Does increased stochasticity speed up extinction?

We consider the impact of increased stochastic fluctuations on the extinction date of an unstructured population subject to either environmental or demographical stochasticity (or both). By modelling the population density as a general linear diffusion, we state a set of typically satisfied conditions under which the decreasing minimal r-excessive mapping (and, therefore, the moment generating function) of the considered diffusion process is convex and, consequently, under which the impact of increased stochastic fluctuations on the expected date at which the density becomes arbitrarily small is unambiguously negative. In other words, we establish a set of sufficient conditions under which increased stochasticity speeds up the extinction process independently of whether stochasticity is environmental or demographic. In this way, we are able to confirm that increased stochasticity is detrimental for population growth. Received: 25 April 2000 / Revised version: 18 April 2001 / Published online: 12 October 2001     

Climate variation and regional gradients in population dynamics of two hole-nesting passerines

Latitudinal gradients in population dynamics can arise through regional variation in the deterministic components of the population dynamics and the stochastic factors. Here, we demonstrate an increase with latitude in the contribution of a large-scale climate pattern, the North Atlantic Oscillation (NAO), to the fluctuations in size of populations of two European hole-nesting passerine species. However, this influence of climate induced different latitudinal gradients in the population dynamics of the two species. In the great tit the proportion of the variability in the population fluctuations explained by the NAO increased with latitude, showing a larger impact of climate on the population fluctuations of this species at higher latitudes. In contrast, no latitudinal gradient was found in the relative contribution of climate to the variability of the pied flycatcher populations because the total environmental stochasticity increased with latitude. This shows that the population ecological consequences of an expected climate change will depend on how climate affects the environmental stochasticity in the population process. In both species, the effects will be larger in those parts of Europe where large changes in climate are expected.     

An evaluation of linear models of population fluctuations

The analysis of non-linear stochastic population models is notoriously difficult and there is little or no data against which they may be tested. It is therefore important to investigate the range of validity of models of fluctuating populations which are linear in the deviation of the population from its mean value. In this paper we demonstrate the robustness of some well-known, linear methods of analysing fluctuations, the power of the methods being evaluated by comparison with numerical results for some non-linear models involving demographic or environmental stochasticity.     

Spatial scales of population synchrony of two competing species: effects of harvesting and strength of competition

Theoretical analyses of single‐species models have revealed that the degree of synchrony in fluctuations of geographically separated populations increases with increasing spatial covariation in environmental fluctuations and increased interchange of individuals, but decreases with local strength of density dependence. Here we extend these results to include interspecific competition between two species as well as harvesting. We show that the effects of interspecific competition on the geographical scale of population synchrony are dependent on the pattern of spatial covariation of environmental variables. If the environmental noise is uncorrelated between the competing species, competition generally increases the spatial scale of population synchrony of both species. Otherwise, if the environmental noises are strongly correlated between species, competition generally increases the spatial scale of population synchrony of at least one, but also often of both species. The magnitude of these spatial scaling effects is, however, strongly influenced by the dispersal capacity of the two competing species. If the species are subject to proportional harvesting, this may synchronise population dynamics over large geographical areas, affecting the vulnerability of harvested species to environmental changes. However, the strength of interspecific competition may strongly modify this effect of harvesting on the spatial scale of population synchrony. For example, harvesting of one species may affect the spatial distribution of competing species that are not subject to harvesting. These analytical results provide an important illustration of the importance of applying an ecosystem rather than a single‐species perspective when developing harvest strategies for a sustainable management of exploited species.     

