Erosion of Heterozygosity in Fluctuating Populations

Abstract: Demographic, environmental, and genetic stochasticity threaten the persistence of isolated populations. The relative importance of these intertwining factors remains unresolved, but a common view is that random demographic and environmental events will usually drive small populations to the brink of extinction before genetic deterioration poses a serious threat. To evaluate the potential importance of genetic factors, we analyzed a model linking demographic and environmental conditions to the loss of genetic diversity in isolated populations undergoing natural levels of fluctuation. Nongenetic processes—environmental stochasticity and population demography—were modeled according to a bounded diffusion process. Genetic processes were modeled by quantifying the rate of drift according to the effective population size, which was predicted from the same parameters used to describe the nongenetic processes. We combined these models to predict the heterozygosity remaining at the time of extinction, as predicted by the nongenetic portion of the model. Our model predicts that many populations will lose most or all of their neutral genetic diversity before nongenetic random events lead to extinction. Given the abundant evidence for inbreeding depression and recent evidence for elevated extinction rates of inbred populations, our findings suggest that inbreeding may be a greater general threat to population persistence than is generally recognized. Therefore, conservation biologists should not ignore the genetic component of extinction risk when assessing species endangerment and developing recovery plans.     

Demographic stochasticity reduces the synchronizing effect of dispersal in predator-prey metapopulations

Dispersal may affect predator-prey metapopulations by rescuing local sink populations from extinction or by synchronizing population dynamics across the metapopulation, increasing the risk of regional extinction. Dispersal is likely influenced by demographic stochasticity, however, particularly because dispersal rates are often very low in metapopulations. Yet the effects of demographic stochasticity on predator-prey metapopulations are not well known. To that end, I constructed three models of a two-patch predator-prey system. The models constitute a hierarchy of complexity, allowing direct comparisons. Two models included demographic stochasticity (pure jump process [PJP] and stochastic differential equations [SDE]), and the third was deterministic (ordinary differential equations [ODE]). One stochastic model (PJP) treated population sizes as discrete, while the other (SDE) allowed population sizes to change continuously. Both stochastic models only produced synchronized predator-prey dynamics when dispersal was high for both trophic levels. Frequent dispersal by only predators or prey in the PJP and SDE spatially decoupled the trophic interaction, reducing synchrony of the non-dispersive species. Conversely, the ODE generated synchronized predator-prey dynamics across all dispersal rates, except when initial conditions produced anti-phase transients. These results indicate that demographic stochasticity strongly reduces the synchronizing effect of dispersal, which is ironic because demographic stochasticity is often invoked post hoc as a driver of extinctions in synchronized metapopulations.     

Integral projection models for finite populations in a stochastic environment

Continuous types of population structure occur when continuous variables such as body size or habitat quality affect the vital parameters of individuals. These structures can give rise to complex population dynamics and interact with environmental conditions. Here we present a model for continuously structured populations with finite size, including both demographic and environmental stochasticity in the dynamics. Using recent methods developed for discrete age-structured models we derive the demographic and environmental variance of the population growth as functions of a continuous state variable. These two parameters, together with the expected population growth rate, are used to define a one-dimensional diffusion approximation of the population dynamics. Thus, a substantial reduction in complexity is achieved as the dynamics of the complex structured model can be described by only three population parameters. We provide methods for numerical calculation of the model parameters and demonstrate the accuracy of the diffusion approximation by computer simulation of specific examples. The general modeling framework makes it possible to analyze and predict future dynamics and extinction risk of populations with various types of structure, and to explore consequences of changes in demography caused by, e.g., climate change or different management decisions. Our results are especially relevant for small populations that are often of conservation concern.     

Demographic Viability of a Relict Population of the Critically Endangered Plant Borderea chouardii

