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.     

Do haddock select habitats to maximize condition?

Haddock Melanogrammus aeglefinus in the North sea increased their distributional range when more abundant, but this density dependent habitat selection (DDHS) explained only a small part of the year‐on‐year variation in distribution patterns. The condition of haddock was examined at 24 sites in the North Sea in August and September 2004 and related to their abundance, to examine if the ideal free distribution theory (IFD), which assumes that organisms select habitats that maximize their rate of food intake, can be used to explain this variation in large scale distribution patterns. At a given temperature, condition (hepato‐somatic index, I _H) was better at stations where haddock were most abundant. Therefore, haddock were not distributed perfectly according to the IFD in 2004. The positive correlation between abundance and I _H, however, indicated there was some habitat selection by haddock, as in the total absence of habitat selection no correlation between I _H and abundance, and no spatial variation in abundance was expected. DDHS may only explain a small part of the yearly variation in the distribution because haddock did not equalize and maximize their fitness at the scale of the North Sea. In addition, stable isotope analysis of muscle samples showed that haddock did not avoid competition for food when at high abundance by feeding at a lower or wider range of trophic levels.     

A piecewise linear modeling approach for testing competing theories of habitat selection: an example with mule deer in northern winter ranges

Habitat selection fundamentally drives the distribution of organisms across landscapes; density-dependent habitat selection (DDHS) is considered a central component of ecological theories explaining habitat use and population regulation. A preponderance of DDHS theories is based on ideal distributions, such that organisms select habitat according to either the ideal free, despotic, or pre-emptive distributions. Models that can be used to simultaneously test competing DDHS theories are desirable to help improve our understanding of habitat selection. We developed hierarchical, piecewise linear models that allow for simultaneous testing of DDHS theories and accommodate densities from multiple habitats and regional populations, environmental covariates, and random effects. We demonstrate the use of these models with data on mule deer (Odocoileus hemionus) abundance and net energy costs in different snow depths within winter ranges of five regional populations in western Idaho, USA. Regional population density explained 40 % of the variation in population growth, and we found that deer were ideal free in winter ranges. Deer occupied habitats with lowest net energy costs at higher densities and at a higher rate than compared to habitats with intermediate and high energy costs. The proportion of a regional population in low energy cost habitat the previous year accounted for a significant amount of variation in population growth (17 %), demonstrating the importance of winter habitat selection in regulating deer populations. These linear models are most appropriate for empirical data collected from centralized habitat patches within the local range of a species where individuals are either year-round residents or migratory (but have already arrived from migration).     

Density-dependent habitat selection alters drivers of population distribution in northern Yellowstone elk

Although it is well established that density dependence drives changes in organismal abundance over time, relatively little is known about how density dependence affects variation in abundance over space. We tested the hypothesis that spatial trade-offs between food and safety can change the drivers of population distribution, caused by opposing patterns of density-dependent habitat selection (DDHS) that are predicted by the multidimensional ideal free distribution. We addressed this using winter aerial survey data of northern Yellowstone elk (Cervus canadensis) spanning four decades. Supporting our hypothesis, we found positive DDHS for food (herbaceous biomass) and negative DDHS for safety (openness and roughness), such that the primary driver of habitat selection switched from food to safety as elk density decreased from 9.3 to 2.0 elk/km². Our results demonstrate how population density can drive landscape-level shifts in population distribution, confounding habitat selection inference and prediction and potentially affecting community-level interactions.     

Spatially heterogeneous stochasticity and the adaptive diversification of dormancy

Diversified bet‐hedging, a strategy that leads several individuals with the same genotype to express distinct phenotypes in a given generation, is now well established as a common evolutionary response to environmental stochasticity. Life‐history traits defined as diversified bet‐hedging (e.g. germination or diapause strategies) display marked differences between populations in spatial proximity. In order to find out whether such differences can be explained by local adaptations to spatially heterogeneous environmental stochasticity, we explored the evolution of bet‐hedging dormancy strategies in a metapopulation using a two‐patch model with patch differences in stochastic juvenile survival. We found that spatial differences in the level of environmental stochasticity, restricted dispersal, increased fragmentation and intermediate survival during dormancy all favour the adaptive diversification of bet‐hedging dormancy strategies. Density dependency also plays a major role in the diversification of dormancy strategies because: (i) it may interact locally with environmental stochasticity and amplify its effects; however, (ii) it can also generate chaotic population dynamics that may impede diversification. Our work proposes new hypotheses to explain the spatial patterns of bet‐hedging strategies that we hope will encourage new empirical studies of this topic.     

