Evolutionary rescue under demographic and environmental stochasticity

Populations suffer two types of stochasticity: demographic stochasticity, from sampling error in offspring number, and environmental stochasticity, from temporal variation in the growth rate. By modelling evolution through phenotypic selection following an abrupt environmental change, we investigate how genetic and demographic dynamics, as well as effects on population survival of the genetic variance and of the strength of stabilizing selection, differ under the two types of stochasticity. We show that population survival probability declines sharply with stronger stabilizing selection under demographic stochasticity, but declines more continuously when environmental stochasticity is strengthened. However, the genetic variance that confers the highest population survival probability differs little under demographic and environmental stochasticity. Since the influence of demographic stochasticity is stronger when population size is smaller, a slow initial decline of genetic variance, which allows quicker evolution, is important for population persistence. In contrast, the influence of environmental stochasticity is population-size-independent, so higher initial fitness becomes important for survival under strong environmental stochasticity. The two types of stochasticity interact in a more than multiplicative way in reducing the population survival probability. Our work suggests the importance of explicitly distinguishing and measuring the forms of stochasticity during evolutionary rescue.     

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。     

Effective size of a fluctuating age-structured population

Previous theories on the effective size of age-structured populations assumed a constant environment and, usually, a constant population size and age structure. We derive formulas for the variance effective size of populations subject to fluctuations in age structure and total population size produced by a combination of demographic and environmental stochasticity. Haploid and monoecious or dioecious diploid populations are analyzed. Recent results from stochastic demography are employed to derive a two-dimensional diffusion approximation for the joint dynamics of the total population size, N, and the frequency of a selectively neutral allele, p. The infinitesimal variance for p, multiplied by the generation time, yields an expression for the effective population size per generation. This depends on the current value of N, the generation time, demographic stochasticity, and genetic stochasticity due to Mendelian segregation, but is independent of environmental stochasticity. A formula for the effective population size over longer time intervals incorporates deterministic growth and environmental stochasticity to account for changes in N.     

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.     

Extinction in relation to demographic and environmental stochasticity in age-structured models

The demographic variance of an age-structured population is defined. This parameter is further split into components generated by demographic stochasticity in each vital rate. The applicability of these parameters are investigated by checking how an age-structured population process can be approximated by a diffusion with only three parameters. These are the deterministic growth rate computed from the expected projection matrix and the environmental and demographic variances. We also consider age-structured populations where the fecundity at any stage is either zero or one, and there is neither environmental stochasticity nor dependence between individual fecundity and survival. In this case the demographic variance is uniquely determined by the vital rates defining the projection matrix. The demographic variance for a long-lived bird species, the wandering albatross in the southwestern part of the Indian Ocean, is estimated. We also compute estimates of the age-specific contributions to the total demographic variance from survival, fecundity and the covariance between survival and fecundity.     

Habitat dependence and correlations between elasticities of long-term growth rates

In population biology, elasticity is a measure of the importance of a demographic rate on population growth. A relatively small amount of stochasticity can substantially impact the dynamics of a population whose growth is a function of deterministic and stochastic processes. Analyses of natural populations frequently neglect the latter. Even in a population that fluctuates substantially with time, the results of a deterministic perturbation analysis correlated strongly with results of a perturbation analysis of the long-run stochastic growth rate. Population growth was, however, not uniformly sensitive to demographic rates across different environmental conditions. The overall correlation between deterministic and stochastic perturbation analysis may be high, but environmental variability can dramatically alter the contributions of demographic rates in different environmental conditions. This potentially informative detail is neglected by deterministic analysis, yet it highlights one difficulty when extrapolating results from long-term analysis to shorter-term environmental change.     

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.     