Abstract:   In addition to human-caused changes in the environment, natural stochasticity may threaten species persistence, and its impact must be taken into account when priorities are established and management plans are designed. Borderea chouardii is a Tertiary relict at risk of extinction that occurs in only one location in the world, where the probability of human disturbance is low. Its persistence, therefore, is mainly linked to its response to natural threats such as stochasticity. Over 8 years I monitored up to 25% of this rupicolous small geophyte. The population had an unbalanced size structure and 90% failure in seed arrival at appropriate microhabitats, which suggests a problem with recruitment. I used matrix models to describe its population dynamics, conducted hand sowings, and performed stochastic simulations to investigate the effect of environmental stochasticity on population trend and viability. I modeled several scenarios to represent a variety of ecological situations, such as population reduction, episodic or persistent disease, and enhancement or decrease of recruitment. Population growth rate (λ) was never significantly different from unity over the study period. The risk of extinction was null over the next five centuries under current conditions. Increase of mortality and decrease of recruitment reduced stochastic population growth rate, but no factor except a persistent increase of 10% mortality resulted in extinction. These results are the consequence of the plant's extremely long life span (over 300 years) and low temporal variability of key vital rates. Even though hand sowing significantly increased the stochastic population growth rate, other approaches may be more important for the persistence of this species. The extremely slow capacity for recovery following disturbances renders habitat preservation essential. In addition, the founding of new populations would reduce the risk associated with habitat destruction.     

Predicting Extinction Times from Environmental Stochasticity and Carrying Capacity

Managers of small populations often need to estimate the expected time to extinction T_e of their charges. Useful models for extinction times must be ecologically realistic and depend on measurable parameters. Many populations become extinct due to environmental stochasticity, even when the carrying capacity K is stable and the expected growth rate is positive. A model is proposed that gives T_e by diffusion analysis of the log population size n_t (= log_e N_t). The model population grows according to the equation N_t+1 = R_tN_t, with K as a ceiling. Application of the model requires estimation of the parameters k = logK, r_d = the expected change in n, v_r = Variance(log R), and ϱ the autocorrelation of the r_t. These are readily calculable from annual census data (r_d is trickiest to estimate). General formulas for T_e are derived. As a special case, when environmental fluctuations overwhelm expected growth (that is r_d 0), T_e = 2n_o(k - n_o/2)/v_r. If the r_t are autocorrelated, then the effective variance is v_re v_r (1 + ϱ)/(1 - ϱ). The theory is applied to populations of checkerspot butterfly, grizzly bear, wolf, and mountain lion.     

Food web structure affects the extinction risk of species in ecological communities

This paper studies the effect of food web structure on the extinction risk of species. We examine 793 different six-species food web structures with different number, position and strength of trophic links and expose them to stochasticity in a model with Lotka–Volterra predator–prey dynamics. The characteristics of species (intrinsic rates of increase as well as intraspecific density dependence) are held constant, but the interactions with other species and characteristics of the food web are varied.     

The Long-Term Effects of Tiger Poaching on Population Viability

Poaching tigers, primarily for their bones, has become the latest threat to the persistence of wild tiger populations throughout the world. Anecdotal information indicates the seriousness of this new threat. It is important, however, to provide a quantitative analysis of poaching as a basis for strong policy action. We therefore created a tiger simulation model to explore the effects of realistic levels of poaching on population viability. The model is an individually based, stochastic spatial model that is based on the extensive data set from Royal Chitwan National Park, Nepal. We found that as poaching continues over time, the probability of population extinction increases sigmoidally; a critical zone exists in which a small, incremental increase in poaching greatly increases the probability of extinction. The implication is that poaching may not at first be seen as a threat but could suddenly become one. Moreover, even if poaching is effectively stopped, tiger populations will still be vulnerable and could go extinct due to demographic and environmental stochasticity. Our model also shows that poaching reduces genetic variability, which could further reduce population viability due to inbreeding depression. The longer poaching is allowed to continue, the more vulnerable a population will be to these stochastic events. At currently reported rates of poaching our analysis indicates that many wild tiger populations will be extirpated during the latter half of the 1990s.     

Four Factors Modifying the Effect of Competition on Carnivore Population Dynamics as Illustrated by African Wild Dogs

Abstract: Limitation of predator populations by prey availability and the effects of predators on prey populations are widely recognized as important ecological processes that affect carnivore conservation. Interspecific competition can also be a strong limiting factor for carnivore populations, and the effects of competition help explain why some carnivore species are prone to extinction. Competition among carnivores is unusual in some ways, so some predictions from traditional models of competition do not hold. For example, an increase in the density of prey can increase the effect of competition among carnivores, rather than weakening it. I used published data from African wild dogs (    Lycaon pictus ) to highlight four complexities that can modify the effects of competition on the population dynamics of carnivores: habitat fragmentation, counterintuitive effects of prey density, predator-prey size ratios, and habitat type.     

