Density-dependent demographic variation determines extinction rate of experimental populations

Understanding population extinctions is a chief goal of ecological theory. While stochastic theories of population growth are commonly used to forecast extinction, models used for prediction have not been adequately tested with experimental data. In a previously published experiment, variation in available food was experimentally manipulated in 281 laboratory populations of Daphnia magna to test hypothesized effects of environmental variation on population persistence. Here, half of those data were used to select and fit a stochastic model of population growth to predict extinctions of populations in the other half. When density-dependent demographic stochasticity was detected and incorporated in simple stochastic models, rates of population extinction were accurately predicted or only slightly biased. However, when density-dependent demographic stochasticity was not accounted for, as is usual when forecasting extinction of threatened and endangered species, predicted extinction rates were severely biased. Thus, an experimental demonstration shows that reliable estimates of extinction risk may be obtained for populations in variable environments if high-quality data are available for model selection and if density-dependent demographic stochasticity is accounted for. These results suggest that further consideration of density-dependent demographic stochasticity is required if predicted extinction rates are to be relied upon for conservation planning.     

Predicting patterns of long‐term adaptation and extinction with population genetics

Population genetics struggles to model extinction; standard models track the relative rather than absolute fitness of genotypes, while the exceptions describe only the short‐term transition from imminent doom to evolutionary rescue. But extinction can result from failure to adapt not only to catastrophes, but also to a backlog of environmental challenges. We model long‐term adaptation to long series of small challenges, where fitter populations reach higher population sizes. The population's long‐term fitness dynamic is well approximated by a simple stochastic Markov chain model. Long‐term persistence occurs when the rate of adaptation exceeds the rate of environmental deterioration for some genotypes. Long‐term persistence times are consistent with typical fossil species persistence times of several million years. Immediately preceding extinction, fitness declines rapidly, appearing as though a catastrophe disrupted a stably established population, even though gradual evolutionary processes are responsible. New populations go through an establishment phase where, despite being demographically viable, their extinction risk is elevated. Should the population survive long enough, extinction risk later becomes constant over time.     

Harvest reserves reduce extinction risk in an experimental microcosm

Overharvesting by humans threatens a substantial fraction of endangered species. Reserves have recently received enormous attention as a means of better conserving harvested resources, despite limited empirical evidence of their efficacy. We used manipulated microcosms to test whether reserves reduce extinction risk in mobile populations of harvested Tetrahymena thermophila , a ciliate. Here we show that patterns of population distribution inside and outside reserves can be accurately predicted on the basis of simple models of diffusion coupled with logistic controls on local population growth. No extinctions occurred in eight experimental trials with reserves, whereas extinction occurred in seven of eight trials without reserves, as predicted by population viability models based on stochastic population processes. These results suggest that marine reserves may be an effective means of improving long-term viability in heavily harvested fish species.     

Population extinction and the genetics of adaptation

Theories of adaptation typically ignore the effect of environmental change on population size. But some environmental challenges--challenges to which populations must adapt--may depress absolute fitness below 1, causing populations to decline. Under this scenario, adaptation is a race; beneficial alleles that adapt a population to the new environment must sweep to high frequency before the population becomes extinct. We derive simple, though approximate, solutions to the probability of successful adaptation (population survival) when adaptation involves new mutations, the standing genetic variation, or a mixture of the two. Our results show that adaptation to such environmental challenges can be difficult when relying on new mutations at one or a few loci, and populations will often decline to extinction.     

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。     

Incorporating spatial variation in density enhances the stability of simple population dynamics models

Simple discrete time models of population growth admit a wide variety of dynamic behaviors, including population cycles and chaos. Yet studies of natural and laboratory populations typically reveal their dynamics to be relatively stable. Many explanations for the apparent rarity of unstable or chaotic behavior in real populations have been developed, including the possible stabilizing roles of migration, refugia, abrupt density-dependence, and genetic variation in sensitivity to density. We develop a theoretical framework for incorporating random spatial variation in density into simple models of population growth, and apply this approach to two commonly used models in ecology: the Ricker and Hassell maps. We show that the incorporation of spatial density variation into both these models has a strong stabilizing influence on their dynamic behavior, and leads to their exhibiting stable point equilibria or stable limit cycles over a relatively much larger range of parameter values. We suggest that one reason why chaotic population dynamics are less common than the simple models indicate is, these models typically neglect the potentially stabilizing role of spatial variation in density.     

