Population dynamics of three Neotropical small mammals: Time series models and the role of delayed density-dependence in population irruptions

It is widely believed that only precipitation levels (through increased primary production) determine irruptions of small mammals in semi-arid areas of western South America. Nevertheless, density-dependent factors may also drive population fluctuations. To test statistically these putative effects we analysed 11 years of population records on three sympatric species of small mammals at two different habitat types in north central Chile. We applied the classical diagnostic tools of time series analysis (the autocorrelation function: ACF) to the observed time series of three neotropical small mammals. We also used simple linear autoregressive time series models to reconstruct the endogenous dynamics of these populations. The analysis strongly suggests that population fluctuations of the three species have an important density-dependent component, with the most irruptive species (Phyllotis darwini, Waterhouse 1837) displaying stronger second order population feedbacks than the other two (Akodon olivaceus, Waterhouse 1837 and Thylamys elegans, Waterhouse 1839). The latter two species showed direct density-dependent feedbacks. We hypothesize that the frequent population outbreaks of P. darwini (and perhaps of other species) in semi-arid regions of western South America, may be the result of population-level (direct density- dependence) and community-level processes (delayed density-dependence), interacting with exogenous perturbations (rainfall and associated primary production).     

Accounting for wildlife life-history strategies when modeling stochastic density-dependent populations: A review

This article briefly reviews and provides discussion on the evidence for, and nature of, density-dependence patterns in r and K-selected species. In this review, I discuss how life-history strategies cause different nonlinear density-dependence patterns and I provide a simple modeling recommendation to incorporate nonlinear density dependence in population growth equations. Second, I discuss the importance of incorporation of environmental stochasticity and local extinction associated with nonlinear density dependence associated with life-history patterns through a novel modeling exercise. Last, I discuss the importance of considering how life-history nonlinear density dependence could affect optimal harvest yields. Though these topics are extensive, this review should spur wildlife biologists and managers to consider more inclusive population models that incorporate life-history strategies and stochasticity in their decision-making processes. © 2012 The Wildlife Society.     

Defining fitness in evolutionary models

The analysis of evolutionary models requires an appropriate definition for fitness. In this paper, I review such definitions in relation to the five major dimensions by which models may be described, namely (i) finite versus infinite (or very large) population size, (ii) type of environment (constant, fixed length, temporally stochastic, temporally predictable, spatially stochastic, spatially predictable and social environment), (iii) density-independent or density-dependent, (iv) inherent population dynamics (equilibrium, cyclical and chaotic), and (v) frequency-dependent or independent. In simple models, the Malthusian parameter ‘r’ or the net reproductive rate R ₀ may be satisfactory, but once density-dependence or complex population dynamics is introduced the invasion exponent should be used. Defining fitness in a social environment or when there is frequency-dependence requires special consideration.     

Consumers limit the abundance and dynamics of a perennial shrub with a seed bank

For nearly 30 years, ecologists have argued that predators of seeds and seedlings seldom have population-level effects on plants with persistent seed banks and density-dependent seedling survival. We parameterized stage-based population models that incorporated density dependence and seed dormancy with data from a 5.5-year experiment that quantified how granivorous mice and herbivorous voles influence bush lupine (Lupinus arboreus) demography. We asked how seed dormancy and density-dependent seedling survival mediate the impacts of these consumers in dune and grassland habitats. In dune habitat, mice reduced analytical lambda (the intrinsic rate of population growth) by 39%, the equilibrium number of aboveground plants by 90%, and the seed bank by 98%; voles had minimal effects. In adjacent grasslands, mice had minimal effects, but seedling herbivory by voles reduced analytical lambda by 15% and reduced both the equilibrium number of aboveground plants and dormant seeds by 63%. A bootstrap analysis demonstrated that these consumer effects were robust to parameter uncertainty. Our results demonstrate that the quantitative strengths of seed dormancy and density-dependent seedling survival--not their mere existence--critically mediate consumer effects. This study suggests that plant population dynamics and distribution may be more strongly influenced by consumers of seeds and seedlings than is currently recognized.     

