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.     

Distance decay of similarity,effects of environmental noise and ecological heterogeneity among species in the spatio‐temporal dynamics of a dispersal‐limited community

The joint spatial and temporal fluctuations in community structure may be due to dispersal, variation in environmental conditions, ecological heterogeneity among species and demographic stochasticity. These factors are not mutually exclusive, and their relative contribution towards shaping species abundance distributions and in causing species fluctuations have been hard to disentangle. To better understand community dynamics when the exchange of individuals between localities is very low, we studied the dynamics of the freshwater zooplankton communities in 17 lakes located in independent catchment areas, sampled at end of summer from 2002 to 2008 in Norway. We analysed the joint spatial and temporal fluctuations in the community structure by fitting the two‐dimensional Poisson lognormal model under a two‐stage sampling scheme. We partitioned the variance of the distribution of log abundance for a random species at a random time and location into components of demographic stochasticity, ecological heterogeneity among species, and independent environmental noise components for the different species. Non‐neutral mechanisms such as ecological heterogeneity among species (20%) and spatiotemporal variation in the environment (75%) explained the majority of the variance in log abundances. Overdispersion relative to Poisson sampling and demographic stochasticity had a small contribution to the variance (5%). Among a set of environmental variables, lake acidity was the environmental variable that was most strongly related to decay of community similarity in space and time.     

Looking for a needle in a haystack: inference about individual fitness components in a heterogeneous population

Studies of wild vertebrates have provided evidence of substantial differences in lifetime reproduction among individuals and the sequences of life history ‘states’ during life (breeding, nonbreeding, etc.). Such differences may reflect ‘fixed’ differences in fitness components among individuals determined before, or at the onset of reproductive life. Many retrospective life history studies have translated this idea by assuming a ‘latent’ unobserved heterogeneity resulting in a fixed hierarchy among individuals in fitness components. Alternatively, fixed differences among individuals are not necessarily needed to account for observed levels of individual heterogeneity in life histories. Individuals with identical fitness traits may stochastically experience different outcomes for breeding and survival through life that lead to a diversity of ‘state’ sequences with some individuals living longer and being more productive than others, by chance alone. The question is whether individuals differ in their underlying fitness components in ways that cannot be explained by observable ‘states’ such as age, previous breeding success, etc. Here, we compare statistical models that represent these opposing hypotheses, and mixtures of them, using data from kittiwakes. We constructed models that accounted for observed covariates, individual random effects (unobserved heterogeneity), first‐order Markovian transitions between observed states, or combinations of these features. We show that individual sequences of states are better accounted for by models incorporating unobserved heterogeneity than by models including first‐order Markov processes alone, or a combination of both. If we had not considered individual heterogeneity, models including Markovian transitions would have been the best performing ones. We also show that inference about age‐related changes in fitness components is sensitive to incorporation of underlying individual heterogeneity in models. Our approach provides insight into the sources of individual heterogeneity in life histories, and can be applied to other data sets to examine the ubiquity of our results across the tree of life.     

Lifetime reproductive output: individual stochasticity,variance, and sensitivity analysis

Lifetime reproductive output (LRO) determines per-generation growth rates, establishes criteria for population growth or decline, and is an important component of fitness. Empirical measurements of LRO reveal high variance among individuals. This variance may result from genuine heterogeneity in individual properties, or from individual stochasticity, the outcome of probabilistic demographic events during the life cycle. To evaluate the extent of individual stochasticity requires the calculation of the statistics of LRO from a demographic model. Mean LRO is routinely calculated (as the net reproductive rate), but the calculation of variances has only recently received attention. Here, we present a complete, exact, analytical, closed-form solution for all the moments of LRO, for age- and stage-classified populations. Previous studies have relied on simulation, iterative solutions, or closed-form analytical solutions that capture only part of the sources of variance. We also present the sensitivity and elasticity of all of the statistics of LRO to parameters defining survival, stage transitions, and (st)age-specific fertility. Selection can operate on variance in LRO only if the variance results from genetic heterogeneity. The potential opportunity for selection is quantified by Crow’s index \(\mathcal {I}\), the ratio of the variance to the square of the mean. But variance due to individual stochasticity is only an apparent opportunity for selection. In a comparison of a range of age-classified models for human populations, we find that proportional increases in mortality have very small effects on the mean and variance of LRO, but large positive effects on \(\mathcal {I}\). Proportional increases in fertility increase both the mean and variance of LRO, but reduce \(\mathcal {I}\). For a size-classified tree population, the elasticity of both mean and variance of LRO to stage-specific mortality are negative; the elasticities to stage-specific fertility are positive.     