Metapopulation dynamics: Does it help to have more of the same?

With the interest in conservation biology shifting towards processes from patterns, and to populations from communities, the theory of metapopulation dynamics is replacing the equilibrium theory of island biogeography as the population ecology paradigm in conservation biology. The simplest models of metapopulation dynamics make predictions about the effects of habitat fragmentation - size and isolation of habitat patches - on metapopulation persistence. The simple models may be enriched by considerations of the effects of demographic and environmental stochasticity on the size and extinction probability of local populations. Environmental stochasticity affects populations at two levels: it makes local extinctions more probable, and it also decreases metapopulation persistence time by increasing the correlation of extinction events across populations. Some controversy has arisen over the significance of correlated extinctions, and how they may affect the optimal subdivision of metapopulations to maximize their persistence time.     

种群统计随机性和环境随机性对种群绝灭的影响

影响种群绝灭的随机干扰可分为种群统计随机性、环境随机性和随机灾害三大类。在相对稳定的环境条件下和相对较短的时间内,以前两类随机干扰对种群绝灭的影响为生态学家关注的焦点。但是,由于自然种群动态及其影响因子的复杂特征,进一步深入研究随机干扰对种群绝灭的作用在理论上和实践上都必须发展新的技术手段。本文回顾了种群统计随机性与环境随机性的概念起源与发展,系统阐述了其分析方法。归纳了两类随机性在种群绝灭研究中的应用范围、作用方式和特点的异同和区别方法。各类随机作用与种群动态之间关系的理论研究与对种群绝灭机理的实践研究紧密相关。根据理论模型模拟和自然种群实际分析两方面的研究现状,作者提出了进一步深入研究随机作用与种群非线性动态方法的策略。指出了随机干扰影响种群绝灭过程的研究的方向：更多的研究将从单纯的定性分析随机干扰对种群动力学简单性质的作用,转向结合特定的种群非线性动态特征和各类随机力作用特点具体分析绝灭极端动态的成因,以期做出精确的预测。     

From individual behavior to metapopulation dynamics: unifying the patchy population and classic metapopulation models

Spatially structured populations in patchy habitats show much variation in migration rate, from patchy populations in which individuals move repeatedly among habitat patches to classic metapopulations with infrequent migration among discrete populations. To establish a common framework for population dynamics in patchy habitats, we describe an individual-based model (IBM) involving a diffusion approximation of correlated random walk of individual movements. As an example, we apply the model to the Glanville fritillary butterfly (Melitaea cinxia) inhabiting a highly fragmented landscape. We derive stochastic patch occupancy model (SPOM) approximations for the IBMs assuming pure demographic stochasticity, uncorrelated environmental stochasticity, or completely correlated environmental stochasticity in local dynamics. Using realistic parameter values for the Glanville fritillary, we show that the SPOMs mimic the behavior of the IBMs well. The SPOMs derived from IBMs have parameters that relate directly to the life history and behavior of individuals, which is an advantage for model interpretation and parameter estimation. The modeling approach that we describe here provides a unified framework for patchy populations with much movements among habitat patches and classic metapopulations with infrequent movements.     

Density-dependent habitat selection: evaluating isoleg theory with a Lotka-Volterra model

Contemporary models of density-dependent habitat selection generally focus on long-term evolutionary consequences of intraspecific or interspecific competition and/or patterns of resource use in patchy environments. A primary goal of such studies often is to elucidate evolutionary stable strategies based on steady-state dynamics of population growth. In contrast, we developed a simulation model to explore short-term movements of interspecific competitors among fine-grained habitats of differing attributes, as might result from field manipulations of habitat quality or population densities. In this model, habitat quality is expressed in terms of mean individual fitness, represented by average per capita growth rate calculated according to the Lotka-Volterra equations describing interspecific competition. This model provides a mechanism for quantifying the effects of habitat quality, patterns of resource use and competition on distributions of individuals. Results demonstrate the heuristic value of this model in corroborating predictions derived from the ideal free distribution and isodar theory, and in generating isolegs to test the predictions of isoleg theory. Results indicate that small changes in model parameters have substantial impacts on patterns of habitat use and co-occurrence between species. The model identifies a variety of conditions under which isolegs for a given type of community organization deviate from predictions of contemporary isoleg theory, potentially expanding the universe of possible interspecific behaviors underlying the development of evolutionary stable strategies.     

