Metapopulation persistence in heterogeneous landscapes: lessons about the effect of stochasticity

This article addresses an important aspect of the analysis of metapopulation persistence. It highlights some consequences of ignoring and including stochasticity in the sequence of extinction and colonization events. The results are based on a comparative analysis of the outcomes of two (one deterministic, one stochastic) spatially realistic metapopulation models and a search for common effects and differences. One key result of the article is that, under certain conditions, there are extra effects of the landscape structure (number and configuration of patches, patch size distribution) on metapopulation persistence if stochasticity is included. In these cases, ignoring or including stochasticity can change conclusions about the persistence status but also ranking orders, relative results, and qualitative trends. A list of conditions is provided under which including stochasticity is vital to prevent counterproductive conclusions about metapopulation persistence. The results of the overall study are condensed in five lessons about the effect of stochasticity. A number of implications for ecological theory and conservation management are discussed. The study demonstrates the potential of three recently published approximation formulas (metapopulation capacity lambdaM, mean lifetime Tm, and effective number of patches N) to serve as tools for ecological analysis and thinking.     

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.     

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。     

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.     

Patterns of neutral diversity under general models of selective sweeps

Two major sources of stochasticity in the dynamics of neutral alleles result from resampling of finite populations (genetic drift) and the random genetic background of nearby selected alleles on which the neutral alleles are found (linked selection). There is now good evidence that linked selection plays an important role in shaping polymorphism levels in a number of species. One of the best-investigated models of linked selection is the recurrent full-sweep model, in which newly arisen selected alleles fix rapidly. However, the bulk of selected alleles that sweep into the population may not be destined for rapid fixation. Here we develop a general model of recurrent selective sweeps in a coalescent framework, one that generalizes the recurrent full-sweep model to the case where selected alleles do not sweep to fixation. We show that in a large population, only the initial rapid increase of a selected allele affects the genealogy at partially linked sites, which under fairly general assumptions are unaffected by the subsequent fate of the selected allele. We also apply the theory to a simple model to investigate the impact of recurrent partial sweeps on levels of neutral diversity and find that for a given reduction in diversity, the impact of recurrent partial sweeps on the frequency spectrum at neutral sites is determined primarily by the frequencies rapidly achieved by the selected alleles. Consequently, recurrent sweeps of selected alleles to low frequencies can have a profound effect on levels of diversity but can leave the frequency spectrum relatively unperturbed. In fact, the limiting coalescent model under a high rate of sweeps to low frequency is identical to the standard neutral model. The general model of selective sweeps we describe goes some way toward providing a more flexible framework to describe genomic patterns of diversity than is currently available.     

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.     

Stochasticity in microbiology: managing unpredictability to reach the Sustainable Development Goals

Pure (single) cultures of microorganisms and mixed microbial communities (microbiomes) have been important for centuries in providing renewable energy, clean water and food products to human society and will continue to play a crucial role to pursue the Sustainable Development Goals. To use microorganisms effectively, microbial engineered processes require adequate control. Microbial communities are shaped by manageable deterministic processes, but also by stochastic processes, which can promote unforeseeable variations and adaptations. Here, we highlight the impact of stochasticity in single culture and microbiome engineering. First, we discuss the concepts and mechanisms of stochasticity in relation to microbial ecology of single cultures and microbiomes. Second, we discuss the consequences of stochasticity in relation to process performance and human health, which are reflected in key disadvantages and important opportunities. Third, we propose a suitable decision tool to deal with stochasticity in which monitoring of stochasticity and setting the boundaries of stochasticity by regulators are central aspects. Stochasticity may give rise to some risks, such as the presence of pathogens in microbiomes. We argue here that by taking the necessary precautions and through clever monitoring and interpretation, these risks can be mitigated.     