Experience versus effort: what explains dynamic heterogeneity with respect to age?

Age‐related patterns of survival and reproduction have been explained by accumulated experience (‘experience hypothesis’), increased effort (‘effort hypothesis’), and intrinsic differences in phenotypes (‘selection hypothesis’). We examined the experience and effort hypotheses using a 40‐year data set in a population of Leach's storm‐petrels Oceanodroma leucorhoa, long‐lived seabirds for which the effect of phenotypic variation has been previously demonstrated. Age was quantified by time since recruitment (‘breeding age’). The best model of adult survival included a positive effect of breeding age (1, 2, 3+ years), sex (male > female), and year. Among‐individuals variation (fixed heterogeneity) accounted for 31.6% of the variance in annual reproductive success. We further examined within‐individual patterns in reproductive success (dynamic heterogeneity) in the subset of individuals with at least five breeding attempts. Three distinct phases characterized reproductive success – early increase, long asymptotic peak, late decline. No effect of early reproductive output on longevity was found, however, early success was positively correlated with lifetime reproductive success. Reproductive success was lower earlier than later in life. Among the few natally philopatric individuals in the population, age of first breeding had no effect on longevity, lifetime reproductive success, or early reproductive success. No support for the effort hypothesis was found in this population. Instead, age‐specific patterns of survival and reproduction in these birds are best explained by the experience hypothesis over and above the effect of intrinsic differences among individuals.     

Interspecific differences in stochastic population dynamics explains variation in Taylor's temporal power law

Taylor’s power law, i.e. that the slope for the increase in variance with mean population size is between 1 and 2 at a logarithmic scale, provides one of the few quantitative relationships in population ecology, yet the underlying ecological mechanisms are only poorly understood. Stochastic theory of population dynamics predicts that demographic and environmental stochasticity will affect the slope differently. In a stable environment under the influence of demographic stochasticity alone the slope will be equal to 1. In large populations in which demographic variance will have a negligible effect on the dynamics the slope will approach 2. In addition, the slope will also be influenced by how the strength of density dependence is related to mean population size. To disentangle the relative contribution of these processes we estimate the mean‐variance relationship for a large number of populations of British birds. The variance in population size of most species decreased with the mean due to decreased influence of demographic stochasticity at larger population sizes. Interspecific differences in demographic stochasticity was the main factor influencing variation in slopes of Taylor’s power law among species through a significant negative relationship between the slope and demographic variance. In addition, slopes were influenced by interspecific variation in life history parameters such as adult survival and clutch size. These analyses show that Taylor’s power law is generated from an interplay between stochastic and density dependent factors, modulated by life history.     