The route to extinction in variable environments

Estimating the extinction risk of natural populations is not only an urgent problem in conservation biology but also involves some profound aspects of population dynamics. Apart from the obvious case of a continuous decrease in a population's carrying capacity, understanding the extinction process necessarily includes environmental and demographic stochasticity. Here, we build from first principles two stochastic, single-population models that can account for various routes to extinction via demographic and environmental variability. The Ricker model of population dynamics generates extinctions from either low or high (around or above carrying capacity) population densities, primarily depending on the growth parameter r . Since extinctions from high densities seem 'unnatural', there is either something wrong with the model or with our intuition. Suitable data are scarce. Environmental variability has its strongest influence on extinction risk via per capita birth rates and is only marginally influencing that risk via per capita death rates if the growth parameter is high. The distribution of the environmental noise and the stochastic structure of the model have quantitative, but not qualitative effects on the estimates of extinction risks. We conclude that to determine the route to extinction and to estimate the extinction risk require a careful choice of both the deterministic component of the population model (e.g., under- or over-compensation) and the structure of the demographic and environmental variabilities.     

Colonization and extinction dynamics of an annual plant metapopulation in an urban environment

The metapopulation framework considers that the spatiotemporal distribution of organisms results from a balance between the colonization and extinction of populations in a suitable and discrete habitat network. Recent spatially realistic metapopulation models have allowed patch dynamics to be investigated in natural populations but such models have rarely been applied to plants. Using a simple urban fragmented population system in which favourable habitat can be easily mapped, we studied patch dynamics in the annual plant Crepis sancta (Asteraceae). Using stochastic patch occupancy models (SPOMs) and multi‐year occupancy data we dissected extinction and colonization patterns in our system. Overall, our data were consistent with two distinct metapopulation scenarios. A metapopulation (sensu stricto) dynamic in which colonization occurs over a short distance and extinction is lowered by nearby occupied patches (rescue effect) was found in a set of patches close to the city centre, while a propagule rain model in which colonization occurs from a large external population was most consistent with data from other networks. Overall, the study highlights the importance of external seed sources in urban patch dynamics. Our analysis emphasizes the fact that plant distributions are governed not only by habitat properties but also by the intrinsic properties of colonization and dispersal of species. The metapopulation approach provides a valuable tool for understanding how colonization and extinction shape occupancy patterns in highly fragmented plant populations. Finally, this study points to the potential utility of more complex plant metapopulation models than traditionally used for analysing ecological and evolutionary processes in natural metapopulations.     

Using ancient DNA and coalescent-based methods to infer extinction

DNA sequences extracted from preserved remains can add considerable resolution to inference of past population dynamics. For example, coalescent-based methods have been used to correlate declines in some arctic megafauna populations with habitat fragmentation during the last ice age. These methods, however, often fail to detect population declines preceding extinction, most likely owing to a combination of sparse sampling, uninformative genetic markers, and models that cannot account for the increasingly structured nature of populations as habitats decline. As ancient DNA research expands to include full-genome analyses, these data will provide greater resolution of the genomic consequences of environmental change and the genetic signatures of extinction.     

Habitat selection reduces extinction of populations subject to Allee effects

Theoretical studies indicate that a single population under an Allee effect will decline to extinction if reduced below a particular threshold, but the existence of multiple local populations connected by random dispersal improves persistence of the global population. An additional process that can facilitate persistence is the existence of habitat selection by dispersers. Using analytic and simulation models of population change, I found that when habitat patches exhibiting Allee effects are connected by dispersing individuals, habitat selection by these dispersers increases the likelihood that patches persist at high densities, relative to results expected by random settlement. Populations exhibiting habitat selection also attain equilibrium more quickly than randomly dispersing populations. These effects are particularly important when Allee effects are large and more than two patches exist. Integrating habitat selection into population dynamics may help address why some studies have failed to find extinction thresholds in populations, despite well-known Allee effects in many species.     