Extinction of populations by inbreeding depression under stochastic environments

Inbreeding depression may induce rapid extinction due to positive feedbacks between inbreeding depression and reduction of population size, which is often referred to as extinction vortex by inbreeding depression. The present analysis has demonstrated that the extinction vortex is likely to happen with realistic parameter values of genomic mutation rate of lethals or semilethals, equilibrium population size, intrinsic rate of natural increase, and rate of population decline caused by nongenetic extrinsic factors. Simulation models incorporating stochastic fluctuations of population size further indicated that extinction by inbreeding depression is facilitated by environmental fluctuations in population size. The results suggest that there is a positive interaction between genetic stochasticity and environmental stochasticity for extinction of populations by inbreeding depression. Received: May 10, 1999 / Accepted: November 5, 1999     

Effects of climate on population fluctuations of ibex

Predicting the effects of the expected changes in climate on the dynamics of populations require that critical periods for climate‐induced changes in population size are identified. Based on time series analyses of 26 Swiss ibex (Capra ibex) populations, we show that variation in winter climate affected the annual changes in population size of most of the populations after accounting for the effects of density dependence and demographic stochasticity. In addition, precipitation during early summer also influenced the population fluctuations. This suggests that the major influences of climate on ibex population dynamics operated either through loss of individuals during winter or early summer, or through an effect on fecundity. However, spatial covariation in these climate variables was not able to synchronize the population fluctuations of ibex over larger distances, probably due to large spatial heterogeneity in the effects of single climate variables on different populations. Such spatial variation in the influence of the same climate variable on the local population dynamics suggests that predictions of influences of climate change need to account for local differences in population dynamical responses to climatic conditions.     

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。     

Relative importance of density-dependent regulation and environmental stochasticity for butterfly population dynamics

The relative contribution of density-dependent regulation and environmental stochasticity to the temporal dynamics of animal populations is one of the central issues of ecology. In insects, the primary role of the latter factor, typically represented by weather patterns, is widely accepted. We have evaluated the impact of density dependence as well as density-independent factors, including weather and mowing regime, on annual fluctuations of butterfly populations. As model species, we used Maculinea alcon and M. teleius living in sympatry and, consequently, we also analysed the effect of their potential competition. Density dependence alone explained 62 and 42% of the variation in the year-to-year trends of M. alcon and M. teleius, respectively. The cumulative Akaike weight of models with density dependence, which can be interpreted as the probability that this factor should be contained in the most appropriate population dynamics model, exceeded 0.97 for both species. In contrast, the impacts of inter-specific competition, mowing regime and weather were much weaker, with their cumulative weights being in the range of 0.08–0.21; in addition, each of these factors explained only 2–5% of additional variation in Maculinea population trends. Our results provide strong evidence for density-dependent regulation in Maculinea, while the influence of environmental stochasticity is rather minor. In the light of several recent studies on other butterflies that detected significant density-dependent effects, it would appear that density-dependent regulation may be more widespread in this group than previously thought, while the role of environmental stochasticity has probably been overestimated. We suggest that this misconception is the result of deficiencies in the design of most butterfly population studies in the past, including (1) a strong focus on adults and a neglect of the larval stage in which density-dependent effects are most likely to occur; (2) an almost exclusive reliance on transect count results that may confound the impact of environmental stochasticity on butterfly numbers with its impact on adult longevity. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Population dynamics of Turbo militaris (Gastropoda: Turbinidae) on rocky shores in a subtropical marine park: implications for management

Understanding the relative impacts of harvesting across an area such as a marine park is vital if the goals of fisheries management are to be met. Given their accessibility, densities of targeted intertidal turbinids should be relatively simple to quantify; however, natural spatial and temporal variability in these populations has hampered this effort. This study aimed to quantify short-term population dynamics of Turbo militaris in relation to current zoning regulations and accessibility. While our results reflected the variability found by other studies, we also detected significantly lower densities at a headland where harvesting is known to occur, and found a trend towards denser aggregations at more remote locations. Importantly, we found that densities at some locations were so low that current fisheries bag limits may be ineffective for protecting populations at local scales. Comparisons between study sites suggest a combination of no-take zoning and inaccessibility may provide the most effective protection for this species. However, a greater understanding of the wider impacts of harvesting, and processes affecting recovery, are essential to ensure sustainable management of this fishery.     

