilibrium simultaneously.