Consequences of heterogeneity in survival probability in a population of Florida scrub-jays

1. Using data on breeding birds from a 35-year study of Florida scrub-jays Aphelocoma coerulescens (Bosc 1795), we show that survival probabilities are structured by age, birth cohort, and maternal family, but not by sex. Using both accelerated failure time (AFT) and Cox proportional hazard models, the data are best described by models incorporating variation among birth cohorts and greater mortality hazard with increasing age. AFT models using Weibull distributions with the shape parameter > 1 were always the best-fitting models. 2. Shared frailty models allowing for family structure greatly reduce model deviance. The best-fitting models included a term for frailty shared by maternal families. 3. To ask how long a data set must be to reach qualitatively the same conclusions, we repeated the analyses for all possible truncated data sets of 2 years in length or greater. Length of the data set affects the parameter estimates, but not the qualitative conclusions. In all but three of 337 truncated data sets the best-fitting models pointed to same conclusions as the full data set. Shared frailty models appear to be quite robust. 4. The data are not adequate for testing hypotheses as to whether variation in frailty is heritable. 5. Substantial structured heterogeneity for survival exists in this population. Such structured heterogeneity has been shown to have substantial effects in reducing demographic stochasticity.     

Are the numbers adding up? Exploiting discrepancies among complementary population models

Large carnivores are difficult to monitor because they tend to be sparsely distributed, sensitive to human activity, and associated with complex life histories. Consequently, understanding population trend and viability requires conservationists to cope with uncertainty and bias in population data. Joint analysis of combined data sets using multiple models (i.e., integrated population model) can improve inference about mechanisms (e.g., habitat heterogeneity and food distribution) affecting population dynamics. However, unobserved or unobservable processes can also introduce bias and can be difficult to quantify. We developed a Bayesian hierarchical modeling approach for inference on an integrated population model that reconciles annual population counts with recruitment and survival data (i.e., demographic processes). Our modeling framework is flexible and enables a realistic form of population dynamics by fitting separate density-dependent responses for each demographic process. Discrepancies estimated from shared parameters among different model components represent unobserved additions (i.e., recruitment or immigration) or removals (i.e., death or emigration) when annual population counts are reliable. In a case study of gray wolves in Wisconsin (1980–2011), concordant with policy changes, we estimated that a discrepancy of 0% (1980–1995), −2% (1996–2002), and 4% (2003–2011) in the annual mortality rate was needed to explain annual growth rate. Additional mortality in 2003–2011 may reflect density-dependent mechanisms, changes in illegal killing with shifts in wolf management, and nonindependent censoring in survival data. Integrated population models provide insights into unobserved or unobservable processes by quantifying discrepancies among data sets. Our modeling approach is generalizable to many population analysis needs and allows for identifying dynamic differences due to external drivers, such as management or policy changes.     

Evolution of a plastic quantitative trait in an age-structured population in a fluctuating environment

We analyze weak fluctuating selection on a quantitative character in an age-structured population not subject to density regulation. We assume that early in the first year of life before selection, during a critical state of development, environments exert a plastic effect on the phenotype, which remains constant throughout the life of an individual. Age-specific selection on the character affects survival and fecundity, which have intermediate optima subject to temporal environmental fluctuations with directional selection in some age classes as special cases. Weighting individuals by their reproductive value, as suggested by Fisher, we show that the expected response per year in the weighted mean character has the same form as for models with no age structure. Environmental stochasticity generates stochastic fluctuations in the weighted mean character following a first-order autoregressive model with a temporally autocorrelated noise term and stationary variance depending on the amount of phenotypic plasticity. The parameters of the process are simple weighted averages of parameters used to describe age-specific survival and fecundity. The "age-specific selective weights" are related to the stable distribution of reproductive values among age classes. This allows partitioning of the change in the weighted mean character into age-specific components.     

Efficiency of using spatial analysis in first-generation coastal Douglas-fir progeny tests in the US Pacific Northwest