Environmental stochasticity and extinction risk in a population of a small songbird, the great tit

Using a long-term demographic data set, we estimated the separate effects of demographic and environmental stochasticity in the growth rate of the great tit population in Wytham Wood, United Kingdom. Assuming logistic density regulation, both the demographic (sigma2d = 0.569) and environmental (sigma2e = 0.0793) variance, with interactions included, were significantly greater than zero. The estimates of the demographic variance seemed to be relatively insensitive to the length of the study period, whereas reliable estimates of the environmental variance required long time series (at least 15 yr of data). The demographic variance decreased significantly with increasing population density. These estimates are used in a quantitative analysis of the demographic factors affecting the risk of extinction of this population. The very long expected time to extinction of this population (approximately 10(19) yr) was related to a relatively large population size (>/=120 pairs during the study period). However, for a given population size, the expected time to extinction was sensitive to both variation in population growth rate and environmental stochasticity. Furthermore, the form of the density regulation strongly affected the expected time to extinction. Time to extinction decreased when the maximum density regulation approached K. This suggests that estimates of viability of small populations should be given both with and without inclusion of density dependence.     

Demographic stochasticity, allee effects, and extinction: the influence of mating system and sex ratio

Demographic stochasticity has a substantial influence on the growth of small populations and consequently on their extinction risk. Mating system is one of several population characteristics that may affect this. We use a stochastic pair-formation model to investigate the combined effects of mating system, sex ratio, and population size on demographic stochasticity and thus on extinction risk. Our model is designed to accommodate a continuous range of mating systems and sex ratios as well as several levels of stochasticity. We show that it is not mating system alone but combinations of mating system and sex ratio that are important in shaping the stochastic dynamics of populations. Specifically, polygyny has the potential to give a high demographic variance and to lower the stochastic population growth rate substantially, thus also shortening the time to extinction, but the outcome is highly dependent on the sex ratio. In addition, population size is shown to be important. We find a stochastic Allee effect that is amplified by polygyny. Our results demonstrate that both mating system and sex ratio must be considered in conservation planning and that appreciating the role of stochasticity is key to understanding their effects.     

Estimating the time to extinction in an island population of song sparrows

We estimated and modelled how uncertainties in stochastic population dynamics and biases in parameter estimates affect the accuracy of the projections of a small island population of song sparrows which was enumerated every spring for 24 years. The estimate of the density regulation in a theta-logistic model (theta = 1.09 suggests that the dynamics are nearly logistic, with specific growth rate r1 = 0.99 and carrying capacity K = 41.54. The song sparrow population was strongly influenced by demographic (ŝigma2(d) = 0.66) and environmental (ŝigma2(d) = 0.41) stochasticity. Bootstrap replicates of the different parameters revealed that the uncertainties in the estimates of the specific growth rate r1 and the density regulation theta were larger than the uncertainties in the environmental variance sigma2(e) and the carrying capacity K. We introduce the concept of the population prediction interval (PPI), which is a stochastic interval which includes the unknown population size with probability (1 - alpha). The width of the PPI increased rapidly with time because of uncertainties in the estimates of density regulation as well as demographic and environmental variance in the stochastic population dynamics. Accepting a 10% probability of extinction within 100 years, neglecting uncertainties in the parameters will lead to a 33% overestimation of the time it takes for the extinction barrier (population size X = 1) to be included into the PPI. This study shows that ignoring uncertainties in population dynamics produces a substantial underestimation of the extinction risk.     

Estimating population trends using population viability analyses for the conservation ofCapra pyrenaica

Large herbivore populations can suffer important oscillations with considerable effects on ecosystem functions and services, yet our capacity to predict population fate is limited and conditional upon the availability of data. This study investigated the interannual variation in the growth rate of populations ofCapra pyrenaica Schinz, 1838, and its extinction risk by comparing the dynamics of populations that were stable for more than two decades (Gredos and Tortosa-Beceite), populations that had increased recently (Tejeda-Almijara), and populations that were in decline (Cazorla-Segura) or extinct (the Pyrenees population; hereafter, bucardo). To estimate quasi-extinction threshold assessments (50% of population extinct in this study), which have implications for the conservation of the species, we used empirical data and the predictions derived from several theoretical models. The results indicate that when variance of log population growth rate reaches a specific threshold, the probability of quasi-extinction increased drastically. ForC. pyrenaica, we recommend keeping population variance < 0.05, which will reduce the likelihood that the irruptive oscillations caused by environmental and demographic stochasticity will put the population at risk. Models to predict the dynamics ofC. pyrenaica populations should incorporate temporal stochasticity because, in this study, it strongly increased the likelihood that a population declined.     

