Building integral projection models with nonindependent vital rates

Population dynamics are functions of several demographic processes including survival, reproduction, somatic growth, and maturation. The rates or probabilities for these processes can vary by time, by location, and by individual. These processes can co‐vary and interact to varying degrees, e.g., an animal can only reproduce when it is in a particular maturation state. Population dynamics models that treat the processes as independent may yield somewhat biased or imprecise parameter estimates, as well as predictions of population abundances or densities. However, commonly used integral projection models (IPMs) typically assume independence across these demographic processes. We examine several approaches for modelling between process dependence in IPMs and include cases where the processes co‐vary as a function of time (temporal variation), co‐vary within each individual (individual heterogeneity), and combinations of these (temporal variation and individual heterogeneity). We compare our methods to conventional IPMs, which treat vital rates independent, using simulations and a case study of Soay sheep (Ovis aries). In particular, our results indicate that correlation between vital rates can moderately affect variability of some population‐level statistics. Therefore, including such dependent structures is generally advisable when fitting IPMs to ascertain whether or not such between vital rate dependencies exist, which in turn can have subsequent impact on population management or life‐history evolution.     

Variability of important vital rates challenges the demographic buffering hypothesis

Selection is assumed to eliminate life-histories showing high variability in vital rates that have the greatest influence on population performance. Therefore, an inverse variability-importance relationship of vital rates is believed to be a universal pattern for diverse life-histories. We tested for such a relationship using multi-year demographic data on a large number of populations of two perennial plant species. Applying different approaches, we first examined the overall variability-importance relationship for the average main vital rates (survival, growth, retrogression, fecundity) per species, and then separately for each population. We found an overall inverse relationship between temporal variation and importance of the average main vital rates for both study species, but these negative species-level correlations were mainly caused by different scales of the examined vital rates. When variability-importance relationships were examined across individual demographic transitions within populations, the abundance of positive and negative correlations depended largely on the method used, and positive correlations were more common after correcting vital rates for sampling variation than when using uncorrected vital rates. Our results cast doubt on the generality of the demographic buffering hypothesis, suggesting that the inverse variability-importance relationship may not be a universal pattern when vital rates are examined for multiple populations of the same plant species.     

Disease transmission models with density-dependent demographics

The models considered for the spread of an infectious disease in a population are of SIRS or SIS type with a standard incidence expression. The varying population size is described by a modification of the logistic differential equation which includes a term for disease-related deaths. The models have density-dependent restricted growth due to a decreasing birth rate and an increasing death rate as the population size increases towards its carrying capacity. Thresholds, equilibria and stability are determined for the systems of ordinary differential equations for each model. The persistence of the infectious disease and disease-related deaths can lead to a new equilibrium population size below the carrying capacity and can even cause the population to become extinct.Research supported in part by Centers for Disease Control contract 200-87-0515     

Coexistence in competition models with density-dependent mortality

We consider a two-competitor/one-prey model in which both competitors exhibit a general functional response and one of the competitors exhibits a density-dependent mortality rate. It is shown that the two competitors can coexist upon the single prey. As an example, we consider a two-competitor/one-prey model with a Holling II functional response. Our results demonstrate that density-dependent mortality in one of the competitors can prevent competitive exclusion. Moreover, by constructing a Liapunov function, the system has a globally stable positive equilibrium.     