Approximate Bayesian estimation of extinction rate in the Finnish Daphnia magna metapopulation

Approximate Bayesian computation (ABC) is useful for parameterizing complex models in population genetics. In this study, ABC was applied to simultaneously estimate parameter values for a model of metapopulation coalescence and test two alternatives to a strict metapopulation model in the well‐studied network of Daphnia magna populations in Finland. The models shared four free parameters: the subpopulation genetic diversity (θ_S), the rate of gene flow among patches (4Nm), the founding population size (N₀) and the metapopulation extinction rate (e) but differed in the distribution of extinction rates across habitat patches in the system. The three models had either a constant extinction rate in all populations (strict metapopulation), one population that was protected from local extinction (i.e. a persistent source), or habitat‐specific extinction rates drawn from a distribution with specified mean and variance. Our model selection analysis favoured the model including a persistent source population over the two alternative models. Of the closest 750 000 data sets in Euclidean space, 78% were simulated under the persistent source model (estimated posterior probability = 0.769). This fraction increased to more than 85% when only the closest 150 000 data sets were considered (estimated posterior probability = 0.774). Approximate Bayesian computation was then used to estimate parameter values that might produce the observed set of summary statistics. Our analysis provided posterior distributions for e that included the point estimate obtained from previous data from the Finnish D. magna metapopulation. Our results support the use of ABC and population genetic data for testing the strict metapopulation model and parameterizing complex models of demography.     

Which functional responses preclude extinction in ecological population-dynamic models?

The growth and loss terms of interacting populations, called functional responses, are known to have a significant impact on the extinction dynamics of ecological models. We are able to construct models that preclude extinction for any parameter value, simply through the use of particular combinations of functional responses. These structural coexistence (SC) models have functional responses where the per capita growth terms remain positive (non-vanishing), while the per capita loss terms tend to zero (vanishing) as the relevant population tends to zero. Any of the commonly used functional responses, such as Holling Types I, II, and III, lead to non-vanishing growth terms for nutrient uptake, while any type of nonlinearity such as Ivlev or density dependent mortality of the population leads to vanishing loss terms. In order for herbivore/carnivore feeding terms to simultaneously be a vanishing loss term for the prey and a non-vanishing growth term for the predator, the exponent on the predator population must be exactly one, whilst the exponent on the prey population must be greater than one (such as a Holling Type III response). Any SC system with at least one autotroph and (possibly many) heterotrophs will always possess an internal equilibrium point. We show that the inclusion of linear mortality terms are, however, sufficient to restore the possibility of population extinctions. This allows for the formulation of ‘mixed’ systems, where some populations are guaranteed to coexist, whilst others are subject to the possibility of extinction. SC models have use in studies of, for example, biogeochemical cycling or the plankton base of fisheries models, where extinction is not desirable or relevant.     

The probability of extinction in a bovine respiratory syncytial virus epidemic model

Backward bifurcation is a relatively recent yet well-studied phenomenon associated with deterministic epidemic models. It allows for the presence of multiple subcritical endemic equilibria, and is generally found only in models possessing a reasonable degree of complexity. One particular aspect of backward bifurcation that appears to have been virtually overlooked in the literature is the potential influence its presence might have on the behaviour of any analogous stochastic model. Indeed, the primary aim of this paper is to investigate this possibility. Our approach is to compare the theoretical probabilities of extinction, calculated via a particular stochastic formulation of a deterministic model exhibiting backward bifurcation, with those obtained from a series of stochastic simulations. We have found some interesting links in the behaviour between the deterministic and stochastic models, and are able to offer plausible explanations for our observations.     

Equilibrium and extinction in stochastic population dynamics

Stochastic models of interacting biological populations, with birth and death rates depending on the population size are studied in the quasi-stationary state. Confidence regions in the state space are constructed by a new method for the numerical, solution of the ray equations. The concept of extinction time, which is closely related to the concept of stability for stochastic systems, is discussed. Results of numerical calculations for two-dimensional stochastic population models are presented.     

Comparing vector–host and SIR models for dengue transmission

Various simple mathematical models have been used to investigate dengue transmission. Some of these models explicitly model the mosquito population, while others model the mosquitoes implicitly in the transmission term. We study the impact of modeling assumptions on the dynamics of dengue in Thailand by fitting dengue hemorrhagic fever (DHF) data to simple vector–host and SIR models using Bayesian Markov chain Monte Carlo estimation. The parameter estimates obtained for both models were consistent with previous studies. Most importantly, model selection found that the SIR model was substantially better than the vector–host model for the DHF data from Thailand. Therefore, explicitly incorporating the mosquito population may not be necessary in modeling dengue transmission for some populations.     

Generation cycles in stage structured populations

Some insect populations exhibit cycles in which successive population peaks may correspond to effectively discrete generations. Motivated by this observation, we investigate the structure of matriarchal generations in five simple, continuous-time, stage structure models in order to determine the proportion of individuals in one population peak who are the offspring of individuals in the pervious peak. We conclude that in certain models (including a model of Nicholson's blowflies) successive population peaks do not correspond to discrete generations, whereas in others (including some models of uniform larval competition) successive peaks may well approximate discrete generations. In all models, however, there is eventually significant overlap of generations.     