Density effects on life-history traits of an island lizard population

Long-term monitoring of life-history traits and the effects of density upon them were studied in an island population of the lizardEumeces okadae. Although life-history traits such as clutch size, egg size and the proportion of mature reproductive females varied little over 7 years in the intact population, manipulation of density to simulate decreased population density enhanced juvenile growth rate, age at first reproduction, frequency of female reproduction and size-specific clutch mass. In particular, the proportion of mature females reproducing annually increased almost 10 times from 5.6% to 53.8% after the removal of some lizards. However, body size at first reproduction and egg size were almost identical under both high and low density conditions. This study suggests that there were strong density-dependent effects on several life-history traits and thatE. okadae attained a density close to the carrying capacity of the environment.     

Elasticity analysis of density-dependent matrix population models: the invasion exponent and its substitutes

In density-independent models, the population growth rate lambda measures population performance, and the perturbation analysis of lambda (its sensitivity and elasticity) plays an important role in demography. In density-dependent models, the invasion exponent lambdaI replaces lambda as a measure of population performance. The perturbation analysis of lambdaI reveals the effects of environmental changes and management actions, gives the direction and intensity of density-dependent natural selection on life history traits, and permits calculation of the sampling variance of the invasion exponent. Because density-dependent models require more data than density-independent models, it is tempting to look for substitutes for the invasion exponent, the sensitivity and elasticity of which can be calculated from a density-independent model. Here we examine the accuracy of two such substitutes: the dominant eigenvalue of the projection matrix evaluated at equilibrium (An) and the dominant eigenvalue of the matrix averaged over the attractor (A). Using a two-stage model that represents a wide range of life history types, we find that the elasticities of An or A often agree to within less than 5% error with those of the invasion exponent, even when population dynamics are chaotic. The exceptions are for semelparous life histories, especially when density-dependence affects fertility. This suggests that the elasticity analysis of density-independent models near equilibrium, or averaged over the attractor, provides useful information about the elasticity of the invasion exponent in density-dependent models.     

Estimation of a Nonlinear Density-Dependence Parameter for Wild Turkey

Abstract: Although previous research and theory has suggested that wild turkey (Meleagris gallopavo) populations may be subject to some form of density dependence, there has been no effort to estimate and incorporate a density-dependence parameter into wild turkey population models. To estimate a functional relationship for density dependence in wild turkey, we analyzed a set of harvest-index time series from 11 state wildlife agencies. We tested for lagged correlations between annual harvest indices using partial autocorrelation analysis. We assessed the ability of the density-dependent theta-Ricker model to explain harvest indices over time relative to exponential or random walk growth models. We tested the homogeneity of the density-dependence parameter estimates (θ) from 3 different harvest indices (spring harvest no. reported harvest/effort, survey harvest/effort) and calculated a weighted average based on each estimate's variance and its estimated covariance with the other indices. To estimate the potential bias in parameter estimates from measurement error, we conducted a simulation study using the theta-Ricker with known values and lognormally distributed measurement error. Partial autocorrelation function analysis indicated that harvest indices were significantly correlated only with their value at the previous time step. The theta-Ricker model performed better than the exponential growth or random walk models for all 3 indices. Simulation of known parameters and measurement error indicated a strong positive upward bias in the density-dependent parameter estimate, with increasing measurement error. The average density-dependence estimate, corrected for measurement error ranged 0.25 ≤ θC ≤ 0.49, depending on the amount of measurement error and assumed spring harvest rate. We infer that density dependence is nonlinear in wild turkey, where growth rates are maximized at 39-42% of carrying capacity. The annual yield produced by density-dependent population growth will tend to be less than that caused by extrinsic environmental factors. This study indicates that both density-dependent and density-independent processes are important to wild turkey population growth, and we make initial suggestions on incorporating both into harvest management strategies.     