Evaluating the demographic buffering hypothesis with vital rates estimated for Weddell seals from 30 years of mark-recapture data

1. Life-history theory predicts that those vital rates that make larger contributions to population growth rate ought to be more strongly buffered against environmental variability than are those that are less important. Despite the importance of the theory for predicting demographic responses to changes in the environment, it is not yet known how pervasive demographic buffering is in animal populations because the validity of most existing studies has been called into question because of methodological deficiencies. 2. We tested for demographic buffering in the southern-most breeding mammal population in the world using data collected from 5558 known-age female Weddell seals over 30 years. We first estimated all vital rates simultaneously with mark-recapture analysis and then estimated process variance and covariance in those rates using a hierarchical Bayesian approach. We next calculated the population growth rate's sensitivity to changes in each of the vital rates and tested for evidence of demographic buffering by comparing properly scaled values of sensitivity and process variance in vital rates. 3. We found evidence of positive process covariance between vital rates, which indicates that all vital rates are affected in the same direction by changes in annual environment. Despite the positive correlations, we found strong evidence that demographic buffering occurred through reductions in variation in the vital rates to which population growth rate was most sensitive. Process variation in vital rates was inversely related to sensitivity measures such that variation was greatest in breeding probabilities, intermediate for survival rates of young animals and lowest for survival rates of older animals. 4. Our work contributes to a small but growing set of studies that have used rigorous methods on long-term, detailed data to investigate demographic responses to environmental variation. The information from these studies improves our understanding of life-history evolution in stochastic environments and provides useful information for predicting population responses to future environmental change. Our results for an Antarctic apex predator also provide useful baselines from a marine ecosystem when its top- and middle-trophic levels were not substantially impacted by human activity.     

Differences in vital demographic rates in three populations of the epiphytic bromeliad,Werauhia sanguinolenta

Since the response to differences in resource availability is most pronounced in smaller individuals of vascular epiphytes such as Werauhia sanguinolenta Cogn. et Marchal (Bromeliaceae), I expected variation in growth and survival of small individuals to play an important role in the dynamics of entire populations. Four annual censuses (2002–2005) of three study populations, which were located across the isthmus of Panama, allowed me to construct stage transition matrices, and to conduct growth analysis and elasticity analysis. Differences between populations were highly consistent through time, but, contrary to expectations, hardly related to the comportment of smaller plants. For example, although average mortality rates were highest at the driest site, close to the Pacific, small plants were not predominantly affected. Similarly, although the highest relative growth rates (RGR) of individuals and the highest population growth rates (λ) were found in the population with the highest moisture input, which was located close to the Atlantic coast, this was not due to a particularly strong stimulation of RGR in small plants. Elasticity analysis indicated rather small differences in the importance of the three demographic processes growth, survival, and reproduction for population growth in the three populations, but invariably identified the survival of large tanks as the single most important process determining λ.     

Incorporating the temporal autocorrelation of demographic rates into structured population models

Population dynamics are typically temporally autocorrelated: population sizes are positively or negatively correlated with past population sizes. Previous studies have found that positive temporal autocorrelation increases the risk of extinction due to ‘inertia’ that prolongs downward fluctuations in population size. However, temporal autocorrelation has not yet been analyzed at the level of life cycle transitions. We developed an R package, colorednoise, which creates stochastic matrix population projections with distinct temporal autocorrelation values for each matrix element. We used it to analyze long-term demographic data on 25 populations from the COMADRE and COMPADRE databases and simulate their stochastic dynamics. We found a broad range of temporal autocorrelation across species, populations and life cycle stages. The number of stage-classes in the matrix strongly affected the temporal autocorrelation of the growth rate. In the plant populations, reproduction transitions had more negative temporal autocorrelation than survival transitions, and matrices dominated by positive temporal autocorrelation had higher extinction risk, while in animal populations transition type was not associated with noise color. Our results indicate that temporal autocorrelation varies across life cycle transitions, even among populations of the same species. We present the colorednoise package for researchers to analyze the temporal autocorrelation of structured demographic rates.     

A juvenile-adult model with periodic vital rates

A global branch of positive cycles is shown to exist for a general discrete time, juvenile-adult model with periodically varying coefficients. The branch bifurcates from the extinction state at a critical value of the mean, inherent fertility rate. In comparison to the autonomous system with the same mean fertility rate, the critical bifurcation value can either increase or decrease with the introduction of periodicities. Thus, periodic oscillations in vital parameter can be either advantageous or deleterious. A determining factor is the phase relationship among the oscillations in the inherent fertility and survival rates.Research supported by NSF grant DMS-0414212.     

