Correctly estimating how environmental stochasticity influences fitness and population growth

Increased temporal variance in life-history traits is generally predicted to decrease individual fitness and population growth. We show that a widely used result of stochastic sensitivity analysis that bolsters this generality is flawed because it ignores the effects of correlations between vital rates. Considering the effects of these correlations (although ignoring autocorrelations), we show that the apparently simple relationship between vital rate variance and fitness can be considerably more complex than previously thought. In particular, the previously estimated negative sensitivities of fitness or population growth to variance in a vital rate can be either enhanced by positive correlations between rates or reversed by negative correlations, even to the point that variability in a rate can increase fitness or population growth. We apply this new sensitivity calculation to data from the desert tortoise and discuss its interpretation in light of the factors generating vital rate correlations.     

Expected relative fitness and the adaptive topography of fluctuating selection

Wright's adaptive topography describes gene frequency evolution as a maximization of mean fitness in a constant environment. I extended this to a fluctuating environment by unifying theories of stochastic demography and fluctuating selection, assuming small or moderate fluctuations in demographic rates with a stationary distribution, and weak selection among the types. The demography of a large population, composed of haploid genotypes at a single locus or normally distributed phenotypes, can then be approximated as a diffusion process and transformed to produce the dynamics of population size, N, and gene frequency, p, or mean phenotype, . The expected evolution of p or is a product of genetic variability and the gradient of the long-run growth rate of the population, , with respect to p or . This shows that the expected evolution maximizes , the mean Malthusian fitness in the average environment minus half the environmental variance in population growth rate. Thus, as a function of p or represents an adaptive topography that, despite environmental fluctuations, does not change with time. The haploid model is dominated by environmental stochasticity, so the expected maximization is not realized. Different constraints on quantitative genetic variability, and stabilizing selection in the average environment, allow evolution of the mean phenotype to undergo a stochastic maximization of . Although the expected evolution maximizes the long-run growth rate of the population, for a genotype or phenotype the long-run growth rate is not a valid measure of fitness in a fluctuating environment. The haploid and quantitative character models both reveal that the expected relative fitness of a type is its Malthusian fitness in the average environment minus the environmental covariance between its growth rate and that of the population.     

Does environmental stochasticity matter? Analysis of red deer life-histories on Rum

Summary Most life-history theory assumes that short-term variation in an organism's environment does not affect the survivorships and fecundities of the organisms. This assumption is rarely met. Here we investigate the population and evolutionary biology of red deer,Cervus elephas, to see if relaxation of this assumption is likely to make significant differences to the predicted evolutionary biology of this species. To do this we used 21 years of data from a population of deer on Rum, Western Isles, Scotland. Population growth rates in a stochastic environment were estimated using Tuljapurkar's small noise approximation, confirmed by bootstrap simulation. Numerical differentiation was used to see if the selection pressures (i.e. sensitivities of population growth rate to changes in the vital rates) differ between the stochastic and deterministic cases. The data also allow the costs of reproduction to be estimated. These costs, incorporated as trade-offs into the sensitivity analysis, allow investigation of evolutionary benefits of different life-history tactics. Environmentally induced stochastic variation in the red deer vital rates causes a slight reduction ( 1%) in the predicted population growth rate and has little impact on the estimated selection pressures on the deer's life-history. We thus conclude that, even though density-independent stochastic effects on the population are marked, the deer's fitness is not markedly affected by these and they are adapted to the average conditions they experience. However, the selected life-history is sensitive to the trade-offs between current fecundity, survivorship and future fecundity and it is likely that the environmental variance will affect these trade-offs and, thus, affect the life-history favoured by selection. We also show that the current average life-history is non-optimal and suggest this is a result of selection pressures exerted by culling and predation, now much reduced. As the use of stochastic or deterministic methods provide similar estimates in this case, the use of the latter is justified. Thus,r (the annual per capita rate of population growth) is an appropriate measure of fitness in a population with stochastic numerical fluctuations. In a population of constant size lifetime reproductive success is the obvious measure of fitness to use.     

The evolution of parental care in stochastic environments

Parental care is of fundamental importance to understanding reproductive strategies and allocation decisions. Here, we explore how parental care strategies evolve in variable environments. Using a set of life-history trait trade-offs, we explore the relative costs and benefits of parental care in stochastic environments. Specifically, we consider the cases in which environmental variability results in varying adult death rates, egg death rates, reproductive rate and carrying capacity. Using a measure of fitness appropriate for stochastic environments, we find that parental care has the potential to evolve over a wide range of life-history characteristics when the environment is variable. A variable environment that affects adult or egg death rates can either increase or decrease the fitness of care relative to that in a constant environment, depending on the specific costs of care. Variability that affects carrying capacity or adult reproductive rate has negligible effects on the fitness associated with care. Increasing parental care across different life-history stages can increase fitness gains in variable environments. Costly investment in care is expected to affect the overall fitness benefits, the fitness optimum and rate of evolution of parental care. In general, we find that environmental variability, the life-history traits affected by such variability and the specific costs of care interact to determine whether care will be favoured in a variable environment and what levels of care will be selected.     

