Impact of developmental variance on behavior of models for insect populations II. Models for populations with density dependent restrictions on growth

The impact of variation in developmental times on behavior of models for insect populations is investigated with special reference to models which include various types of intraspecific competition. At low densities, increases in developmental variation led to decreases in reproductive rate. At high densities, increases in developmental variation led to increases in reproductive rate. There was little change in the relationship between developmental variance and generation time as density increased.     

Developmental quantitative genetic models of evolutionary change

Discussions about evolutionary change in developmental processes or morphological structures are predicated on specific quantitative genetic models whose parameters predict whether evolutionary change can occur, its relative rate and direction, and if correlated change will occur in other related and unrelated structures. The appropriate genetic model should reflect the relevant genetical and developmental biology of the organisms, yet be simple enough in its parameters so that deductions can be made and hypotheses tested. As a consequence, the choice of the most appropriate genetic model for polygenically controlled traits is a complex tissue and the eventual choice of model is often a compromise between completeness of the model and computational expediency. Herein, we discuss several developmental quantitative genetic models for the evolution of development and morphology. The models range from the classical direct effects model to complex epigenetic models. Further, we demonstrate the algebraic equivalency of the Cowley and Atchley epigenetic model and Wagner's developmental mapping model. Finally, we propose a new multivariate model for continuous growth trajectories. The relative efficacy of these various models for understanding evolutionary change in developmental and morphological traits is discussed. © 1994 Wiley-Liss, Inc.     

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.     

Estimating developmental and mortality rates and stage recruitment from insect stage-frequency data

A model for the analysis of insect stage-frequency data is developed which includes stage-specific variable developmental periods and stage-specific daily survival rates. The model can predict the development of an insect population through its developmental stages and consequently may form the basis for a simulation model of the population.     

Stochastic modelling of environmental variation for biological populations

We examine stochastic effects, in particular environmental variability, in population models of biological systems. Some simple models of environmental stochasticity are suggested, and we demonstrate a number of analytic approximations and simulation-based approaches that can usefully be applied to them. Initially, these techniques, including moment-closure approximations and local linearization, are explored in the context of a simple and relatively tractable process. Our presentation seeks to introduce these techniques to a broad-based audience of applied modellers. Therefore, as a test case, we study a natural stochastic formulation of a non-linear deterministic model for nematode infections in ruminants, proposed by Roberts and Grenfell (1991). This system is particularly suitable for our purposes, since it captures the essence of more complicated formulations of parasite demography and herd immunity found in the literature. We explore two modes of behaviour. In the endemic regime the stochastic dynamic fluctuates widely around the non-zero fixed points of the deterministic model. Enhancement of these fluctuations in the presence of environmental stochasticity can lead to extinction events. Using a simple model of environmental fluctuations we show that the magnitude of this system response reflects not only the variance of environmental noise, but also its autocorrelation structure. In the managed regime host-replacement is modelled via periodic perturbation of the population variables. In the absence of environmental variation stochastic effects are negligible, and we examine the system response to a realistic environmental perturbation based on the effect of micro-climatic fluctuations on the contact rate. The resultant stochastic effects and the relevance of analytic approximations based on simple models of environmental stochasticity are discussed.     

Demographic stochasticity and temporal autocorrelation in the dynamics of structured populations

Populations can show temporal autocorrelation in the dynamics arising from different mechanisms, including fluctuations in the demographic structure. This autocorrelation is often treated as a complicating factor in the analyses of stochastic population growth and extinction risk. However, it also reflects important information about the demographic structure. Here, we consider how temporal autocorrelation is related to demographic stochasticity in structured populations. Demographic stochasticity arises from inherent randomness in the demographic processes of individuals, like survival and reproduction, and the resulting impact on population growth is measured by the demographic variance. Earlier studies have shown that population structure have positive or negative effects on the demographic variance compared to a model where the structure is ignored. Here, we derive a new expression for the demographic variance of a structured population, using the temporal autocorrelation function of the population growth rate. We show that the relative difference in demographic variance when the structure is included or ignored (the effect of structure on demographic variance) is approximately twice the sum of the autocorrelations. We demonstrate the result for a simple hypothetical example, as well as a set of empirical examples using age‐structured models of 24 mammals from the demographic database COMADRE. In the empirical examples, the sum of the autocorrelation function was negative in all cases, indicating that age structure generally has a negative effect on the demographic variance (i.e. the demographic variance is lower compared to that of a model where the structure is ignored). Other kinds of structure, such as spatial heterogeneity affecting fecundity, can have positive effects on the demographic variance, and the sum of the autocorrelations will then be positive. These results yield new insights into the complex interplay between population structure, demographic variance, and temporal autocorrelation, that shapes the population dynamics and extinction risk of populations.     