Vaccination in density-dependent epidemic models

This paper examines the effect of vaccination for an epidemic model where the death rate depends on the number of individuals in the population. The basic model which is described is based on measles or other childhood diseases in developing countries or viral diseases such as rabies in animal populations. An equilibrium analysis of the model and the local stability of small perturbations about the equilibrium values are discussed. The biological implications of these results are examined and similar results presented for modifications of the basic model.     

Differences in vital demographic rates in three populations of the epiphytic bromeliad, Werauhia sanguinolenta

Since the response to differences in resource availability is most pronounced in smaller individuals of vascular epiphytes such as Werauhia sanguinolenta Cogn. et Marchal (Bromeliaceae), I expected variation in growth and survival of small individuals to play an important role in the dynamics of entire populations. Four annual censuses (2002–2005) of three study populations, which were located across the isthmus of Panama, allowed me to construct stage transition matrices, and to conduct growth analysis and elasticity analysis. Differences between populations were highly consistent through time, but, contrary to expectations, hardly related to the comportment of smaller plants. For example, although average mortality rates were highest at the driest site, close to the Pacific, small plants were not predominantly affected. Similarly, although the highest relative growth rates (RGR) of individuals and the highest population growth rates (λ) were found in the population with the highest moisture input, which was located close to the Atlantic coast, this was not due to a particularly strong stimulation of RGR in small plants. Elasticity analysis indicated rather small differences in the importance of the three demographic processes growth, survival, and reproduction for population growth in the three populations, but invariably identified the survival of large tanks as the single most important process determining λ.     

Optimal rates of bisexual reproduction in cyclical parthenogens with density-dependent growth

This work explores theoretical patterns of reproduction that maximize the production of resting eggs and the long-term fitness of genotypes in cyclical parthenogens. Our focus is on density-dependent reproduction as it influences the consequences of a trade-off between producing amictic daughters – which reproduce parthenogenetically and subitaneously – and producing mictic daughters – which undergo meiosis and bisexual reproduction. Amictic females increase competitive ability and allow the population to achieve a larger size; mictic females directly contribute to population survival through harsh periods by producing resting eggs. Although morphologically indistinguishable, the two types of females differ greatly in their ecological and reproductive roles. What factors underlie the differential allocation of resources to produce amictic and mictic females? Using a demographic model based on readily accessible parameters we demonstrate the existence of a frequency of mictic females that will maximize the population's long-term fitness. This frequency, termed the optimal mictic ratio, m_o, is 1 ? (q/b)^1/2, where q is the mortality rate and b is the maximum birth rate. Using computer simulation we compared the fitness of a population with this constant mictic ratio with populations having multiple switches from complete parthenogenetic growth to complete allocation in mixis (mictic ratio either 0 or 1). Two important conclusions for optimal mixis in density-dependent growth conditions are: (1) intermediate mictic ratios are optimal, and (2) optimal mictic ratios are higher when habitat conditions are better. Physiological cues responding to differences in birth and death rates are common so that it is possible that populations may adjust their relative rates of mictic and amictic female production in response to environmentally induced changes to the optimum mictic ratio. Our analysis demonstrates that different patterns of mixis are expected in different type of habitats. Since the optimal mictic ratio is sensitive to the effects of a variety of environmental challenges, our model makes possible a new means to evaluate life history evolution in cyclical parthenogens.     