Habitat Templets and the Changing Worldview of Ecology

Habitat templets are graphical-qualitative models which describe the development of life-history strategies in specific environmental conditions. In the context of the previous models of life-history strategies, life-history theorists focused on the density-dependent factors as the factors determining life-history strategies. With the use of habitat templets, the focus is oriented towards the environmental causal factors, considering density-dependent phenomena as by-products of the environmental impact. This implies an important shift in causality as well as in the worldview of life-history theorists: population is not considered as a closed system isolated from the environment. The object of study is the organism-in-its-environment, as a complex multilevel system. This shift has also methodological consequences: Life-history theory combines holistic and reductionistic insights, using a variety of heuristic models. This imposes a new conception of generality as well as of the structure of scientific theories.     

Survival-variation within and between functional categories of the African multimammate rat

1. By identifying ecological factors specific to functional categories of individuals, it may be possible to understand the mechanisms underlying life-history evolution and population dynamics. While empirical analyses within the field of population biology have focused on changes in population size, theoretical models assuming differential sensitivities of population growth rate or fitness to demographic parameters have mostly been untested, particularly against data on small mammals.
2. Statistical modelling of capture–mark–recapture data on the multimammate rat ( Mastomys natalensis ) from Tanzania shows that: (i) females survive slightly better than males and subadults survive much better than adults; (ii) temporal variation of survival of all individuals is similarly related to the rainfall of the month; (iii) subadults exhibit a strongly density-dependent low persistence rate in the population immediately after their first capture; (iv) subadults survival in later months is, however, positively related to density; and (v) adult survival shows negative density-dependence.
3. Both density-dependent and density-independent factors simultaneously determine stage-dependent survival variation of the multimammate rat. Whereas environmental factors in this population seem to affect survival rates of all individuals in a similar manner, density-dependent relationships are more complex.
4. The patterns of survival variation in small mammals may be different from those observed in large mammals.
5. Further studies of demography in small mammals should aim at understanding how much of the variability in population growth rate is accounted for by the variability of the demographic rates resulting from limiting (density-independent) and regulating (density-dependent) factors, respectively. This study emphasizes the use of robust and accurate statistical methods as well as stage- or age-structured population modelling.     

Population variability among three small mammal species in the semiarid Neotropics: the role of density-dependent and density-independent factors

We addressed the role of density-dependent (direct and delayed) and density-independent (precipitation) factors in shaping the dynamics of fluctuating populations of three small mammal species. Using a stepwise regression procedure, we tested the effects of nonlagged population density (log₁₀ N_t-1), lagged population density (log₁₀ N_t-2), and annual precipitation on the per capita rate of population change of Phyllotis darwini, Akodon olivaceus , and Thylamys elegans in two habitat types of a semiarid region of Chile. The most irruptive species ( P. darwini ) showed direct and delayed density-dependent effects in equatorial subpopulation, and only direct density-dependence in polar subpopulation. The per capita rates of population change of A. olivaceus showed direct density-dependent and precipitation effects in both habitats types, while T. elegans showed direct density-dependence and precipitation effects in the equatorial subpopulation but only a marginal effect of direct density-dependence in the polar subpopulation. The presence of delayed density-dependent strongly suggests the importance of biological interactions in shaping the dramatic irruptions exhibited by P. darwini.     

Lost in time, lonely, and single: reproductive asynchrony and the Allee effect

Identifying linkages between life-history traits and small population processes is essential to effective multispecies conservation. Reproductive asynchrony, which occurs when individuals are reproductively active for only a portion of the population-level breeding period, may provide one such link. Traditionally, reproductive asynchrony has been considered from evolutionary perspectives as an advantageous bet-hedging strategy in temporally unpredictable environments. Here, we explore the dynamic consequences of reproductive asynchrony as a density-dependent life-history trait. To examine how asynchrony affects population growth rate and extinction risk, we used a general model of reproductive timing to quantify the temporal overlap of opposite-sex individuals and to simulate population dynamics over a range of initial densities and empirical estimates of reproductive asynchrony. We also considered how protandry, a sexually selected life-history strategy that often accompanies asynchrony, modulates the population-level effects of reproductive asynchrony. We found that asynchrony decreases the number of males a female overlaps with, decreases the average probability of mating per male/female pair that does overlap, and leaves some females completely isolated in time. This loss of reproductive potential, which is exacerbated by protandry, reduces population growth rate at low density and can lead to extinction via an Allee effect. Thus reproductive asynchrony and protandry, both of which can be evolutionarily advantageous at higher population densities, may prove detrimental when population density declines.     

