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.     

Simple assumptions on age composition lead to erroneous conclusions on the nature of density dependence in age-structured populations

Ecologists have debated the nature of density dependence in natural populations for decades, and efforts to detect density dependence from time series of abundance data have paralleled these debates. Yet due to the correlative nature of time series data, these undertakings have been statistically problematic. Most analyses of density dependence have focused on simple population models (i.e., non-overlapping generations), but in reality most vertebrates exhibit more complex life histories, and this complexity has been incorporated into population models in a variety of ways. Unfortunately, adding complexity to population models can further exacerbate efforts to detect density dependence. We examined the effect of adding age structure when inadequate data exist in support; to demonstrate this effect, we adopted Pacific salmon (Oncorhynchus spp.) as our study organism. Most salmon populations are semelparous and have variable age at maturity. Salmon populations (and many other fish species populations) are typically modeled in terms of numbers of recruits arising from spawners in a given brood year. Recruits are enumerated as they return as adults to spawn, and proper assignment of recruits to brood year requires age information. Unfortunately, while adult counts are common, detailed age information is not. A common practice is to apply long-term averages of age composition to returning adults to "reconstruct" time series of recruits. Here, by conducting simulations and analyzing data from natural populations, we demonstrated that this practice leads to a biased portrayal of density dependence by overestimating recruits from small spawning classes and underestimating recruits from large spawning classes. Also, productivity was overestimated and variance was underestimated, which could lead to overly optimistic predictions of extinction risk or overharvesting.     

An experimental field study of delayed density dependence in natural populations of Aedes albopictus

Aedes albopictus, a species known to transmit dengue and chikungunya viruses, is primarily a container-inhabiting mosquito. The potential for pathogen transmission by Ae. albopictus has increased our need to understand its ecology and population dynamics. Two parameters that we know little about are the impact of direct density-dependence and delayed density-dependence in the larval stage. The present study uses a manipulative experimental design, under field conditions, to understand the impact of delayed density dependence in a natural population of Ae. albopictus in Raleigh, North Carolina. Twenty liter buckets, divided in half prior to experimentation, placed in the field accumulated rainwater and detritus, providing oviposition and larval production sites for natural populations of Ae. albopictus. Two treatments, a larvae present and larvae absent treatment, were produced in each bucket. After five weeks all larvae were removed from both treatments and the buckets were covered with fine mesh cloth. Equal numbers of first instars were added to both treatments in every bucket. Pupae were collected daily and adults were frozen as they emerged. We found a significant impact of delayed density-dependence on larval survival, development time and adult body size in containers with high larval densities. Our results indicate that delayed density-dependence will have negative impacts on the mosquito population when larval densities are high enough to deplete accessible nutrients faster than the rate of natural food accumulation.     

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.     

Cowpox virus infection in natural field vole Microtus agrestis populations: delayed density dependence and individual risk

1. Little is known about the dynamics of pathogen (microparasite) infection in wildlife populations, and less still about sources of variation in the risk of infection. Here we present the first detailed analysis of such variation. 2. Cowpox virus is an endemic sublethal pathogen circulating in populations of wild rodents. Cowpox prevalence was monitored longitudinally for 2 years, in populations of field voles exhibiting multiannual cycles of density in Kielder Forest, UK. 3. The probability that available susceptible animals seroconverted in a given trap session was significantly positively related to host density with a 3-month time lag. 4. Males were significantly more likely to seroconvert than females. 5. Despite most infection being found in young animals (because transmission rates were generally high) mature individuals were more likely to seroconvert than immature ones, suggesting that behavioural or physiological changes associated with maturity contribute to variation in infection risk. 6. Hence, these analyses confirm that there is a delayed numerical response of cowpox infection to vole density, supporting the hypothesis that endemic pathogens may play some part in shaping vole cycles.     

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.     

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.     

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.     