The influence of variation and of developmental constraints on the rate of multivariate phenotypic evolution

A model of multivariate phenotypic evolution is analysed under the assumption that all characters have the same variance or at least constant ratios of variance. The rate of evolution is examined as a function of the amount of phenotypic variance in a variety of adaptive landscapes (fitness functions). It is demonstrated that the effect of variation depends on the type of adaptive landscape. In “well behaved” adaptive landscapes the rate of evolution can theoretically increase without limits, depending on the amount of heritable phenotypic variation. However, in other adaptive landscapes there are upper limits to the rate of evolution which cannot be exceeded if phenotypic variation is developmentally unconstrained, i. e. if it is the same for all characters. Further it is shown that the maximal rate of evolution becomes small if the number of characters becomes large. Fitness functions of this type are called malignant. It is argued that malignant fitness functions are more adequate models for the evolution of typical organismic systems, because they are models of functionally interdependent characters. It is concluded that there are upper limits to the rate of phenotypic evolution if the variation of functionally interdependent characters is developmentally unconstrained. The possible role of developmental constraints in adaptive phenotypic evolution is discussed.     

Demographic variance in heterogeneous populations: matrix models and sensitivity analysis

The demographic consequences of stochasticity in processes such as survival and reproduction are modulated by the heterogeneity within the population. Therefore, to study effects of stochasticity on population growth and extinction risk, it is critical to use structured population models in which the most important sources of heterogeneity (e.g. age, size, developmental stage) are incorporated as i‐state variables. Demographic stochasticity in heterogeneous populations has often been studied using one of two approaches: multitype branching processes and diffusion approximations. Here, we link these approaches, through the demographic stochasticity in age‐ or stage‐structured matrix population models. We derive the demographic variance, σ²_d, which measures the per capita contribution to the variance in population growth increment, and we show how it can be decomposed into contributions from transition probabilities and fertility across ages or stages. Furthermore, using matrix calculus we derive the sensitivity of σ²_d to age‐ or stage‐specific mortality and fertility. We apply the methods to an extensive set of data from age‐classified human populations (long‐term time‐series for Sweden, Japan and the Netherlands; two hunter–gatherer populations, and the high‐fertility Hutterites), and to a size‐classified population of the herbaceous plant Calathea ovandensis. For the human populations our analysis reveals substantial temporal changes in the demographic variance as well as its main components across age. These new methods provide a powerful approach for calculating the demographic variance for any structured model, and for analyzing its main components and sensitivities. This will make possible new analyses of demographic variance across different kinds of heterogeneity in different life cycles, which will in turn improve our understanding of mechanisms underpinning extinction risk and other important biological outcomes.     

Song learning as an indicator mechanism: modelling the developmental stress hypothesis

The ‘developmental stress hypothesis’ attempts to provide a functional explanation of the evolutionary maintenance of song learning in songbirds. It argues that song learning can be viewed as an indicator mechanism that allows females to use learned features of song as a window on a male's early development, a potentially stressful period that may have long-term phenotypic effects. In this paper we formally model this hypothesis for the first time, presenting a population genetic model that takes into account both the evolution of genetic learning preferences and cultural transmission of song. The models demonstrate that a preference for song types that reveal developmental stress can evolve in a population, and that cultural transmission of these song types can be stable, lending more support to the hypothesis.     