Scales and costs of habitat selection in heterogeneous landscapes

Summary Two scales of habitat selection are likely to influence patterns of animal density in heterogeneous landscapes. At one scale, habitat selection is determined by the differential use of foraging locations within a home range. At a larger scale, habitat selection is determined by dispersal and the ability to relocate the home range. The limits of both scales must be known for accurate assessments of habitat selection and its role in effecting spatial patterns in abundance. Isodars, which specify the relationships between population density in two habitats such that the expected reproductive success of an individual is the same in both, allow us to distinguish the two scales of habitat selection because each scale has different costs. In a two-habitat environment, the cost of rejecting one of the habitats within a home range can be expressed as a devaluation of the other, because, for example, fine-grained foragers must travel through both. At the dispersal scale, the cost of accepting a new home range in a different habitat has the opposite effect of inflating the value of the original habitat to compensate for lost evolutionary potential associated with relocating the home range. These costs produce isodars at the foraging scale with a lower intercept and slope than those at the dispersal scale.Empirical data on deer mice occupying prairie and badland habitats in southern Alberta confirm the ability of isodar analysis to differentiate between foraging and dispersal scales. The data suggest a foraging range of approximately 60 m, and an effective dispersal distance near 140 m. The relatively short dispersal distance implies that recent theories may have over-emphasized the role of habitat selection on local population dynamics. But the exchange of individuals between habitats sharing irregular borders may be substantial. Dispersal distance may thus give a false impression of the inability of habitat selection to help regulate population density.     

Cycles, stochasticity and density dependence in pink salmon population dynamics

Complex dynamics of animal populations often involve deterministic and stochastic components. A fascinating example is the variation in magnitude of 2-year cycles in abundances of pink salmon (Oncorhynchus gorbuscha) stocks along the North Pacific rim. Pink salmon have a 2-year anadromous and semelparous life cycle, resulting in odd- and even-year lineages that occupy the same habitats but are reproductively isolated in time. One lineage is often much more abundant than the other in a given river, and there are phase switches in dominance between odd- and even-year lines. In some regions, the weak line is absent and in others both lines are abundant. Our analysis of 33 stocks indicates that these patterns probably result from stochastic perturbations of damped oscillations owing to density-dependent mortality caused by interactions between lineages. Possible mechanisms are cannibalism, disease transmission, food depletion and habitat degradation by which one lineage affects the other, although no mechanism has been well-studied. Our results provide comprehensive empirical estimates of lagged density-dependent mortality in salmon populations and suggest that a combination of stochasticity and density dependence drives cyclical dynamics of pink salmon stocks.     

Comparative population dynamics of large and small mammals in the Northern Hemisphere: deterministic and stochastic forces

Deterministic feedbacks within populations interact with extrinsic, stochastic processes to generate complex patterns of animal abundance over time and space. Animals inherently differ in their responses to fluctuating environments due to differences in body sizes and life history traits. However, controversy remains about the relative importance of deterministic and stochastic forces in shaping population dynamics of large and small mammals. We hypothesized that effects of environmental stochasticity and density dependence are stronger in small mammal populations relative to their effects in large mammal populations and thus differentiate the patterns of population dynamics between them. We conducted an extensive, comparative analysis of population dynamics in large and small mammals to test our hypothesis, using seven population parameters to describe general dynamic patterns for 23 (14 species) time series of observations of abundance of large mammals and 38 (21 species) time series for small mammals. We used state‐space models to estimate the strength of direct and delayed density dependence as well as the strength of environmental stochasticity. We further used phylogenetic comparative analysis to detect differences in population dynamic patterns and individual population parameters, respectively, between large and small mammals. General population dynamic patterns differed between large and small mammals. However, the strength of direct and delayed density dependence was comparable between large and small mammals. Moreover, the variances of population growth rates and environmental stochasticity were greater in small mammals than in large mammals. Therefore, differences in population response to stochastic forces and strength of environmental stochasticity are the primary factor that differentiates population dynamic patterns between large and small mammal species.     