Long-term demographic analysis in goshawk <Emphasis Type="Italic">Accipiter gentilis</Emphasis>: the role of density dependence and stochasticity

Density dependence and environmental stochasticity are both potentially important processes influencing population demography and long-term population growth. Quantifying the importance of these two processes for population growth requires both long-term population as well as individual-based data. I use a 30-year data set on a goshawk Accipiter gentilis population from Eastern Westphalia, Germany, to describe the key vital rate elements to which the growth rate is most sensitive and test how environmental stochasticity and density dependence affect long-term population growth. The asymptotic growth rate of the fully age-structured mean matrix model was very similar to the observed one (0.7% vs. 0.3% per annum), and population growth was most elastic to changes in survival rate at age classes 1-3. Environmental stochasticity led only to a small change in the projected population growth rate (between -0.16% and 0.67%) and did not change the elasticities qualitatively, suggesting that the goshawk life history of early reproduction coupled with high annual fertility buffers against a variable environment. Age classes most crucial to population growth were those in which density dependence seemed to act most strongly. This emphasises the importance of density dependence as a regulatory mechanism in this goshawk population. It also provides a mechanism that might enable the population to recover from population lows, because a mean matrix model incorporating observed functional responses of both vital rates to population density coupled with environmental stochasticity reduced long-term extinction risk of 30% under density-independent environmental stochasticity and 60% under demographic stochasticity to zero.     

Pattern of variation in avian population growth rates

A central question in population ecology is to understand why population growth rates differ over time. Here, we describe how the long-term growth of populations is not only influenced by parameters affecting the expected dynamics, for example form of density dependence and specific population growth rate, but is also affected by environmental and demographic stochasticity. Using long-term studies of fluctuations of bird populations, we show an interaction between the stochastic and the deterministic components of the population dynamics: high specific growth rates at small densities r(1) are typically positively correlated with the environmental variance sigma(e)(2). Furthermore, theta, a single parameter describing the form of the density regulation in the theta-logistic density-regulation model, is negatively correlated with r(1). These patterns are in turn correlated with interspecific differences in life-history characteristics. Higher specific growth rates, larger stochastic effects on the population dynamics and stronger density regulation at small densities are found in species with large clutch sizes or high adult mortality rates than in long-lived species. Unfortunately, large uncertainties in parameter estimates, as well as strong stochastic effects on the population dynamics, will often make even short-term population projections unreliable. We illustrate that the concept of population prediction interval can be useful in evaluating the consequences of these uncertainties in the population projections for the choice of management actions.     

Extinctions in competitive communities forced by coloured environmental variation

Understanding the relationships between environmental fluctuations, population dynamics and species interactions in natural communities is of vital theoretical and practical importance. This knowledge is essential in assessing extinction risks in communities that are, for example, pressed by changing environmental conditions and increasing exploitation. We developed a model of density dependent population renewal, in a Lotka–Volterra competitive community context, to explore the significance of interspecific interactions, demographic stochasticity, population growth rate and species abundance on extinction risk in populations under various autocorrelation (colour) regimes of environmental forcing. These factors were evaluated in two cases, where either a single species or the whole community was affected by the external forcing. Species' susceptibility to environmental noise with different autocorrelation structure depended markedly on population dynamics, species' position in the abundance hierarchy and how similarly community members responded to external forcing. We also found interactions between demographic stochasticity and environmental noise leading to a reversal in extinction probabilities from under- to overcompensatory dynamics. We compare our results with studies of single species populations and contrast possible mechanisms leading to extinctions. Our findings indicate that abundance rank, the form of population dynamics, and the colour of environmental variation interact in affecting species extinction risk. These interactions are further modified by interspecific interactions within competitive communities as the interactions filter and modulate the environmental noise.     