Demographic factors and genetic variation influence population persistence under environmental change

Population persistence has been studied in a conservation context to predict the fate of small or declining populations. Persistence models have explored effects on extinction of random demographic and environmental fluctuations, but in the face of directional environmental change they should also integrate factors affecting whether a population can adapt. Here, we examine the population‐size dependence of demographic and genetic factors and their likely contributions to extinction time under scenarios of environmental change. Parameter estimates were derived from experimental populations of the rainforest species, Drosophila birchii, held in the lab for 10 generations at census sizes of 20, 100 and 1000, and later exposed to five generations of heat‐knockdown selection. Under a model of directional change in the thermal environment, rapid extinction of populations of size 20 was caused by a combination of low growth rate (r) and high stochasticity in r. Populations of 100 had significantly higher reproductive output, lower stochasticity in r and more additive genetic variance (V_A) than populations of 20, but they were predicted to persist less well than the largest size class. Even populations of 1000 persisted only a few hundred generations under realistic estimates of environmental change because of low V_A for heat‐knockdown resistance. The experimental results document population‐size dependence of demographic and adaptability factors. The simulations illustrate a threshold influence of demographic factors on population persistence, while genetic variance has a more elastic impact on persistence under environmental change.     

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.     

Genetic structuring in fluctuating populations

The effect of genetic drift in spatially distributed dispersal-linked and density-regulated populations is studied in a classical one-locus two-allele system. We analyse emergence of genetic differentiation assuming random drift only, where the noise-like variability is due to demographic stochasticity. We find emergence of clusters of sub-units with local allele fixation and persistence of both alleles in lengthy simulations. We demonstrate that local allele fixation (extending over a number of adjoining spatial sub-units) – without global loss of alleles – may occur when the carrying capacities of local patches are small, under a full range population dynamic regimes, when dispersal rate is small, and when redistribution (through dispersal) does not act as global mixer. These results are novel. The key to the observations is that drift is simultaneously influenced by distance-dependent dispersal, demographic stochasticity and autocorrelated population fluctuations due to delayed-density dependence. These are standard elements of contemporary population models in spatially structured context. With stable large populations, no stochasticity and dispersal limited to neighbours only, our model collapses to the stepping-stone model, while with dispersal being random and global, the model collapses to Wright's island model.     

Estimating the growth of a newly established moose population using reproductive value

Estimating the population growth rate and environmental stochasticity of long-lived species is difficult because annual variation in population size is influenced by temporal autocorrelations caused by fluctuations in the age-structure. Here we use the dynamics of the reproductive value to estimate the long-term growth rate s and the environmental variance of a moose population that recently colonized the island of Vega in northern Norway. We show that the population growth rate was high (ŝ=0.26). The major stochastic influences on the population dynamics were due to demographic stochasticity, whereas the environmental variance was not significantly different from 0. This supports the suggestion that population growth rates of polytocous ungulates are high, and that demographic stochasticity must be assessed when estimating the growth of small ungulate populations.     

Effective size of density‐dependent two‐sex populations: the effect of mating systems

Density dependence in vital rates is a key feature affecting temporal fluctuations of natural populations. This has important implications for the rate of random genetic drift. Mating systems also greatly affect effective population sizes, but knowledge of how mating system and density regulation interact to affect random genetic drift is poor. Using theoretical models and simulations, we compare N_e in short‐lived, density‐dependent animal populations with different mating systems. We study the impact of a fluctuating, density‐dependent sex ratio and consider both a stable and a fluctuating environment. We find a negative relationship between annual N_e/N and adult population size N due to density dependence, suggesting that loss of genetic variation is reduced at small densities. The magnitude of this decrease was affected by mating system and life history. A male‐biased, density‐dependent sex ratio reduces the rate of genetic drift compared to an equal, density‐independent sex ratio, but a stochastic change towards male bias reduces the N_e/N ratio. Environmental stochasticity amplifies temporal fluctuations in population size and is thus vital to consider in estimation of effective population sizes over longer time periods. Our results on the reduced loss of genetic variation at small densities, particularly in polygamous populations, indicate that density regulation may facilitate adaptive evolution at small population sizes.     