Does extrinsic mortality accelerate the pace of life? A bare-bones approach

It is commonly asserted that when extrinsic mortality is high, individuals should invest early in reproduction. This intuition thrives in the literature on life-history theory and human behavior, yet it has been criticized repeatedly on the basis of mathematical models. The intuition is indeed wrong; but a recent theoretical criticism has confused the reason why it is wrong, thereby obscuring earlier and sounder criticisms. In the present article, based on the simplest possible model, we sought to clarify these issues. We confirm earlier findings that extrinsic mortality can affect the evolution of pace of life, not because it leaves little time to reproduce, but through its effects on density-dependent competition. This result highlights the importance of accounting for density-dependence in theoretical models and data analyses. Further, we find little support for the recent claim that the direction of selection on a reaction norm in a variable environment cannot be easily inferred from models made in homogeneous environments. In conclusion, although life-history theory is still imperfect, it has provided simple results that deserve to be understood.     

Population extinction and quasi-stationary behavior in stochastic density-dependent structured models

Density-independent and density-dependent, stochastic and deterministic, discrete-time, structured models are formulated, analysed and numerically simulated. A special case of the deterministic, density-independent, structured model is the well-known Leslie age-structured model. The stochastic, density-independent model is a multitype branching process. A review of linear, density-independent models is given first, then nonlinear, density-dependent models are discussed. In the linear, density-independent structured models, transitions between states are independent of time and state. Population extinction is determined by the dominant eigenvalue λ of the transition matrix. If λ ≤ 1, then extinction occurs with probability one in the stochastic and deterministic models. However, if λ > 1, then the deterministic model has exponential growth, but in the stochastic model there is a positive probability of extinction which depends on the fixed point of the system of probability generating functions. The linear, density-independent, stochastic model is generalized to a nonlinear, density-dependent one. The dependence on state is in terms of a weighted total population size. It is shown for small initial population sizes that the density-dependent, stochastic model can be approximated by the density-independent, stochastic model and thus, the extinction behavior exhibited by the linear model occurs in the nonlinear model. In the deterministic models there is a unique stable equilibrium. Given the population does not go extinct, it is shown that the stochastic model has a quasi-stationary distribution with mean close to the stable equilibrium, provided the population size is sufficiently large. For small values of the population size, complete extinction can be observed in the simulations. However, the persistence time increases rapidly with the population size. This author received partial support by the National Science Foundation grant # DMS-9626417.     

Effects of body size and temperature on population growth

For at least 200 years, since the time of Malthus, population growth has been recognized as providing a critical link between the performance of individual organisms and the ecology and evolution of species. We present a theory that shows how the intrinsic rate of exponential population growth, rmax, and the carrying capacity, K, depend on individual metabolic rate and resource supply rate. To do this, we construct equations for the metabolic rates of entire populations by summing over individuals, and then we combine these population-level equations with Malthusian growth. Thus, the theory makes explicit the relationship between rates of resource supply in the environment and rates of production of new biomass and individuals. These individual-level and population-level processes are inextricably linked because metabolism sets both the demand for environmental resources and the resource allocation to survival, growth, and reproduction. We use the theory to make explicit how and why rmax exhibits its characteristic dependence on body size and temperature. Data for aerobic eukaryotes, including algae, protists, insects, zooplankton, fishes, and mammals, support these predicted scalings for rmax. The metabolic flux of energy and materials also dictates that the carrying capacity or equilibrium density of populations should decrease with increasing body size and increasing temperature. Finally, we argue that body mass and body temperature, through their effects on metabolic rate, can explain most of the variation in fecundity and mortality rates. Data for marine fishes in the field support these predictions for instantaneous rates of mortality. This theory links the rates of metabolism and resource use of individuals to life-history attributes and population dynamics for a broad assortment of organisms, from unicellular organisms to mammals.     