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.     

Dynamics of age- and size-structured populations in fluctuating environments: Applications of stochastic matrix models to natural populations

Recent developments of the theory of stochastic matrix modeling have made it possible to estimate general properties of age- and size-structured populations in fluctuating environments. However, applications of the theory to natural populations are still few. The empirical studies which have used stochastic matrix models are reviewed here to examine whether predictions made by the theory can be generally found in wild populations. The organisms studied include terrestrial grasses and herbs, a seaweed, a fish, a reptile, a deer and some marine invertebrates. In all the studies, the stochastic population growth rate (ln λ _s ) was no greater than the deterministic population growth rate determined using average vital rates, suggesting that the model based only on average vital rates may overestimate growth rates of populations in fluctuating environments. Factors affecting ln λ _s include the magnitude of variation in vital rates, probability distribution of random environments, fluctuation in different types of vital rates, covariances between vital rates, and autocorrelation between successive environments. However, comprehensive rules were hardly found through the comparisons of the empirical studies. Based on shortcomings of previous studies, I address some important subjects which should be examined in future studies.     

Demographic strategies of plant invaders in temporally varying environments

Plant populations may have evolved different demographic strategies to cope with temporal environmental variation. According to the demographic buffering hypothesis, vital rates that are most critical to population persistence are buffered against environmental variation and vary little over time, whereas the demographic lability hypothesis suggests that populations may track and benefit from environmental variation. While the hypotheses of demographic strategies have been widely tested in plant and animal species, they have not been explicitly examined for invasive plants, or in relation to different modelling methods (deterministic vs. stochastic). Here, we tested the demographic buffering and lability hypotheses for 23 populations of eight invasive plant species in relation to life form (woody vs. herbaceous species) and population growth rate using deterministic and stochastic modelling methods, and absolute and relative scales. We found that conclusions of demographic strategies depended on scale, with an absolute scale resulting in stronger negative correlations between the variability and importance of vital rates (i.e., buffering) than a relative scale. Conclusions of demographic strategies were also affected by life form that interacted with method. The populations of woody invaders exhibited buffering regardless of the method used, while for the populations of herbaceous species, deterministic calculations suggested buffering and stochastic calculations suggested lability. Overall, our findings emphasise the role of life form and methodological issues that need to be considered when exploring demographic strategies in fluctuating environments.     

Accounting for stochasticity in demographic compensation along the elevational range of an alpine plant

Demographic compensation arises when vital rates change in opposite directions across populations, buffering the variation in population growth rates, and is a mechanism often invoked to explain the stability of species geographic ranges. However, studies on demographic compensation have disregarded the effects of temporal variation in vital rates and their temporal correlations, despite theoretical evidence that stochastic dynamics can affect population persistence in temporally varying environments. We carried out a seven‐year‐long demographic study on the perennial plant Arabis alpina (L.) across six populations encompassing most of its elevational range. We discovered demographic compensation in the form of negative correlations between the means of plant vital rates, but also between their temporal coefficients of variation, correlations and elasticities. Even if their contribution to demographic compensation was small, this highlights a previously overlooked, but potentially important, role of stochastic processes in stabilising population dynamics at range margins.     

Mapping the environmental fitness landscape of a synthetic gene circuit

Gene expression actualizes the organismal phenotypes encoded within the genome in an environment-dependent manner. Among all encoded phenotypes, cell population growth rate (fitness) is perhaps the most important, since it determines how well-adapted a genotype is in various environments. Traditional biological measurement techniques have revealed the connection between the environment and fitness based on the gene expression mean. Yet, recently it became clear that cells with identical genomes exposed to the same environment can differ dramatically from the population average in their gene expression and division rate (individual fitness). For cell populations with bimodal gene expression, this difference is particularly pronounced, and may involve stochastic transitions between two cellular states that form distinct sub-populations. Currently it remains unclear how a cell population's growth rate and its subpopulation fractions emerge from the molecular-level kinetics of gene networks and the division rates of single cells. To address this question we developed and quantitatively characterized an inducible, bistable synthetic gene circuit controlling the expression of a bifunctional antibiotic resistance gene in Saccharomyces cerevisiae. Following fitness and fluorescence measurements in two distinct environments (inducer alone and antibiotic alone), we applied a computational approach to predict cell population fitness and subpopulation fractions in the combination of these environments based on stochastic cellular movement in gene expression space and fitness space. We found that knowing the fitness and nongenetic (cellular) memory associated with specific gene expression states were necessary for predicting the overall fitness of cell populations in combined environments. We validated these predictions experimentally and identified environmental conditions that defined a "sweet spot" of drug resistance. These findings may provide a roadmap for connecting the molecular-level kinetics of gene networks to cell population fitness in well-defined environments, and may have important implications for phenotypic variability of drug resistance in natural settings.     