Shifting dynamic forces in fish stock fluctuations triggered by age truncation?

Accumulating evidence shows that environmental fluctuations and exploitation jointly affect marine fish populations, and understanding their interaction is a key issue for fisheries ecology. In particular, it has been proposed that age truncation induced by fisheries exploitation may increase the population's sensitivity to climate. In this study, we use unique long‐term abundance data for the Northeast Arctic stock of cod (Gadus morhua) and the Norwegian Spring‐Spawning stock of herring (Clupea harengus), which we analyze using techniques based on age‐structured population matrices. After identifying time periods with different age distributions in the spawning stock, we use linear models to quantify the relative effect of exploitation and temperature on the population growth rates. For the two populations, age truncation was found to be associated with an increasing importance of temperature and a relatively decreasing importance of exploitation, while the population growth rate became increasingly sensitive to recruitment variations. The results suggested that the removal of older age classes reduced the buffering capacity of the population, thereby making the population growth rate more dependent on recruitment than adult survival and increasing the effect of environmental fluctuations. Age structure appeared as a key characteristic that can affect the response of fish stocks to climate variations and its consequences may be of key importance for conservation and management.     

Epidemiology of anther-smut disease (Microbotryum violaceum) and numeric regulation of populations ofSilene dioica

Summary The interaction between the anther smutMicrobotryum violaceum and its hostSilene dioica was studied during 1985–1990 in 47 populations of different ages, sizes and densities, in an archipelago area in northern Sweden. The sizes of these populations had also been surveyed in the early 1970s. Our results indicate that establishment ofMicrobotryum violaceum is host-size and density dependent. Firstly, young populations ofSilene dioica that became diseased during the study were larger and tended to be more dense than young populations that remained healthy. Secondly, populations diseased in both 1985 and 1990 were found to be larger and tended to be more dense than populations healthy in both years. We were able to document that the pathogen actually failed to establish in two small young populations (diseased plants died shortly after they appeared) and did go extinct in one small old population. Disease incidences within populations did not show large fluctuations between years. The highest increases in disease incidence during the study were found in three relatively young populations that were disease-free at the start of the study. Older populations highly diseased at the start showed less of an increase. Our study indicates thatMicrobotryum violaceum acts as a regulatory factor influencing the rate of increase inSilene dioica populations, once they are sufficiently large to maintain the pathogen. Firstly, seedling density decreased with increased incidence of disease and a seed addition experiment indicated seed-limited recuritment in highly diseased populations. Secondly, those populations that reached very large sizes or densities were either healthy or had very low incidences, indicating the potential for populations that for one reason or another escape an epidemic. However, in the comparison of changes in population size over 16–18 years there was no simple correlation between expansion rate and disease incidence.     

The extended Moran effect and large-scale synchronous fluctuations in the size of great tit and blue tit populations

1. Synchronous fluctuations of geographically separated populations are in general explained by the Moran effect, i.e. a common influence on the local population dynamics of environmental variables that are correlated in space. Empirical support for such a Moran effect has been difficult to provide, mainly due to problems separating out effects of local population dynamics, demographic stochasticity and dispersal that also influence the spatial scaling of population processes. Here we generalize the Moran effect by decomposing the spatial autocorrelation function for fluctuations in the size of great tit Parus major and blue tit Cyanistes caeruleus populations into components due to spatial correlations in the environmental noise, local differences in the strength of density regulation and the effects of demographic stochasticity. 2. Differences between localities in the strength of density dependence and nonlinearity in the density regulation had a small effect on population synchrony, whereas demographic stochasticity reduced the effects of the spatial correlation in environmental noise on the spatial correlations in population size by 21.7% and 23.3% in the great tit and blue tit, respectively. 3. Different environmental variables, such as beech mast and climate, induce a common environmental forcing on the dynamics of central European great and blue tit populations. This generates synchronous fluctuations in the size of populations located several hundred kilometres apart. 4. Although these environmental variables were autocorrelated over large areas, their contribution to the spatial synchrony in the population fluctuations differed, dependent on the spatial scaling of their effects on the local population dynamics. We also demonstrate that this effect can lead to the paradoxical result that a common environmental variable can induce spatial desynchronization of the population fluctuations. 5. This demonstrates that a proper understanding of the ecological consequences of environmental changes, especially those that occur simultaneously over large areas, will require information about the spatial scaling of their effects on local population dynamics.     