Population rate of change in the leaf-eared mouse: The role of density-dependence,seasonality and rainfall

We analysed statistically the influence of density-dependent regulation, seasonality and precipitation on the realized population rate of change of the Neotropical rodent Phyllotis darwini (Waterhouse 1837) at an intra-annual time scale. We used four years of continuous live trapping at a semiarid locality of north central Chile. Results showed that density-dependence, seasonal effects and precipitation were important factors influencing population growth rates in this species. An empirical population model including a sine function for seasonal effects, a linear form for density-dependence, and precipitation was fitted to the full data set and to the data set with the first year removed (after an outbreak). The empirical model explained 33% and 48% of the variance in population growth. The natural rate of population increase, estimated from the empirical model, was r_max = 2.51 or 5.06 years^?1. These estimates indicate a great potential for population increase and may explain the capability of this species to undergo large irruptions. We propose that merging empirical and theoretical modelling with field research is the most promising avenue to understand the outbreaks experienced by some rodent species in western South America.     

A modeling exercise to show why population models should incorporate distinct life histories of dispersers

Dispersal is an important form of movement influencing population dynamics, species distribution and gene flow between populations. In population models, dispersal is often included in a simplified manner by removing a random proportion of the population. Many ecologists now argue that models should be formulated at the level of individuals instead of the population level. To fully understand the effects of dispersal on natural systems, it is therefore necessary to incorporate individual-level differences in dispersal behavior in population models. Here, we parameterized an integral projection model, which allows for studying how individual life histories determine population-level processes, using bulb mites, Rhizoglyphus robini, to assess to what extent dispersal expression (frequency of individuals in the dispersal stage) and dispersal probability affect the proportion of successful dispersers and natal population growth rate. We find that allowing for life-history differences between resident phenotypes and disperser phenotypes shows that multiple combinations of dispersal probability and dispersal expression can produce the same proportion of leaving individuals. Additionally, a given proportion of successful dispersing individuals result in different natal population growth rates. The results highlight that dispersal life histories, and the frequency with which disperser phenotypes occur in the natal population, significantly affect population-level processes. Thus, biological realism of dispersal population models can be increased by incorporating the typically observed life-history differences between resident phenotypes and disperser phenotypes, and we here present a methodology to do so.     

Quantile regression estimates of animal population trends

Ecologists often estimate population trends of animals in time series of counts using linear regression to estimate parameters in a linear transformation of multiplicative growth models, where logarithms of rates of change in counts in time intervals are used as response variables. We present quantile regression estimates for the median (0.50) and interquartile (0.25, 0.75) relationships as an alternative to mean regression estimates for common density-dependent and density-independent population growth models. We demonstrate that the quantile regression estimates are more robust to outliers and require fewer distributional assumptions than conventional mean regression estimates and can provide information on heterogeneous rates of change ignored by mean regression. We provide quantile regression trend estimates for 2 populations of greater sage-grouse (Centrocercus urophasianus) in Wyoming, USA, and for the Crawford population of Gunnison sage-grouse (Centrocercus minimus) in southwestern Colorado, USA. Our selected Gompertz models of density dependence for both populations of greater sage-grouse had smaller negative estimates of density-dependence terms and less variation in corresponding predicted growth rates (λ) for quantile than mean regression models. In contrast, our selected Gompertz models of density dependence with piecewise linear effects of years for the Crawford population of Gunnison sage-grouse had predicted changes in λ across years from quantile regressions that varied more than those from mean regression because of heterogeneity in estimated λs that were both less and greater than mean estimates. Our results add to literature establishing that quantile regression provides better behaved estimates than mean regression when there are outlying growth rates, including those induced by adjustments for zeros in the time series of counts. The 0.25 and 0.75 quantiles bracketing the median provide robust estimates of population changes (λ) for the central 50% of time series data and provide a 50% prediction interval for a single new prediction without making parametric distributional assumptions or assuming homogeneous λs. Compared to mean estimates, our quantile regression trend estimates for greater sage-grouse indicated less variation in density-dependent λs by minimizing sensitivity to outlying values, and for Gunnison sage-grouse indicated greater variation in density-dependent λs associated with heterogeneity among quantiles.     