Speed of expansion and extinction in experimental populations

Although the causes of population extinction are well understood, the speed at which populations decline to extinction is not. A testable, counter-intuitive prediction of stochastic population theory is that, on average, for any interior interval of the domain of biologically attainable population sizes, the expected duration of increase equals the expected duration of decline. Here we report the first empirical tests of this hypothesis. Using data from two experiments in which replicate populations of Daphnia magna were observed to go extinct under different experimental conditions, we failed to reject the null hypothesis of no difference between the growth and decline phases in populations under constant conditions and conditions with modest environmental variability, but find strong evidence to reject equal first passage time in highly variable environments. These results confirm the prediction of equal passage times entailed by diffusion models of population dynamics, supporting continued application in both population theory and conservation decision making under the restricted conditions where the approximation can be expected to hold.     

Deterministic extinction effect of parasites on host populations

 Experimental studies have shown that parasites can reduce host density and even drive host population to extinction. Conventional mathematical models for parasite-host interactions, while can address the host density reduction scenario, fail to explain such deterministic extinction phenomena. In order to understand the parasite induced host extinction, Ebert et al. (2000) formulated a plausible but ad hoc epidemiological microparasite model and its stochastic variation. The deterministic model, resembles a simple SI type model, predicts the existence of a globally attractive positive steady state. Their simulation of the stochastic model indicates that extinction of host is a likely outcome in some parameter regions. A careful examination of their ad hoc model suggests an alternative and plausible model assumption. With this modification, we show that the revised parasite-host model can exhibit the observed parasite induced host extinction. This finding strengthens and complements that of Ebert et al. (2000), since all continuous models are likely break down when all population densities are small. This extinction dynamics resembles that of ratio-dependent predator-prey models. We report here a complete global study of the revised parasite-host model. Biological implications and limitations of our findings are also presented. Received: 30 October 2001 / Revised version: 11 February 2002 / Published online: 17 October 2002 Work is partially supported by NSF grant DMS-0077790 Mathematics Subject Classification (2000): 34C25, 34C35, 92D25. Keywords or phrases: Microparasite model – Ratio-dependent predator-prey model – Host extinction – Global stability – Biological control     

Virus population extinction via ecological traps

Populations are at risk of extinction when unsuitable or when sink habitat exceeds a threshold frequency in the environment. Sinks that present cues associated with high-quality habitats, termed ecological traps, have especially detrimental effects on net population growth at metapopulation scales. Ecological traps for viruses arise naturally, or can be engineered, via the expression of viral-binding sites on cells that preclude viral reproduction. We present a model for virus population growth in a heterogeneous host community, parameterized with data from populations of the RNA bacteriophage Φ6 presented with mixtures of suitable host bacteria and either neutral or trap cells. We demonstrate that viruses can sustain high rates of population growth in the presence of neutral non-hosts as long as some host cells are present, whereas trap cells dramatically reduce viral fitness. In addition, we demonstrate that the efficacy of traps for viral elimination is frequency dependent in spatially structured environments such that population viability is a nonlinear function of habitat loss in dispersal-limited virus populations. We conclude that the ecological concepts applied to species conservation in altered landscapes can also contribute to the development of trap cell therapies for infectious human viruses.     

Localization and extinction of bacterial populations under inhomogeneous growth conditions

The transition from localized to systemic spreading of bacteria, viruses, and other agents is a fundamental problem that spans medicine, ecology, biology, and agriculture science. We have conducted experiments and simulations in a simple one-dimensional system to determine the spreading of bacterial populations that occurs for an inhomogeneous environment under the influence of external convection. Our system consists of a long channel with growth inhibited by uniform ultraviolet (UV) illumination except in a small "oasis", which is shielded from the UV light. To mimic blood flow or other flow past a localized infection, the oasis is moved with a constant velocity through the UV-illuminated "desert". The experiments are modeled with a convective reaction-diffusion equation. In both the experiment and model, localized or extinct populations are found to develop, depending on conditions, from an initially localized population. The model also yields states where the population grows everywhere. Further, the model reveals that the transitions between localized, extended, and extinct states are continuous and nonhysteretic. However, it does not capture the oscillations of the localized population that are observed in the experiment.