Cultivation Fosters Plant Naturalization by Reducing Environmental Stochasticity

A large fraction of the immigrant (or founder) populations of terrestrial plants are small (< 10⁴) and are acutely sensitive to environmental stochasticity. As a result, they undergo radical size fluctuations during a prolonged lag phase that almost always result in their extirpation. Naturalizations are those rare examples in which an immigrant population increases above a threshold size such that the consequences of environmental stochasticity are markedly lower. The likelihood that a non-indigenous population will reach this threshold size would be enhanced substantially through either deliberate or inadvertent cultivation. Cultivation (e.g. protection from predators, parasites, drought, frost) shields small immigrant populations from the extreme expressions of environmental stochasticity. In addition, cultivation can preserve through seed storage a residual non-indigenous population from which new populations can be established. As disseminules are spread locally, and even regionally, immigrant populations sample a wide variety of micro-habitats, thus increasing the likelihood that some plants will survive even without cultivation. Origins of naturalized floras in Australia and the US reveal a strong circumstantial link between cultivation and subsequent naturalization: the single largest group of naturalized species was deliberately introduced as either crops, forage spp., or ornamentals. Another group was introduced inadvertently as contaminants in crop seeds. This correspondence between cultivation and subsequent naturalization provides a common demographic explanation for non-indigenous plant persistence that largely transcends species’ attributes and the commonly ascribed features of communities that are vulnerable to the entry of non-indigenous plants. Humans have played a more profound role in fostering plant naturalizations than by acting simply as plant dispersers; their post-immigration cultivation fosters much naturalization. This revised version was published online in July 2006 with corrections to the Cover Date.     

The Allee effect, stochastic dynamics and the eradication of alien species

Previous treatments of the population biology of eradication have assumed that eradication can only be achieved via 100% removal of the alien population. However, this assumption appears to be incorrect because stochastic dynamics and the Allee effect typically contribute to the extinction of very low‐density populations. We explore a model that incorporates Allee dynamics and stochasticity to observe how these two processes contribute to the extinction of isolated populations following eradication treatments of varying strength (percentage mortality). As a case study, we used historical data on the dynamics of isolated gypsy moth, Lymantria dispar, populations to fit parameters to this model. The parameterized model was then used in simulations that evaluated the efficacy of various eradication strategies. The results indicated that eradication of isolated gypsy moth populations could be easily achieved following a treatment of >80% mortality as long as populations were relatively low (indicated by <100 males captured in pheromone traps).     

Pushed beyond the brink: Allee effects,environmental stochasticity,and extinction

To understand the interplay between environmental stochasticity and Allee effects, we analyse persistence, asymptotic extinction, and conditional persistence for stochastic difference equations. Our analysis reveals that persistence requires that the geometric mean of fitness at low densities is greater than one. When this geometric mean is less than one, asymptotic extinction occurs with high probability for low initial population densities. Additionally, if the population only experiences positive density-dependent feedbacks, conditional persistence occurs provided the geometric mean of fitness at high population densities is greater than one. However, if the population experiences both positive and negative density-dependent feedbacks, conditional persistence only occurs if environmental fluctuations are sufficiently small. We illustrate counter-intuitively that environmental fluctuations can increase the probability of persistence when populations are initially at low densities, and can cause asymptotic extinction of populations experiencing intermediate predation rates despite conditional persistence occurring at higher predation rates.