Single-trial and across-trial spatial analyses using autoregressive error structures were conducted for growth traits based on 1,146 data sets from 275 Douglas-fir [Pseudotsuga menziesii (Mirb.) Franco] progeny trials in 45 first-generation breeding zones in the US Pacific Northwest. The breeding zones encompassed a wide range of latitude, longitude, and elevation. Efficiency of using spatial analysis in reducing variation due to site heterogeneity, estimating genetic parameters, and increasing prediction accuracy was compared among different experimental designs, traits, assessment ages, and tree spacings. More than 97% of the data sets showed significant model improvement with spatial analysis, and height showed more improvement than diameter or volume. Spatial analysis on average removed 14~34% of residual variance due to spatial heterogeneity, which resulted in an up to 20% increase in accuracy of breeding value prediction. The coefficient of variation decreased substantially due to spatial adjustment. Rank correlation between predicted gains before and after spatial analysis was about 0.96, and spatial analysis had little effect on the average predicted gain of the top 20% of parents. We did not observe substantial geographic trends in improvements due to spatial adjustment. Across-site spatial analysis had almost no effect on genotype-by-environment interaction but tended to increase among-trial heterogeneity of residual variance. Two different methods for across-trial spatial analysis were compared and discussed.     

Can life history predict the effect of demographic stochasticity on extinction risk?

Demographic stochasticity is important in determining extinction risks of small populations, but it is largely unknown how its effect depends on the life histories of species. We modeled effects of demographic stochasticity on extinction risk in a broad range of generalized life histories, using matrix models and branching processes. Extinction risks of life histories varied greatly in their sensitivity to demographic stochasticity. Comparing life histories, extinction risk generally increased with increasing fecundity and decreased with higher ages of maturation. Effects of adult survival depended on age of maturation. At lower ages of maturation, extinction risk peaked at intermediate levels of adult survival, but it increased along with adult survival at higher ages of maturation. These differences were largely explained by differences in sensitivities of population growth to perturbations of life-history traits. Juvenile survival rate contributed most to total demographic variance in the majority of life histories. Our general results confirmed earlier findings, suggesting that empirical patterns can be explained by a relatively simple model. Thus, basic life-history information can be used to assign life-history-specific sensitivity to demographic stochasticity. This is of great value when assessing the vulnerability of small populations.     

Quantifying individual heterogeneity and its influence on life‐history trajectories: different methods for different questions and contexts

Heterogeneity among individuals influences the life‐history trajectories we observe at the population level because viability selection, selective immigration and emigration processes, and ontogeny change the proportion of individuals with specific trait values with increasing age. Here, we review the two main approaches that have been proposed to account for these processes in life‐history trajectories, contrasting how they quantify ontogeny and selection, and proposing ways to overcome some of their limitations. Nearly all existing approaches to model individual heterogeneity assume either a single normal distribution or a priori known groups of individuals. Ontogenetic processes, however, can vary across individuals through variation in life‐history tactics. We show the usefulness of describing ontogenetic processes by modelling trajectories with a mixture model that focuses on heterogeneity in life‐history tactics. Additionally, most methods examine individual heterogeneity in a single trait, ignoring potential correlations among multiple traits caused by latent common sources of individual heterogeneity. We illustrate the value of using a joint modelling approach to assess the presence of a shared latent correlation and its influence on life‐history trajectories. We contrast the strengths and limitations of different methods for different research questions, and we exemplify the differences among methods using empirical data from long‐term studies of ungulates.     

Nonstochastic variation of species-level diversification rates within angiosperms

Variations in the origination and extinction rates of species over geological time often are linked with a range of factors, including the evolution of key innovations, changes in ecosystem structure, and environmental factors such as shifts in climate and physical geography. Before hypothesizing causality of a single factor, it is critical to demonstrate that the observed variation in diversification is significantly greater than one would expect due to natural stochasticity in the evolutionary branching process. Here, we use a likelihood-ratio test to compare taxonomic rate heterogeneity to a neutral birth-death model, using data on well-supported sister pairs of taxa and their species richness. We test the likelihood that the distribution of extant species among angiosperm genera and families could be the result of constant diversification rates. Results strongly support the conclusion that there is significantly more heterogeneity in diversity at the species level within angiosperms than would be expected due to stochastic processes. This result is consistent in datasets of genus pairs and family pairs and is not affected significantly by degrading pairs to simulate inaccuracy in the assumption of simultaneous origin of sister taxa. When we parse taxon pairs among higher groups of angiosperms, results indicate that a constant rates model is not rejected by rosid and basal eudicot pairs but is rejected by asterid and eumagnoliid pairs. These results provide strong support for the hypothesis that species-level rates of origination and/or extinction have varied nonrandomly within angiosperms and that the magnitude of heterogeneity varies among major groups within angiosperms.     