Demographic stochasticity alters expected outcomes in experimental and simulated non‐neutral communities

Theory has shown that the effects of demographic stochasticity on communities may depend on the magnitude of fitness differences between species. In particular, it has been suggested that demographic stochasticity has the potential to significantly alter competitive outcomes when fitness differences are small (nearly neutral), but that it has negligible effects when fitness differences are large (highly non‐neutral). Here we test such theory experimentally and extend it to examine how demographic stochasticity affects exclusion frequency and mean densities of consumers in simple, but non‐neutral, consumer–resource communities. We used experimental microcosms of protists and rotifers feeding on a bacterial resource to test how varying absolute population sizes (a driver of demographic stochasticity) affected the probability of competitive exclusion of the weakest competitor. To explore whether demographic stochasticity could explain our experimental results, and to generalize beyond our experiment, we paired the experiment with a continuous‐time stochastic model of resource competition, which we simulated for 11 different fitness inequalities between competiting consumers. Consistent with theory, in both our experiments and our simulations we found that demographic stochasticity altered competitive outcomes in communities where fitness differences were small. However, we also found that demographic stochasticity alone could affect communities in other ways, even when fitness differences between competitors were large. Specifically, demographic stochasticity altered mean densities of both weak and strong competitors in experimental and simulated communities. These findings highlight how demographic stochasticity can change both competitive outcomes in non‐neutral communities and the processes underlying overall community dynamics.     

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.     

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.     

Intraindividual variability in behavior shapes fitness landscapes

Although intraindividual variability (IIV) in behavior is fundamental to ecological dynamics, the factors that contribute to the expression of IIV are poorly understood. Using an individual‐based model, this study examined the effects of stochasticity on the evolution of IIV represented by the residual variability of behavior. The model describes a population of prey with nonoverlapping generations, in which prey take refuge upon encountering a predator. The strategy of a prey is characterized by the mean and IIV (i.e., standard deviation) of hiding duration. Prey with no IIV will spend the same duration hiding in a refuge at each predator encounter, while prey with IIV will have variable hiding durations among encounters. For the sources of stochasticity, within‐generation stochasticity (represented by random predator encounters) and between‐generation stochasticity (represented by random resource availability) were considered. Analysis of the model indicates that individuals with high levels of IIV are maintained in a population in the presence of between‐generation stochasticity even though the optimal strategy in each generation is a strategy with no IIV, regardless of the presence or absence of within‐generation stochasticity. This contradictory pattern emerges because the mean behavioral trait and IIV do not independently influence fitness (e.g., the sign of the selection gradient with respect to IIV depends on the mean trait). Consequently, even when evolution eventually leads toward a strategy with no IIV (i.e., the optimal strategy), greater IIV may be transiently selected. Between‐generation stochasticity consistently imposes such transient selection and maintain high levels of IIV in a population.     

Population demography and the evolution of helping behaviors

Limited dispersal may favor the evolution of helping behaviors between relatives as it increases their relatedness, and it may inhibit such evolution as it increases local competition between these relatives. Here, we explore one way out of this dilemma: if the helping behavior allows groups to expand in size, then the kin-competition pressure opposing its evolution can be greatly reduced. We explore the effects of two kinds of stochasticity allowing for such deme expansion. First, we study the evolution of helping under environmental stochasticity that may induce complete patch extinction. Helping evolves if it results in a decrease in the probability of extinction or if it enhances the rate of patch recolonization through propagules formed by fission of nonextinct groups. This mode of dispersal is indeed commonly found in social species. Second, we consider the evolution of helping in the presence of demographic stochasticity. When fecundity is below its value maximizing deme size (undersaturation), helping evolves, but under stringent conditions unless positive density dependence (Allee effect) interferes with demographic stochasticity. When fecundity is above its value maximizing deme size (oversaturation), helping may also evolve, but only if it reduces negative density-dependent competition.     