The size distribution of conspecific populations: the peoples of New Guinea

The size distribution of the language populations in New Guinea, which represent over 15% of the world's languages, is analysed using models analogous to the resource division models of species abundance distribution in ecological communities. A model distribution of resource segments reflecting population size is created by repeated selection of an existing resource segment and its division into two. We found that any dependency of the selection probability on the size of the segment generated negatively skewed abundance distributions after log transformation. Asymmetric segment division further exacerbated the negative skewness. Size-independent selection produced lognormal abundance distributions, irrespective of the segment division method. Size-dependent selection and asymmetric division were deemed reasonable assumptions since large language populations are more likely to generate isolates, which develop into new populations, than small ones, and these isolates are likely to be small relative to the progenitor population. A negatively skewed distribution of the log-transformed population sizes was therefore expected. However, the observed distributions were lognormal, scale invariant for areas containing between 100 and over 1000 language populations. The dynamics of language differentiation, as reflected by the models, may therefore be unimportant relative to the effect of variable growth rates among populations. All lognormal distributions from resource division models had a higher variance than the observed one, where half of the 1053 populations had between 350 and 3000 individuals. The possible mechanisms maintaining such a low variance around a modal population size of 1000 are discussed.     

Evolving populations with overlapping generations

In this paper, we extend a previously published model of an evolving finite population of multi-locus organisms, where the dynamics of the evolving system are described by the cumulants of the population efficacy distribution. We consider the case of overlapping generations and compare it to the previously studied case where generations are discrete. In the weak selection limit, we can solve the dynamics analytically and show that the changes in population genetic variance due to stochastic effects-genetic drift-is twice as great when generations overlap. The comparison of the dynamics of the two models shows many of the features seen in the comparison, performed by Moran, of simple one-locus genetic models with overlapping and non-overlapping generations. Studying the dynamics of the two models gives some insights into these comparisons.     

A cytokinetic model for heterogeneous mammalian cell populations. I. Cell growth and cell death

A general model is proposed for describing the growth behavior of mammalian cell populations, which features:(a) a cell cycle time distribution function with properties such that mean and variance increase with increasing population size; (b) maturation age and maturation rate functions which constrain the maturational pathways of individual cells; and (c) a death rate function, where cell death is construed as irreparable damage to a cell's reproductive apparatus. The biological implications of the model are discussed, and methods for relating the model to real cell systems by means of commonly used experimental techniques are described. The model is compared with earlier models.     

Patchy populations in stochastic environments: critical number of patches for persistence

We introduce a model for the dynamics of a patchy population in a stochastic environment and derive a criterion for its persistence. This criterion is based on the geometric mean (GM) through time of the spatial-arithmetic mean of growth rates. For the population to persist, the GM has to be >/=1. The GM increases with the number of patches (because the sampling error is reduced) and decreases with both the variance and the spatial covariance of growth rates. We derive analytical expressions for the minimum number of patches (and the maximum harvesting rate) required for the persistence of the population. As the magnitude of environmental fluctuations increases, the number of patches required for persistence increases, and the fraction of individuals that can be harvested decreases. The novelty of our approach is that we focus on Malthusian local population dynamics with high dispersal and strong environmental variability from year to year. Unlike previous models of patchy populations that assume an infinite number of patches, we focus specifically on the effect that the number of patches has on population persistence. Our work is therefore directly relevant to patchily distributed organisms that are restricted to a small number of habitat patches.     

Estimates of direct and indirect effects for early juvenile survival in captive populations maintained for conservation purposes: the case of Cuvier's gazelle

Together with the avoidance of any negative impact of inbreeding, preservation of genetic variability for life‐history traits that could undergo future selective pressure is a major issue in endangered species management programmes. However, most of these programmes ignore that, apart from the direct action of genes on such traits, parents, as contributors of offspring environment, can influence offspring performance through indirect parental effects (when parental genotype and phenotype exerts environmental influences on offspring phenotype independently of additive genetic effects). Using quantitative genetic models, we estimated the additive genetic variance for juvenile survival in a population of the endangered Cuvier's gazelle kept in captivity since 1975. The dataset analyzed included performance recording for 700 calves and a total pedigree of 740 individuals. Results indicated that in this population juvenile survival harbors significant additive genetic variance. The estimates of heritability obtained were in general moderate (0.115–0.457) and not affected by the inclusion of inbreeding in the models. Maternal genetic contribution to juvenile survival seems to be of major importance in this gazelle's population as well. Indirect genetic and indirect environmental effects assigned to mothers (i.e., maternal genetic and maternal permanent environmental effects) roughly explain a quarter of the total variance estimated for the trait analyzed. These findings have major evolutionary consequences for the species as show that offspring phenotypes can evolve strictly through changes in the environment provided by mothers. They are also relevant for the captive breeding programme of the species. To take into account, the contribution that mothers have on offspring phenotype through indirect genetic effects when designing pairing strategies might serve to identify those females with better ability to recruit, and, additionally, to predict reliable responses to selection in the captive population.     