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.     

Stochastic model for analysis of longitudinal data on aging and mortality

Aging-related changes in a human organism follow dynamic regularities, which contribute to the observed age patterns of incidence and mortality curves. An organism's 'optimal' (normal) physiological state changes with age, affecting the values of risks of disease and death. The resistance to stresses, as well as adaptive capacity, declines with age. An exposure to improper environment results in persisting deviation of individuals' physiological (and biological) indices from their normal state (due to allostatic adaptation), which, in turn, increases chances of disease and death. Despite numerous studies investigating these effects, there is no conceptual framework, which would allow for putting all these findings together, and analyze longitudinal data taking all these dynamic connections into account. In this paper we suggest such a framework, using a new version of stochastic process model of aging and mortality. Using this model, we elaborated a statistical method for analyses of longitudinal data on aging, health and longevity and tested it using different simulated data sets. The results show that the model may characterize complicated interplay among different components of aging-related changes in humans and that the model parameters are identifiable from the data.     

Life history evolution and demographic stochasticity

Summary Can demographic stochasticity bias the evolution of life history traits? Under a neutral version of the Cole-Charnov-Schaffer model, variance in offspring number for both annuals and perennials depends on the precise values of fitness components. Either annuals or perennials may have the larger variance (for equal ), depending on the importance of random survivalversus fixed reproduction. By extension, the variance in offspring number should generally depend on whether is mainly composed of highly variable elements or elements with limited variation. Thus, data about the variability of demographic parameters may be as important as data about their mean values.This result concerns only one source of demographic stochasticity, the probabilistic nature of demographic processes like survival. The other source of demographic stochasticity is the fact that populations are composed of whole numbers of individuals (integer arithmetic). Integer arithmetic without probabilistic demography (or environmental variation) can make it difficult for rare invaders to persist in populations even when selection would favour the invaders in a deterministic model. Integer arithmetic can also cause population coexistence when the equivalent deterministic model leads to exclusion. This effect disappears when demography is probabilistic, and probably also when there is environmental variation. Thus probabilistic demography and environmental variation may make some population patterns more, rather than less, understandable.     

Fluctuating Environments and Phytoplankton Community Structure: A Stochastic Model

Spatial heterogeneity in organism and resource distributions can generate temporal heterogeneity in resource access for simple organisms like phytoplankton. The role of temporal heterogeneity as a structuring force for simple communities is investigated via models of phytoplankton with contrasting life histories competing for a single fluctuating resource. A stochastic model in which environmental and demographic stochasticity are treated separately is compared with a model with deterministic resource variation to assess the importance of stochasticity. When compared with the deterministic model, the stochastic model allows for coexistence over a wider range of parameter values (or life-history types). The model suggests that demographic stochasticity alone is far more important in increasing the possibility of coexistence than environmental stochasticity alone. However, the combined effects of both types of stochasticity produce the largest likelihood of coexistence. Finally, the influence of relative nutrient levels and nutrient pulse frequency on these results is addressed. We relate our findings to variable environment theory with evidence for both relative nonlinearity and the storage effect acting in this model. We show for the first time that temporal dynamics generated by demographic stochasticity may operate like the storage effect at particular spatial scales.     