Predicting fluctuations of reintroduced ibex populations: the importance of density dependence, environmental stochasticity and uncertain population estimates

1. Development of population projections requires estimates of observation error, parameters characterizing expected dynamics such as the specific population growth rate and the form of density regulation, the influence of stochastic factors on population dynamics, and quantification of the uncertainty in the parameter estimates. 2. Here we construct a Population Prediction Interval (PPI) based on Bayesian state space modelling of future population growth of 28 reintroduced ibex populations in Switzerland that have been censused for up to 68 years. Our aim is to examine whether the interpopulation variation in the precision of the population projections is related to differences in the parameters characterizing the expected dynamics, in the effects of environmental stochasticity, in the magnitude of uncertainty in the population parameters, or in the observation error. 3. The error in the population censuses was small. The median coefficient of variation in the estimates across populations was 5.1%. 4. Significant density regulation was present in 53.6% of the populations, but was in general weak. 5. The width of the PPI calculated for a period of 5 years showed large variation among populations, and was explained by differences in the impact of environmental stochasticity on population dynamics. 6. In spite of the high accuracy in population estimates, the uncertainty in the parameter estimates was still large. This uncertainty affected the precision in the population predictions, but it decreased with increasing length of study period, mainly due to higher precision in the estimates of the environmental variance in the longer time-series. 7. These analyses reveal that predictions of future population fluctuations of weakly density-regulated populations such as the ibex often become uncertain. Credible population predictions require that this uncertainty is properly quantified.     

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.     

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.     

Comparative statics and stochastic dynamics of age-structured populations

Arguments from the comparative statics of populations with fixed vital rates are of limited use in studying age-structured populations subject to stochastically varying vital rates. In an age-structured population that experiences a sequence of independently and identically distributed Leslie matrices, the expectation of the Malthusian parameters of the Leslie matrices has no exact interpretation either as the ensemble average of the long-run rate of growth of each sample path of the population (Eq. (3)) or as the long-run rate of growth of the ensemble average of total population size (Eq. (4)). On the other hand, the Malthusian parameter of the expectation of a sequence of Leslie matrices is exactly the logarithm of the finite growth rate of the ensemble average of total population size when Leslie matrices are independently and identically distributed (though not in general when Leslie matrices are sequentially dependent). These observations appear to contradict the claims of a recent study using computer simulation of age-structured populations with stochastically varying vital rates.     

A perfect storm: the combined effects on population fluctuations of autocorrelated environmental noise, age structure, and density dependence

While it is widely appreciated that climate can affect the population dynamics of various species, a mechanistic understanding of how climate interacts with life-history traits to influence population fluctuations requires development. Here we build a general density-dependent age-structured model that accounts for differential responses in life-history traits to increasing population density. We show that as the temporal frequency of favorable environmental conditions increases, population fluctuations also increase provided that unfavorable environmental conditions still occur. As good years accumulate and the number of individuals in a population increases, successive life-history traits become vulnerable to density dependence once a return to unfavorable conditions prevails. The stronger this ratcheting of density dependence in life-history traits by autocorrelated climatic conditions, the larger the population fluctuations become. Highly fecund species, and those in which density dependence occurs in juvenile and adult vital rates at similar densities, are most sensitive to increases in the frequency of favorable conditions. Understanding the influence of global warming on temporal correlation in regional environmental conditions will be important in identifying those species liable to exhibit increased population fluctuations that could lead to their extinction.     

Population cycles produced by delayed density dependence in an annual plant

In contrast to insect and animal populations, little attention has been directed to the study of cycles in plant populations. It has been argued on theoretical grounds that plants present stable dynamics. Nevertheless, there are examples where plant populations appear to exhibit oscillatory dynamics, but the oscillatory signal is variable and comes from very short time series data. Using a combination of time series, models, and empirical results, we present evidence of population cycles for Descurania sophia in a 16-year field experiment. Endogenous and exogenous causal mechanisms were studied to identify processes underlying this temporal dynamic. Our results show a 4-year cycle produced by delayed density dependence. We suggest that high nutrient levels might be responsible for the observed dynamics of D. sophia. Our results suggest that although plant population dynamics may be stabilized by direct density dependence, delayed density dependence could destabilize dynamics.