Extinction Dynamics in the American Marten (Martes americana)

We constructed a model of marten population dynamics and used it to investigate extinction processes across a wide range of parameter values. The model was based on rules governing the behavior and physiology of individual martens and focused on energy balance. Spatial dynamics and demographic and environmental stochasticity were incorporated. The outcome was the probability of extinction and quasiextinction (20 females remaining) over 500 years. Three qualitative forms of extinction were delineated. The first was deterministic extinction, associated with those parameter combinations leading to a negative population growth rate. The second was probabilistic extinction in systems with a strong positive growth rate but restricted population size due to habitat constraint. The transition from 100% persistence to 100% quasiextinction, as the input habitat size was decreased, was abrupt. The final form of extinction was in systems with a growth rate of approximately zero. Prey availability maintained an upper limit on these populations, but otherwise fluctuations in population size were essentially random, leading to nontrivial probabilities of extinction in even relatively large populations. A number of issues requiring further empirical research were identified. These included the relationship between habitat quality and marten reproduction, dispersal patterns and dispersal mortality, the effect of habitat edge on marten reproduction and mortality, and the characterization of the severity and frequency of catastrophic mortality as experienced by marten populations.     

The Impact of Increased Environmental Stochasticity Due to Climate Change on the Dynamics of Asiatic Wild Ass

Abstract:  Theory proposes that increased environmental stochasticity negatively impacts population viability. Thus, in addition to the directional changes predicted for weather parameters under global climate change (GCC), the increase in variance of these parameters may also have a negative effect on biodiversity. As a case study, we assessed the impact of interannual variance in precipitation on the viability of an Asiatic wild ass ( Equus hemionus ) population reintroduced in Makhtesh Ramon Nature Reserve, Israel. We monitored the population from 1985 to 1999 to determine what environmental factors affect reproductive success. Annual precipitation during the year before conception, drought conditions during gestation, and population size determined reproductive success. We used the parameters derived from this model to assess population performance under various scenarios in a Leslie matrix type model with demographic and environmental stochasticity. Specifically, we used a change in the precipitation regime in our study area to formulate a GCC scenario and compared the simulated dynamics of the population with a no-change scenario. The coefficient of variation in population size under the global change scenario was 30% higher than under the no-change scenario. Minor die-offs (≥15%) following droughts increased extinction probability nearly 10-fold. Our results support the idea that an increase in environmental stochasticity due to GCC may, in itself, pose a significant threat to biodiversity.     

Incorporating landscape stochasticity into population viability analysis.

The importance of incorporating landscape dynamics into population viability analysis (PVA) has previously been acknowledged, but the need to repeat the landscape generation process to encapsulate landscape stochasticity in model outputs has largely been overlooked. Reasons for this are that (1) there is presently no means for quantifying the relative effects of landscape stochasticity and population stochasticity on model outputs, and therefore no means for determining how to allocate simulation time optimally between the two; and (2) the process of generating multiple landscapes to incorporate landscape stochasticity is tedious and user-intensive with current PVA software. Here we demonstrate that landscape stochasticity can be an important source of variance in model outputs. We solve the technical problems with incorporating landscape stochasticity by deriving a formula that gives the optimal ratio of population simulations to landscape simulations for a given model, and by providing a computer program that incorporates the formula and automates multiple landscape generation in a dynamic landscape metapopulation (DLMP) model. Using a case study of a bird population, we produce estimates of DLMP model output parameters that are up to four times more precise than those estimated from a single landscape in the same amount of total simulation time. We use the DLMP modeling software RAMAS Landscape to run the landscape and metapopulation models, though our method is general and could be applied to any PVA platform. The results of this study should motivate DLMP modelers to consider landscape stochasticity in their analyses.     

Sensitivity to assumptions in models of generalist predation on a cyclic prey

Ecological theory predicts that generalist predators should damp or suppress long-term periodic fluctuations (cycles) in their prey populations and depress their average densities. However, the magnitude of these impacts is likely to vary depending on the availability of alternative prey species and the nature of ecological mechanisms driving the prey cycles. These multispecies effects can be modeled explicitly if parameterized functions relating prey consumption to prey abundance, and realistic population dynamical models for the prey, are available. These requirements are met by the interaction between the Hen Harrier (Circus cyaneus) and three of its prey species in the United Kingdom, the Meadow Pipit (Anthus pratensis), the field vole (Microtus agrestis), and the Red Grouse (Lagopus lagopus scoticus). We used this system to investigate how the availability of alternative prey and the way in which prey dynamics are modeled might affect the behavior of simple trophic networks. We generated cycles in one of the prey species (Red Grouse) in three different ways: through (1) the interaction between grouse density and macroparasites, (2) the interaction between grouse density and male grouse aggressiveness, and (3) a generic, delayed density-dependent mechanism. Our results confirm that generalist predation can damp or suppress grouse cycles, but only when the densities of alternative prey are low. They also demonstrate that diametrically opposite indirect effects between pairs of prey species can occur together in simple systems. In this case, pipits and grouse are apparent competitors, whereas voles and grouse are apparent facilitators. Finally, we found that the quantitative impacts of the predator on prey density differed among the three models of prey dynamics, and these differences were robust to uncertainty in parameter estimation and environmental stochasticity.     