Stage,age and individual stochasticity in demography

Demography is the study of the population consequences of the fates of individuals. Individuals are differentiated on the basis of age or, in general, life cycle stages. The movement of an individual through its life cycle is a random process, and although the eventual destination (death) is certain, the pathways taken to that destination are stochastic and will differ even between identical individuals; this is individual stochasticity. A stage‐classified demographic model contains implicit age‐specific information, which can be analyzed using Markov chain methods. The living stages in the life cycles are transient states in an absorbing Markov chain; death is an absorbing state. This paper presents Markov chain methods for computing the mean and variance of the lifetime number of visits to any transient state, the mean and variance of longevity, the net reproductive rate R₀, and the cohort generation time. It presents the matrix calculus methods needed to calculate the sensitivity and elasticity of all these indices to any life history parameters. These sensitivities have many uses, including calculation of selection gradients. It is shown that the use of R₀ as a measure of fitness or an invasion exponent gives erroneous results except when R₀=λ=1. The Markov chain approach is then generalized to variable environments (deterministic environmental sequences, periodic environments, iid random environments, Markovian environments). Variable environments are analyzed using the vec‐permutation method to create a model that classifies individuals jointly by the stage and environmental condition. Throughout, examples are presented using the North Atlantic right whale (Eubaleana glacialis) and an endangered prairie plant (Lomatium bradshawii) in a stochastic fire environment.     

MicroRNA predictors of longevity in Caenorhabditis elegans

Neither genetic nor environmental factors fully account for variability in individual longevity: genetically identical invertebrates in homogenous environments often experience no less variability in lifespan than outbred human populations. Such variability is often assumed to result from stochasticity in damage accumulation over time; however, the identification of early-life gene expression states that predict future longevity would suggest that lifespan is least in part epigenetically determined. Such "biomarkers of aging," genetic or otherwise, nevertheless remain rare. In this work, we sought early-life differences in organismal robustness in unperturbed individuals and examined the utility of microRNAs, known regulators of lifespan, development, and robustness, as aging biomarkers. We quantitatively examined Caenorhabditis elegans reared individually in a novel apparatus and observed throughout their lives. Early-to-mid-adulthood measures of homeostatic ability jointly predict 62% of longevity variability. Though correlated, markers of growth/muscle maintenance and of metabolic by-products ("age pigments") report independently on lifespan, suggesting that graceful aging is not a single process. We further identified three microRNAs in which early-adulthood expression patterns individually predict up to 47% of lifespan differences. Though expression of each increases throughout this time, mir-71 and mir-246 correlate with lifespan, while mir-239 anti-correlates. Two of these three microRNA "biomarkers of aging" act upstream in insulin/IGF-1-like signaling (IIS) and other known longevity pathways, thus we infer that these microRNAs not only report on but also likely determine longevity. Thus, fluctuations in early-life IIS, due to variation in these microRNAs and from other causes, may determine individual lifespan.     

The best in all possible worlds? A quantitative genetic study of geographic variation in the meadow vole,Microtus pennsylvanicus

The meadow vole, Microtus pennsylvanicus , is the most widely distributed Microtus species in North America. Across its range, it shows marked demographic differences, experiences a large range of climatic conditions, and varies considerably in body size and life-history characteristics. To study the genetic basis of the geographic variation in size and life history of this species, we subjected three populations, one from central Canada and two from eastern Canada, to quantitative genetic analysis in the lab. We studied the variance and covariance of several size and growth variables as well as age and size at maturity by means of population crosses, full-sib analysis, and parent-offspring regressions. We found that the phenotypic differences among these populations are almost entirely due to environmental effects. However, within populations, additive genetic and maternal effects explain most of the variation. We discuss possible explanations for the lack of genetic differences among the populations and speculate that a similar reaction norm is maintained in all populations through heterogeneity in the temporal or spatial environment that the populations experience. The heterogeneity may be mediated through population density fluctuations, climatic variation, or variation in site productivity. Thus, we hypothesize that M. pennsylvanicus has evolved to be the best in all possible worlds rather than in one actual world. This study highlights the crucial importance of maternal and environmental effects on the size, growth, and life history of small rodents.