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.     

Threshold and stability results for an age-structured epidemic model

We study a mathematical model for an epidemic spreading in an age-structured population with age-dependent transmission coefficient. We formulate the model as an abstract Cauchy problem on a Banach space and show the existence and uniqueness of solutions. Next we derive some conditions which guarantee the existence and uniqueness for non-trivial steady states of the model. Finally the local and global stability for the steady states are examined.     

A Monte Carlo simulation model for studying evolution in age-structured populations

This model provides for any number of genotypes defined by age-specific survival and fecundity rates in a population with completely overlapping generations and growing under the control of density-governing functions affecting survival or fecundity. It is tested in situations involving two alleles at one locus. Nonselection populations at Hardy–Weinberg equilibrium obey the ecogenetic law; i.e., each genotype follows Lotka's law regarding rate of increase and stable age distribution as if it were an independent true-breeding population. Populations experiencing age- and density-independent selection approximate this situation, and the changes in gene frequency are predicted by relative fitnesses bases on λ, the finite rate of increase of the genotypes. Polymorphic gene equilibria occurring at steady-state population sizes are determined by fitnesses based on R, the net reproductive rate. In examples involving differences in generation time produced by age-dependent differences in fecundity, the allele associated with longer generation time may be favored or opposed by selection, depending on whether the density-governing factor controlling population size affects survival or fecundity. If such genotypes have similar R's, a genetic equilibrium may be established if the population is governed by a density function acting upon fecundity. Received: August 23, 1999 / Accepted: July 13, 2000     

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.     

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.     

Matrix population model with density-dependent recruitment for assessment of age-structured wildlife populations

A logistic density-dependent matrix model is developed in which the matrices contain only parameters and recruitment is a function of adult population density. The model was applied to simulate introductions of white-tailed deer into an area; the fitted model predicted a carrying capacity of 215 deer, which was close to the observed carrying capacity of 220 deer. The rate of population increase depends on the dominant eigenvalue of the Leslie matrix, and the age structure of the simulated population approaches a stable age distribution at the carrying capacity, which was similar to that generated by the Leslie matrix. The logistic equation has been applied to study many phenomena, and the matrix model can be applied to these same processes. For example, random variation can be added to life history parameters, and population abundances generated with random effects on fecundity show both the affect of annual variation in fecundity and a longer-term pattern resulting from the age structure.     

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.     

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.     

Exponential growth in age-structured two-sex populations

We consider a continuous age-structured two-sex population model which is given by a semilinear system of partial differential equations with nonlocal boundary conditions and is a simpler case of Fredrickson-Hoppensteadt model. The non-linearity is introduced by a source term, called from its physical meaning, the marriage function. The explicit form of the marriage function is not known; however, there is an understanding among the demographers about the properties it should satisfy. We have shown that the homogeneity property of the non-linearity leads to the fact that the system supports exponentially growing persistent solutions using a general form of the marriage function and its properties. This suggests that the model can be viewed as a possible extension of the one-sex stable population theory to monogamously mating two-sex populations.     

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.     

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.     

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.     

Existence, multiplicity and stability of endemic states for an age-structured S-I epidemic model

We study an S-I type epidemic model in an age-structured population, with mortality due to the disease. A threshold quantity is found that controls the stability of the disease-free equilibrium and guarantees the existence of an endemic equilibrium. We obtain conditions on the age-dependence of the susceptibility to infection that imply the uniqueness of the endemic equilibrium. An example with two endemic equilibria is shown. Finally, we analyse numerically how the stability of the endemic equilibrium is affected by the extra-mortality and by the possible periodicities induced by the demographic age-structure.     

Time-dependent solution to age-structured equations for sexual populations

The continuous age-time theory of a two-sex population, due to Fredrickson and based on M'Kendrick's equation, has been solved by means of the generation expansion. Difficulties due to the nonlinear birth condition which are encountered when the theory is cast in the form of ordinary differential equations for the total male and female populations are thereby avoided. The expansion can be expected to be useful when the time span over which the solution is required is of the order of only a few generations. Thus, it is a “short time” solution, supplementary to the “long time” solution, the steady state of exponential growth.     

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.     

A parity-structured matrix model for tsetse populations

A matrix model is used to describe the dynamics of a population of female tsetse flies structured by parity (i.e., by the number of larvae laid). For typical parameter values, the intrinsic growth rate of the population is zero when the adult daily survival rate is 0.970, corresponding to an adult life expectancy of 1/0.030 = 33.3 days. This value is plausible and consistent with results found earlier by others. The intrinsic growth rate is insensitive to the variance of the interlarval period. Temperature being a function of the time of the year, a known relationship between temperature and mean pupal and interlarval times was used to produce a time-varying version of the model which was fitted to temperature and (estimated) population data. With well-chosen parameter values, the modeled population replicated at least roughly the population data. This illustrates dynamically the abiotic effect of temperature on population growth. Given that tsetse flies are the vectors of trypanosomiasis ("sleeping sickness") the model provides a framework within which future transmission models can be developed in order to study the impact of altered temperatures on the spread of this deadly disease.