Habitat-dependent foraging in a classic predator–prey system: a fable from snowshoe hares

Current research contrasting prey habitat use has documented, with virtual unanimity, habitat differences in predation risk. Relatively few studies have considered, either in theory or in practice, simultaneous patterns in prey density. Linear predator–prey models predict that prey habitat preferences should switch toward the safer habitat with increasing prey and predator densities. The density‐dependent preference can be revealed by regression of prey density in safe habitat versus that in the riskier one (the isodar). But at this scale, the predation risk can be revealed only with simultaneous estimates of the number of predators, or with their experimental removal. Theories of optimal foraging demonstrate that we can measure predation risk by giving‐up densities of resource in foraging patches. The foraging theory cannot yet predict the expected pattern as predator and prey populations covary. Both problems are solved by measuring isodars and giving‐up densities in the same predator–prey system. I applied the two approaches to the classic predator–prey dynamics of snowshoe hares in northwestern Ontario, Canada. Hares occupied regenerating cutovers and adjacent mature‐forest habitat equally, and in a manner consistent with density‐dependent habitat selection. Independent measures of predation risk based on experimental, as well as natural, giving‐up densities agreed generally with the equal preference between habitats revealed by the isodar. There was no apparent difference in predation risk between habitats despite obvious differences in physical structure. Complementary studies contrasting a pair of habitats with more extreme differences confirmed that hares do alter their giving‐up densities when one habitat is clearly superior to another. The results are thereby consistent with theories of adaptive behaviour. But the results also demonstrate, when evaluating differences in habitat, that it is crucial to let the organisms we study define their own habitat preference.     

Optimal age of maturity in fluctuating environments under r‐ and K‐selection

Optimality models for evolution of life histories have shown that increased environmental stochasticity promotes early age of maturity. Here we argue that if r‐selection for early maturation implies a tradeoff making those phenotypes more sensitive to a change in population size than phenotypes maturing at older ages, K‐selection can favor delayed onset of maturation. We analyze a general stochastic Leslie‐matrix model with a simplified density regulation affecting all survivals equally through a function of the population vector, often called the ‘critical age class’. We show that the outcome of such an age‐dependent r‐ and K‐selection is that the expected value of the ‘critical age class’ is maximized by evolution, a strategy strongly influenced by the magnitude of the environmental stochasticity. We also demonstrate that evolution caused by such density‐dependent selection influences the population dynamics, showing a possible reciprocal effect between ecology and evolution in age‐structured populations. This modeling approach reveals that changes in population size affecting the fitness of phenotypes with different age of maturity may be an important selective agent for variation in onset of reproduction in fluctuating environments. This provides a testable hypothesis for how patterns in the population dynamics should affect life history variation.     

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.     

Natural selection and population dynamics

To what extent, and under which circumstances, are population dynamics influenced by concurrent natural selection? Density dependence and environmental stochasticity are generally expected to subsume any selective modulation of population growth rate, but theoretical considerations point to conditions under which selection can have an appreciable impact on population dynamics. By contrast, empirical research has barely scratched the surface of this fundamental question in population biology. Here, we present a diverse body of mostly empirical evidence that demonstrates how selection can influence population dynamics, including studies of small populations, metapopulations, cyclical populations and host-pathogen interactions. We also discuss the utility, in this context, of inferences from molecular genetic data, placing them within the broader framework of quantitative genetics and life-history evolution.     