Current caveats and further directions in the analysis of density-dependent population regulation

An important issue in population ecology is to disentangle different density-dependent mechanisms that may limit or regulate animal populations. This goal is further complicated when studying long-lived species for which experimental approaches are not feasible, in whose cases density-dependence hypotheses are tested using long-term monitored populations. Here we respond to some criticisms and identify additional problems associated with these kinds of observational studies. Current caveats are related to the temporal and spatial scales covered by population monitoring data, which may question its suitability for density-dependence tests, and to statistical flaws such as the incorrect control for confounding variables, low statistical power, the distribution of demographic variables, the interpretation of spurious correlations, and the often used stepwise series of univariate analyses. Generalised linear mixed models are recommended over other more traditional approaches, since they help to solve the above statistical problems and, more importantly, allow to properly test several hypotheses simultaneously. Finally, several management actions aimed to recover endangered species, such as supplementary feeding, might be considered as field experiments for further testing density-dependence hypotheses in long-lived study models. We expect these opportunities, together with the most adequate statistical tools now available, will help to better our understanding of density-dependent effects in wild populations.     

Density-dependent processes in the life history of fishes: evidence from laboratory populations of zebrafish Danio rerio

Population regulation is fundamental to the long-term persistence of populations and their responses to harvesting, habitat modification, and exposure to toxic chemicals. In fish and other organisms with complex life histories, regulation may involve density dependence in different life-stages and vital rates. We studied density dependence in body growth and mortality through the life-cycle of laboratory populations of zebrafish Danio rerio. When feed input was held constant at population-level (leading to resource limitation), body growth was strongly density-dependent in the late juvenile and adult phases of the life-cycle. Density dependence in mortality was strong during the early juvenile phase but declined thereafter and virtually ceased prior to maturation. Provision of feed in proportion to individual requirements (easing resource limitation) removed density dependence in growth and substantially reduced density dependence in mortality, thus indicating that 'bottom-up' effects act on growth as well as mortality, but most strongly on growth. Both growth and mortality played an important role in population regulation, with density-dependent growth having the greater impact on population biomass while mortality had the greatest impact on numbers. We demonstrate a clear ontogenic pattern of change in density-dependent processes within populations of a very small (maximum length 5 mm) fish, maintained in constant homogeneous laboratory conditions. The patterns are consistent with those distilled from studies on wild fish populations, indicating the presence of broad ontogenic patterns in density-dependent processes that are invariant to maximum body size and hold in homogeneous laboratory, as well as complex natural environments.     

Carry-over effects and habitat quality in migratory populations

Determining the factors that influence migratory population abundance has been constrained by the inability to connect events in different periods of the annual cycle. Carry-over effects are events that occur in one season but influence individual success the following season and recent empirical evidence suggests that they may play an important role in migratory population dynamics. Using a long distance migratory shorebird as an example, I incorporate carry-over effects and changes in the relative amount of habitat quality into a density-dependent equilibrium population model. The model uses the example where the quality of habitat on the wintering grounds (nonbreeding season) influences breeding output the following summer (breeding season). Carry-over effects, however, may be manifested in a number of other ways that could influence population dynamics. In the simulations, population declines occur when habitat is lost on the wintering grounds. However, results show that carry-over effects can magnify these declines when a disproportionate amount of high quality habitat is lost the previous winter. Simulations also show that carry-over effects can have a relative, positive impact on population size when the majority of habitat that is lost in the previous season is low quality. In this case, the carry-over interacts with density-dependence the following season producing an additive and positive effect, buffering the population from severe declines. To predict changes in population size of migratory animals, it will be important to determine (i) which demographic factors in which season produce strong carry-over effects and, (ii) not just the amount, but the relative quality of habitat that is lost. If carry-over effects are significant, they could potentially mitigate 'seasonal compensation effects' from density-dependence, leading to exacerbated population declines.