Stochastic gene expression in fluctuating environments

Stochastic mechanisms can cause a group of isogenic bacteria, each subject to identical environmental conditions, to nevertheless exhibit diverse patterns of gene expression. The resulting phenotypic subpopulations will typically have distinct growth rates. This behavior has been observed in several contexts, including sugar metabolism and pili phase variation. Under fixed environmental conditions, the net growth rate of the population is maximized when all cells are of the fastest growing phenotype, so it is unclear what fitness advantage is conferred by population heterogeneity. However, unlike ideal laboratory conditions, natural environments tend to fluctuate, either periodically or randomly. Here we use a stochastic population model to show that, during growth in such fluctuating environments, a dynamically heterogenous bacterial population can sometimes achieve a higher net growth rate than a homogenous one. By using stochastic mechanisms to sample several distinct phenotypes, the bacteria are able to anticipate and take advantage of sudden changes in their environment. However, this heterogeneity is beneficial only if the bacterial response rate is sufficiently low. Our results could be useful in the design of artificial evolution experiments and in the optimization of fermentation processes.     

Stochastic demography and population dynamics in the red kangaroo Macropus rufus

1.  Many organisms inhabit strongly fluctuating environments but their demography and population dynamics are often analysed using deterministic models and elasticity analysis, where elasticity is defined as the proportional change in population growth rate caused by a proportional change in a vital rate. Deterministic analyses may not necessarily be informative because large variation in a vital rate with a small deterministic elasticity may affect the population growth rate more than a small change in a less variable vital rate having high deterministic elasticity.
2.  We analyse a stochastic environment model of the red kangaroo ( Macropus rufus ), a species inhabiting an environment characterized by unpredictable and highly variable rainfall, and calculate the elasticity of the stochastic growth rate with respect to the mean and variability in vital rates.
3.  Juvenile survival is the most variable vital rate but a proportional change in the mean adult survival rate has a much stronger effect on the stochastic growth rate.
4.  Even if changes in average rainfall have a larger impact on population growth rate, increased variability in rainfall may still be important also in long-lived species. The elasticity with respect to the standard deviation of rainfall is comparable to the mean elasticities of all vital rates but the survival in age class 3 because increased variation in rainfall affects both the mean and variability of vital rates.
5.  Red kangaroos are harvested and, under the current rainfall pattern, an annual harvest fraction of c . 20% would yield a stochastic growth rate about unity. However, if average rainfall drops by more than c . 10%, any level of harvesting may be unsustainable, emphasizing the need for integrating climate change predictions in population management and increase our understanding of how environmental stochasticity translates into population growth rate.     

Selective consequences of catastrophes for growth rates in a stream-dwelling salmonid

Optimal life histories in a fluctuating environment are likely to differ from those that are optimal in a constant environment, but we have little understanding of the consequences of bounded fluctuations versus episodic massive mortality events. Catastrophic disturbances, such as floods, droughts, landslides and fires, substantially alter the population dynamics of affected populations, but little has been done to investigate how catastrophes may act as a selective agent for life-history traits. We use an individual-based model of population dynamics of the stream-dwelling salmonid marble trout (Salmo marmoratus) to investigate how trade-offs between the growth and mortality of individuals and density-dependent body growth can lead to the maintenance of a wide or narrow range of individual variation in body growth rates in environments that are constant (i.e., only demographic stochasticity), variable (i.e., environmental stochasticity), or variable with catastrophic events that cause massive mortalities (e.g., flash floods). We find that occasional episodes of massive mortality can substantially reduce persistent variability in individual growth rates. Lowering the population density reduces density dependence and allows for higher fitness of more opportunistic strategies (rapid growth and early maturation) during the recovery period.     