Resampling Method for Applying Density-Dependent Habitat Selection Theory to Wildlife Surveys

Isodar theory can be used to evaluate fitness consequences of density-dependent habitat selection by animals. A typical habitat isodar is a regression curve plotting competitor densities in two adjacent habitats when individual fitness is equal. Despite the increasing use of habitat isodars, their application remains largely limited to areas composed of pairs of adjacent habitats that are defined a priori. We developed a resampling method that uses data from wildlife surveys to build isodars in heterogeneous landscapes without having to predefine habitat types. The method consists in randomly placing blocks over the survey area and dividing those blocks in two adjacent sub-blocks of the same size. Animal abundance is then estimated within the two sub-blocks. This process is done 100 times. Different functional forms of isodars can be investigated by relating animal abundance and differences in habitat features between sub-blocks. We applied this method to abundance data of raccoons and striped skunks, two of the main hosts of rabies virus in North America. Habitat selection by raccoons and striped skunks depended on both conspecific abundance and the difference in landscape composition and structure between sub-blocks. When conspecific abundance was low, raccoons and striped skunks favored areas with relatively high proportions of forests and anthropogenic features, respectively. Under high conspecific abundance, however, both species preferred areas with rather large corn-forest edge densities and corn field proportions. Based on random sampling techniques, we provide a robust method that is applicable to a broad range of species, including medium- to large-sized mammals with high mobility. The method is sufficiently flexible to incorporate multiple environmental covariates that can reflect key requirements of the focal species. We thus illustrate how isodar theory can be used with wildlife surveys to assess density-dependent habitat selection over large geographic extents.     

Searching for mechanisms of synchrony in spatially structured gamebird populations

1.  Time series data on five species of gamebird from the Dolomitic Alps were used to examine the relative importance of dispersal and common stochastic events in causing synchrony between spatially structured populations.
2.  Cross-correlation analysis of detrended time series was used to describe the spatial pattern of fluctuations in abundance, while standardized time series were used to describe both fluctuations and the trend in abundance. There were large variations in synchrony both within and between species and only weak negative relationships with distance.
3.  Species in neighbouring habitats were more likely to be in synchrony than species separated by several habitats. Species with similar density-dependent structure were more likely to be in synchrony.
4.  In order to estimate the relative importance of dispersal and environmental stochasticity, we modelled the spatial dynamics of each species using two different approaches. First, we used estimating functions and bootstrapping of time series data to calculate the relative importance of dispersal and stochastic effects for each species. Second, we estimated the intensity of environmental stochasticity from climatic records during the breeding season and then modelled the dispersal rate and dispersal distance for each species. The two models exhibited similar results for rock ptarmigan, black grouse, hazel grouse and rock partridge, while contrasting patterns were observed for capercaillie.
5.  The results suggest that environmental stochasticity plays the dominant role in synchronizing the fluctuations of these galliform species, although there will also be some dispersal between populations.     

Resource availability determines the importance of niche‐based versus stochastic community assembly in grasslands

Niche‐based selection and stochastic processes can operate simultaneously to generate spatial and temporal variation in species composition. Yet, the conditions under which ecological dynamics are dominated by niche‐based versus stochastic processes are poorly understood. Using a field experiment in early‐successional temperate grassland and null models of beta diversity, this study investigates the effects of soil nutrient supply on the relative importance of niche‐based selection versus stochastic dynamics for variation in species composition among sites. Nutrient availability was manipulated experimentally, individual seed mixtures with 25 species were sown in each experimental plot, and then stochastic and deterministic niche‐based assembly processes were allowed to happen. We found that compositional variation among grassland plots with low nutrient supply was driven by stochastic immigration and extinctions. In contrast, nutrient enrichment reduced the importance of stochasticity and imposed a deterministic environmental filter that homogenized communities through the selection of few species with greater competitive ability for light. This demonstrates that soil nutrient availability is a critical environmental feature that dictates the degree to which terrestrial plant communities are controlled by niche‐based selection versus stochastic assembly processes. Our study shows further that alternative states of eutrophic grasslands emerge from initial stochastic variation in the composition of a particular functional group of species that can become dominant at high nutrient supply. We discuss potential mechanisms underlying the shift from stochastic to niche‐driven dynamics along soil nutrient gradients.     

Stochastic population theory faces reality in the laboratory

Understanding the factors that affect most severely the extinction risk of populations is crucial for maintaining biodiversity. An important general pattern derived from stochastic population theory is that time to extinction should decrease with increasing environmental stochasticity. Drake and Lodge recently provided one of the first pieces of experimental support for this simple prediction by artificially manipulating the dynamics of populations of Daphnia. A future challenge will be to include both demographic stochasticity and environmental stochasticity in such studies.