Assessing the spatial variability of density dependence in waterfowl populations

Population models commonly assume that the demographic parameters are spatially invariant, but there is considerable evidence that population growth rate (r) and the strength of density dependence (β) can vary over a species' range. To address this issue we developed a spatially explicit Gompertz population model based on the spatially varying coefficients approach to assess the spatial variation in population drivers. The model was fit to spatially stratified time series population estimates of the mallard Anas platyrhynchos in western North America. We included precipitation during the previous year and spring maximum temperature in the current year as environmental factors in the density dependent population model. Because density dependent models can give biased estimates for time series of abundance data, we fit a naïve model without informative priors and a model where we constrained the mean and variance of r to biologically realistic values that were derived via a comparative demography approach. In the naïve model, r and β were not separately identifiable and their values were overestimated, leading to unrealistic population growth. The naïve model also implied spatial variation in population r and the return time to equilibrium [1?(– β)] across the survey area. In contrast, in the informative model, r and the return time to equilibrium did not vary markedly among populations and were generally equal across populations. The effects of the climatic factors were similar across models. Population growth rates in the Prairie‐pothole region were positively correlated with precipitation, while in Alaska rates were positively correlated with spring temperature. Although it has been argued in the past that adding ecological realism could help avoid the pitfalls associated with density dependent models, our results demonstrate that imposing constraints on the population parameters is still the best course of action.     

Interspecific interaction: The analysis of complex structures in carnivorous zooplankton populations

Linear unidirectional dependence of one population upon another is rather atypical among zooplankton species, as carnivory is not only characteristic of high trophic levels but also part of omnivory in many species of copepods. The feeding on its predatorjuveniles may be important for the survival of a copepod population, more in regard to the impact on the predator than to the nutritional benefit obtained. Such complex interrelationships are important for predictive ecosystem models. Ctenophore-copepod interrelationships have been analysed in experiments, and the results produced have been used for constructing simulation models. Details of experimentation and of modelling are described and discussed.     

Recruitment variability and age structure in harvested animal populations

The sensitivity to harvesting in a discrete-time, age-structured model of an animal population is considered. Density dependence and stochasticity are confined to the first year of the life cycle. Sensitivity is characterized by (a) the characteristic return time and (b) the relative variance (coefficient of variation) of recruitment and yield. Harvesting affects the return time and relative variance through two mechanisms: (1) a displacement of the equilibrium down the stock-recruitment curve and (2) a change in the shape of the “fecundity profile.” The first mechanism occurs in models without age structure and usually has the effect of increasing both return time and relative variance. The second mechanism has opposite effects on return time and relative variance—reducing the former while in most cases increasing the latter. For this model it can be concluded that the overall effect of harvesting is to increase the relative variance of recruitment and yield. The behavior of the return time is ambiguous and is in poor agreement with that of the relative variance.     

A general mechanistic model for admixture histories of hybrid populations

Admixed populations have been used for inferring migrations, detecting natural selection, and finding disease genes. These applications often use a simple statistical model of admixture rather than a modeling perspective that incorporates a more realistic history of the admixture process. Here, we develop a general model of admixture that mechanistically accounts for complex historical admixture processes. We consider two source populations contributing to the ancestry of a hybrid population, potentially with variable contributions across generations. For a random individual in the hybrid population at a given point in time, we study the fraction of genetic admixture originating from a specific one of the source populations by computing its moments as functions of time and of introgression parameters. We show that very different admixture processes can produce identical mean admixture proportions, but that such processes produce different values for the variance of the admixture proportion. When introgression parameters from each source population are constant over time, the long-term limit of the expectation of the admixture proportion depends only on the ratio of the introgression parameters. The variance of admixture decreases quickly over time after the source populations stop contributing to the hybrid population, but remains substantial when the contributions are ongoing. Our approach will facilitate the understanding of admixture mechanisms, illustrating how the moments of the distribution of admixture proportions can be informative about the historical admixture processes contributing to the genetic diversity of hybrid populations.