Persistence in population models with demographic fluctuations

A persistence and extinction theory is developed through analytical studies of deterministic population models. Under hypotheses that require demographic parameters to fluctuate temporally, the populations may or may not oscillaate. Extinction, when it occurs, is asymptotic. An hierarchy of persistence criteria, based upon fluctuations measured by time average means, is derived. In some situations a threshold value is found to separate persistent population models from those that tend to extinction. Application of the persistence-extinction theory is to the problem of assessing effects of a toxic substance on a population when toxicant inputs to the environment and to resources are oscillatory.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

Stability analysis of a predator-prey system with mutual interference and density-dependent death rates

A two species predator-prey model is proposed incorporating the notions of mutual interference among predators as well as a density-dependent predator death rate. The latter leads to a curved predator isocline. Conditions for an interior equilibrium are given, and the stability of this equilibrium is analyzed. Certain critical cases, some of which cannot occur in the usual model are also discussed.     

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.     

Using statistics to design and estimate vital rates in matrix population models for a perennial herb

Matrix population models are widely used to assess population status and to inform management decisions. Despite existing theories for building such models, model construction is often partially based on expert opinion. So far, model structure has received relatively little attention, although it may affect estimates of population dynamics. Here, we assessed the consequences of two published matrix structures (a 4 × 4 matrix based on expert opinion and a 10 × 10 matrix based on statistical modeling) for estimates of vital rates and stochastic population dynamics of the long-lived herb Astragalus scaphoides. We explored the ways in which choice of model structure alters the accuracy (i.e., mean) and precision (i.e., variance) of predicted population dynamics. We found that model structure had a negligible effect on the accuracy and precision of vital rates and stochastic stage distribution. However, the 10 × 10 matrix produced lower estimates of stochastic population growth rates than the 4 × 4 matrix, and more accurately predicted the observed trends in population abundance for three out of four study populations. Moreover, estimates of realized variation in population growth rate due to fluctuations in population stage structure over time were occasionally sensitive to matrix structure, suggesting differential roles of transient dynamics. Our study indicates that statistical modeling for choosing categories in matrix models might be preferable over expert opinion to accurately predict population trends and can provide a more objective way for model construction when the biological knowledge of the species is limited.     

Low demographic variability in wild primate populations: fitness impacts of variation, covariation, and serial correlation in vital rates

In a stochastic environment, long-term fitness can be influenced by variation, covariation, and serial correlation in vital rates (survival and fertility). Yet no study of an animal population has parsed the contributions of these three aspects of variability to long-term fitness. We do so using a unique database that includes complete life-history information for wild-living individuals of seven primate species that have been the subjects of long-term (22-45 years) behavioral studies. Overall, the estimated levels of vital rate variation had only minor effects on long-term fitness, and the effects of vital rate covariation and serial correlation were even weaker. To explore why, we compared estimated variances of adult survival in primates with values for other vertebrates in the literature and found that adult survival is significantly less variable in primates than it is in the other vertebrates. Finally, we tested the prediction that adult survival, because it more strongly influences fitness in a constant environment, will be less variable than newborn survival, and we found only mixed support for the prediction. Our results suggest that wild primates may be buffered against detrimental fitness effects of environmental stochasticity by their highly developed cognitive abilities, social networks, and broad, flexible diets.     

Coexistence of multiple pathogen strains in stochastic epidemic models with density-dependent mortality

Stochastic differential equations that model an SIS epidemic with multiple pathogen strains are derived from a system of ordinary differential equations. The stochastic model assumes there is demographic variability. The dynamics of the deterministic model are summarized. Then the dynamics of the stochastic model are compared to the deterministic model. In the deterministic model, there can be either disease extinction, competitive exclusion, where only one strain persists, or coexistence, where more than one strain persists. In the stochastic model, all strains are eventually eliminated because the disease-free state is an absorbing state. However, if the population size and the initial number of infected individuals are sufficiently large, it may take a long time until all strains are eliminated. Numerical simulations of the stochastic model show that coexistence cases predicted by the deterministic model are an unlikely occurrence in the stochastic model even for short time periods. In the stochastic model, either disease extinction or competitive exclusion occur. The initial number of infected individuals, the basic reproduction numbers, and other epidemiological parameters are important determinants of the dominant strain in the stochastic epidemic model.