Introgressive hybridization as a mechanism for species rescue

Rapid evolution on ecological time scales can play a key role in species responses to environmental change. One dynamic that has the potential to generate the diversity necessary for evolution rapid enough to allow response to sudden environmental shifts is introgressive hybridization. However, if distinct sub-species exist before an environmental shift, mechanisms that impede hybridization, such as assortative mating and hybrid inferiority, are likely to be present. Here we explore the theoretical potential for introgressive hybridization to play a role in response to environmental change. In particular, we incorporate assortative mating, hybrid inferiority, and demographic stochasticity into a two-locus, two-allele population genetic model of two interacting species where one locus identifies the species and the other determines how fitness depends on the changing environment. Simulation results indicate that moderately high values for the strength of assortative mating will allow enough hybridization events to outweigh demographic stochasticity but not so many that continued hybridization outweighs backcrossing and introgression. Successful introgressive hybridization also requires intermediate relative fitness at the allele negatively affected by environmental change such that hybrid survivorship outweighs demographic stochasticity but selection remains strong enough to affect the genetic dynamics. The potential for successful introgression instead of extinction with greater environmental change is larger with monogamous rather than promiscuous mating due to lower stochasticity in mating events. These results suggest species characteristics (e.g., intermediate assortative mating and mating systems with low variation in mating likelihood) which indicate a potential for rapid evolution in response to environmental change via introgressive hybridization.     

Sources of PCR-induced distortions in high-throughput sequencing data sets

PCR permits the exponential and sequence-specific amplification of DNA, even from minute starting quantities. PCR is a fundamental step in preparing DNA samples for high-throughput sequencing. However, there are errors associated with PCR-mediated amplification. Here we examine the effects of four important sources of error—bias, stochasticity, template switches and polymerase errors—on sequence representation in low-input next-generation sequencing libraries. We designed a pool of diverse PCR amplicons with a defined structure, and then used Illumina sequencing to search for signatures of each process. We further developed quantitative models for each process, and compared predictions of these models to our experimental data. We find that PCR stochasticity is the major force skewing sequence representation after amplification of a pool of unique DNA amplicons. Polymerase errors become very common in later cycles of PCR but have little impact on the overall sequence distribution as they are confined to small copy numbers. PCR template switches are rare and confined to low copy numbers. Our results provide a theoretical basis for removing distortions from high-throughput sequencing data. In addition, our findings on PCR stochasticity will have particular relevance to quantification of results from single cell sequencing, in which sequences are represented by only one or a few molecules.     

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.     

Setting the absolute tempo of biodiversity dynamics

Neutral biodiversity theory has the potential to contribute to our understanding of how macroevolutionary dynamics influence contemporary biodiversity, but there are issues regarding its dynamical predictions that must first be resolved. Here we address these issues by extending the theory in two ways using a novel analytical approach: (1) we set the absolute tempo of biodiversity dynamics by explicitly incorporating population-level stochasticity in abundance; (2) we allow new species to arise with more than one individual. Setting the absolute tempo yields quantitative predictions on biodiversity dynamics that can be tested using contemporary and fossil data. Allowing incipient-species abundances greater than one individual yields predictions on how these dynamics, and the form of the species-abundance distribution, are affected by multiple speciation modes. We apply this new model to contemporary and fossil data that encompass 30 Myr of macroevolution for planktonic foraminifera. By synthesizing the model with these empirical data, we present evidence that dynamical issues with neutral biodiversity theory may be resolved by incorporating the effects of environmental stochasticity and incipient-species abundance on biodiversity dynamics.     

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.     

种群生存力分析研究进展和趋势

种群生存力分析（PVA）是正在迅速发展的新方法,已成为保护生物学研究的热点。它主要研究随机干扰对小种群绝灭的影响,其目的是制定最小可存活种群（MVP）,把绝灭减少到可接受的水平。随机干扰可分四类;统计随机性,环境随机性,自然灾害和遗传随机性。确定MVP的方法有三种：理论模型,模拟模型,模拟模型和岛屿生物地理学方法。理论模型主要研究理想或特定条件下随机因素对种群的影响;模拟模型是利用计算机模拟种群绝灭过程;岛屿生物地理学方法主要分析岛屿物种的分布和存活,证实分析模型和模拟模型。已有大量的文献研究统计随机性,环境随机性和自然灾害的行为特征,但遗传因素与种群生存力之间的关系还不清楚。建立包括四种随机性的综合性模型,广泛地检验PVA模型,系统地研制目标种的遗传和生态特性以及MVP的实际应用是PVA的发展趋势。     

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.