Experimental demography and the vital rates of generalist and specialist insect herbivores on native and novel host plants

1. Colonization success of species when confronted with novel environments is of interest in ecological, evolutionary and conservation contexts. Such events may represent the first step for ecological diversification. They also play an important role in adaptive divergence and speciation. 2. A species that is able to do well across a range of environments has a higher plasticity than one whose success is restricted to a single or few environments. The breadth of environments in which a species can succeed is ultimately determined by the full pattern of its vital rates in each environment. 3. Examples of organisms colonizing novel environments are insect herbivores expanding their diets to novel host plants. One expectation for insect herbivores is that species with specialized diets may display less plasticity when faced with novel hosts than generalist species. 4. We examine this hypothesis for two generalist and two specialist neotropical beetles (genus Cephaloleia: Chrysomelidae) currently expanding their diets from native to novel plants of the order Zingiberales. Using an experimental approach, we estimated changes in vital rates, life-history traits and lifetime fitness for each beetle species when feeding on native or novel host plants. 5. We did not find evidence supporting more plasticity for generalists than for specialists. Instead, we found similar patterns of survival and fecundity for all herbivores. Larvae survived worse on novel hosts; adults survived at least as well or better, but reproduced less on the novel host than on natives. 6. Some of the novel host plants represent challenging environments where population growth was negative. However, in four novel plant-herbivore interactions, instantaneous population growth rates were positive. 7. Positive instantaneous population growth rates during initial colonization of novel host plants suggest that both generalist and specialist Cephaloleia beetles may be pre-adapted to feed on some novel hosts. This plasticity in host use is a key factor for successful colonization of novel hosts. Future success or failure in the colonization of these novel hosts will depend on the demographic rates described in this research, natural selection and the evolutionary responses of life-history traits in novel environments.     

Demography and population dynamics of the mouse opossum (Thylamys elegans) in semi-arid Chile: seasonality, feedback structure and climate

Here we present, to the authors' knowledge for the very first time for a small marsupial, a thorough analysis of the demography and population dynamics of the mouse opossum (Thylamys elegans) in western South America. We test the relative importance of feedback structure and climatic factors (rainfall and the Southern Oscillation Index) in explaining the temporal variation in the demography of the mouse opossum. The demographic information was incorporated into a stage-structured population dynamics model and the model's predictions were compared with observed patterns. The mouse opossum's capture rates showed seasonal (within-year) and between-year variability, with individuals having higher capture rates during late summer and autumn and lower capture rates during winter and spring. There was also a strong between-year effect on capture probabilities. The reproductive (the fraction of reproductively active individuals) and recruitment rates showed a clear seasonal and a between-year pattern of variation with the peak of reproductive activity occuring during winter and early spring. In addition, the fraction of reproductive individuals was positively related to annual rainfall, while population density and annual rainfall positively influenced the recruitment rate. The survival rates were negatively related to annual rainfall. The average finite population growth rate during the study period was estimated to be 1.011 +/- 0.0019 from capture-recapture estimates. While the annual growth rate estimated from the seasonal linear matrix models was 1.026, the subadult and adult survival and maturation rates represent between 54% (winter) and 81% (summer) of the impact on the annual growth rate.     

Evolution in the real world: stochastic variation and the determinants of fitness in Carlina vulgaris

Empirical studies of life histories often ignore stochastic variation, despite theoretical demonstrations of its potential impact on life-history evolution. Here we use a novel approach to explore the effects of stochastic variation on life-history evolution and estimate the selection pressures operating on the monocarpic perennial Carlina vulgaris, in which flowering may be delayed by up to eight years. The approach is novel in that we use modern theoretical techniques to estimate selection pressures and the fitness landscape from a fully parameterised individual-based model. These approaches take into account temporal variation in demographic rates and density dependence. Analysis of 16 years' data revealed significant temporal variation in growth, mortality, and recruitment in our study population. Flowering was strongly size dependent and, unusually for such a species, also age dependent. Individual-based models of the flowering strategy, parameterized using field data, consistently underestimated the size at flowering, when temporal variation in demographic rates was ignored. In contrast, models that incorporated temporal variation in growth, mortality, and recruitment predicted sizes at flowering not significantly different from those observed in the field. Temporal variation in mortality, which had the largest effect on the flowering strategy, selected for increased size at flowering. An analytical approximation is presented to explain this result, extending the "1-year look-ahead criterion" presented in Rees et al. (2000). A fitness landscape generated by following the fate of rare mutant invaders with a broad range of alternative flowering strategies demonstrated that the observed parameters were adaptive. However, the fitness landscape reveals that approximately equal fitness is achieved by a broad range of strategies, providing a mechanism for the maintenance of genetic variation. To understand how the different parameters that defined our models determine the fitness of rare mutants, we numerically estimated the elasticities and sensitivities of mutant fitness. This demonstrated strong selection on a number of the parameters. Elasticities and sensitivities estimated in constant and random environments were significantly positively correlated, and both were negatively related to the standard error of the parameter. This last result is surprising and, we argue, reflects the genetic and phenotypic responses to selection.     