Consequences of Environmental Fluctuations on Taylor’s Power Law and Implications for the Dynamics and Persistence of Populations

Conservation Biologists have found that demographic stochasticity causes the mean time to extinction to increase exponentially with population size. This has proved helpful in analyses determining extinction times and characterizing the pathway to extinction. The aim of this investigation is to explore the possible interactions between environmental/demographic noises and the scaling effect of the mean population size with its variance, which is expected to follow Taylor’s power law relationship. We showed that the combined effects of environmental/demographic noises and the scaling of population size variability interact with the population dynamics and affect the mean time to extinction.     

Density dependence and stochastic variation in a newly established population of a small songbird

Models describing fluctuations in population size should include both density dependence and stochastic effects. We examine the relative contribution of variation in parameters of the expected dynamics as well as demographic and environmental stochasticity to fluctuations in a population of a small passerine bird, the pied flycatcher, that was newly established in a Dutch study area. Using the theta-logistic model of density regulation, we demonstrate that the estimated quasi-stationary distribution including demographic stochasticity is close to the stationary distribution ignoring demographic stochasticity, indicating a long expected time to extinction. We also show that the variance in the estimated quasi-stationary distribution is especially sensitive to variation in the density regulation function. Reliable population projections must therefore account for uncertainties in parameter estimates which we do by using the population prediction interval (PPI). After 2 years the width of the 90% PPI was already larger than the corresponding estimated range of variation in the quasi-stationary distribution. More precise prediction of future population size than can be derived from the quasi-stationary distribution could only be made for a time span less than about five years.     

Prolonged diapause: a trait increasing invasion speed?

Invasive species are considered to be the second cause of biodiversity erosion, and one challenge is to determine the life history traits that cause an increased invasion capacity. Prolonged diapause is a major trait in evolution and insect population dynamics, but its effects on invasion speed remain unknown. From a recently developed mathematical approach (integro-difference equations) applied to the insect dormancy, we show that despite a dispersal cost, bet-hedging diapause strategies with low (0.1-0.2) prolonged diapause frequency (emergence after 1 or 2 years) can have a higher invasion speed than a simple diapause strategy (emergence after 1 year) when the environmental stochasticity is sufficiently high. In such conditions, prolonged diapause is a trait supporting invasion capacity by increasing population stochastic growth rate. This conclusion, which applies to a large range of demographic parameters, is in opposition to the usual view that prolonged dormancy is an alternative strategy to dispersal. However, prolonged diapause does not support invasion if the level of environmental stochasticity is low. Therefore, conclusion about its influence on invasion ability needs a good knowledge of environmental stochasticity in the introduction area of considered species.     

Predicting extinction risks for plants: environmental stochasticity can save declining populations

An emerging generalization from theoretical and empirical studies on conservation biology is that high levels of environmental stochasticity increase the likelihood of population extinction. However, coexistence theory has illustrated that there are circumstances under which environmental stochasticity can increase the chance of population persistence. These theoretical studies have shown that the sign of the effect of environmental stochasticity on population persistence is determined by interactions between life history and environmental stochasticity. These interactions mean that the stochastic and deterministic rates of population growth might differ fundamentally. Although difficult to demonstrate in real systems, observed life histories and variance in the vital rates of populations suggest that this phenomenon is likely to be common, and is therefore of much relevance to conservation biologists.     

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.     

demoniche – an R‐package for simulating spatially‐explicit population dynamics

demoniche is a freely available R‐package which simulates stochastic population dynamics in multiple populations of a species. A demographic model projects population sizes utilizing several transition matrices that can represent impacts on species growth. The demoniche model offers options for setting demographic stochasticity, carrying capacity, and dispersal. The demographic projection in each population is linked to spatially‐explicit niche values, which affect the species growth. With the demoniche package it is possible to compare the influence of scenarios of environmental changes on future population sizes, extinction probabilities, and range shifts of species.