EVOLUTION IN FLUCTUATING ENVIRONMENTS: DECOMPOSING SELECTION INTO ADDITIVE COMPONENTS OF THE ROBERTSON–PRICE EQUATION

We analyze the stochastic components of the Robertson–Price equation for the evolution of quantitative characters that enables decomposition of the selection differential into components due to demographic and environmental stochasticity. We show how these two types of stochasticity affect the evolution of multivariate quantitative characters by defining demographic and environmental variances as components of individual fitness. The exact covariance formula for selection is decomposed into three components, the deterministic mean value, as well as stochastic demographic and environmental components. We show that demographic and environmental stochasticity generate random genetic drift and fluctuating selection, respectively. This provides a common theoretical framework for linking ecological and evolutionary processes. Demographic stochasticity can cause random variation in selection differentials independent of fluctuating selection caused by environmental variation. We use this model of selection to illustrate that the effect on the expected selection differential of random variation in individual fitness is dependent on population size, and that the strength of fluctuating selection is affected by how environmental variation affects the covariance in Malthusian fitness between individuals with different phenotypes. Thus, our approach enables us to partition out the effects of fluctuating selection from the effects of selection due to random variation in individual fitness caused by demographic stochasticity.     

Temporal autocorrelation and stochastic population growth

How much does environmental autocorrelation matter to the growth of structured populations in real life contexts? Interannual variances in vital rates certainly do, but it has been suggested that between‐year correlations may not. We present an analytical approximation to stochastic growth rate for multistate Markovian environments and show that it is accurate by testing it in two empirically based examples. We find that temporal autocorrelation has sizeable effect on growth rates of structured populations, larger in many cases than the effect of interannual variability. Our approximation defines a sensitivity to autocorrelated variability, showing how demographic damping and environmental pattern interact to determine a population's stochastic growth rate.     

Evolution and population dynamics in stochastic environments

Inter-generational temporal variability of the environment is important in the evolution and adaptation of phenotypic traits. We discuss a population-dynamic approach which plays a central role in the analysis of evolutionary processes. The basic principle is that the phenotypes with the greatest long-term average growth rate will dominate the entire population. The calculation of longterm average growth rates for populations under temporal stochasticity can be highly cumbersome. However, for a discrete non-overlapping population, it is identical to the geometric mean of the growth rates (geometric mean fitness), which is usually different from the simple arithmetic mean of growth rates. Evolutionary outcomes based on geometric mean fitness are often very different from the predictions based on the usual arithmetic mean fitness. In this paper we illustrate the concept of geometric mean fitness in a few simple models. We discuss its implications for the adaptive evolution of phenotypes, e.g. foraging under predation risks and clutch size. Next, we present an application: the risk-spreading egg-laying behaviour of the cabbage white butterfly, and develop a two-patch population dynamic model to show how the optimal solution diverges from the ssual arithmetic mean approach. The dynamics of these stochastic models cannot be predicted from the dynamics of simple deterministic models. Thus the inclusion of stochastic factors in the analyses of populations is essential to the understanding of not only population dynamics, but also their evolutionary dynamics.     

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.     

Generation time: a reliable metric to measure life-history variation among mammalian populations

Oli and Dobson proposed that the ratio between the magnitude and the onset of reproduction (F/ alpha ratio) allows one to predict the relative importance of vital rates on population growth rate in mammalian populations and provides a reliable measure of the ranking of mammalian species on the slow-fast continuum of life-history tactics. We show that the choice of the ratio F/ alpha is arbitrary and is not grounded in demographic theory. We estimate the position on the slow-fast continuum using the first axis of a principal components analysis of all life-history variables studied by Oli and Dobson and show that most individual vital rates perform as well as the F/ alpha ratio. Finally, we find, in agreement with previous studies, that the age of first reproduction is a reliable predictor of the ranking of mammalian populations along the slow-fast continuum and that both body mass and phylogeny markedly influence the generation time of mammalian species. We conclude that arbitrary ratios such as F/ alpha correlate with life-history types in mammals simply because life-history variables are highly correlated in response to allometric, phylogenetic, and environmental influences. We suggest that generation time is a reliable metric to measure life-history variation among mammalian populations and should be preferred to any arbitrary combination between vital rates.