Influence of prey reserve capacity on predator–prey dynamics

In this article the dynamics of a predator and prey population has been modelled when a reserve is created to protect a certain number of prey population from predation. This investigation is essential to derive some guiding principles to conserve the prey population and also to understand the behaviour and dependence of predator population on the reserve capacity. The present study concerns analysis of a fairly general model and hence some of the existing results based on specific explicit models, like Lotka–Volterra model and Rosenzweig–MacArthur model, can be derived from this work. In this study, conditions have been derived for coexistence of predator and prey, and extinction of predators. Results are obtained for global stability of required equilibrium of the model. Also, a method is suggested to compute reserve capacity in order to drive the ecosystem state to a required level. The key results developed in this article have been illustrated using numerical simulation. These results can be interpreted in different contexts like resource conservation, pest management, bio-economics of a renewable resource, etc.     

Recolonizing wolves and mesopredator suppression of coyotes: impacts on pronghorn population dynamics.

Food web theory predicts that the loss of large carnivores may contribute to elevated predation rates and, hence, declining prey populations, through the process of mesopredator release. However, opportunities to test predictions of the mesopredator release hypothesis are rare, and the extent to which changes in predation rates influence prey population dynamics may not be clear due to a lack of demographic information on the prey population of interest. We utilized spatial and seasonal heterogeneity in wolf distribution and abundance to evaluate whether mesopredator release of coyotes (Canis latrans), resulting from the extirpation of wolves (Canis lupus) throughout much of the United States, contributes to high rates of neonatal mortality in ungulates. To test this hypothesis, we contrasted causes of mortality and survival rates of pronghorn (Antilocapra americana) neonates captured at wolf-free and wolf-abundant sites in western Wyoming, USA, between 2002 and 2004. We then used these data to parameterize stochastic population models to heuristically assess the impact of wolves on pronghorn population dynamics due to changes in neonatal survival. Coyote predation was the primary cause of mortality at all sites, but mortality due to coyotes was 34% lower in areas utilized by wolves (P < 0.001). Based on simulation modeling, the realized population growth rate was 0.92 based on fawn survival in the absence of wolves, and 1.06 at sites utilized by wolves. Thus, wolf restoration is predicted to shift the trajectory of the pronghorn population from a declining to an increasing trend. Our results suggest that reintroductions of large carnivores may influence biodiversity through effects on prey populations mediated by mesopredator suppression. In addition, our approach, which combines empirical data on the population of interest with information from other data sources, demonstrates the utility of using simulation modeling to more fully evaluate ecological theories by moving beyond estimating changes in vital rates to analyses of population-level impacts.     

A comparison of models for predicting population persistence

We consider a range of models that may be used to predict the future persistence of populations, particularly those based on discrete-state Markov processes. While the mathematical theory of such processes is very well-developed, they may be difficult to work with when attempting to estimate parameters or expected times to extinction. Hence, we focus on diffusion and other approximations to these models, presenting new and recent developments in parameter estimation for density dependent processes, and the calculation of extinction times for processes subject to catastrophes. We illustrate these and other methods using data from simulated and real time series. We give particular attention to a procedure, due to Ross et al. [Ross, J.V., Taimre, T., Pollett, P.K. On parameter estimation in population models, Theor. Popul. Biol., in press], for estimating the parameters of the stochastic SIS logistic model, and demonstrate ways in which these parameters may be used to estimate expected extinction times. Although the stochastic SIS logistic model is strictly density dependent and allows only for birth and death events, it nonetheless may be used to predict extinction times with some accuracy even for populations that are only weakly density dependent, or that are subject to catastrophes.     

Extinction-effective population index: incorporating life-history variations in population viability analysis

Viability status of populations is a commonly used measure for decision-making in the management of populations. One of the challenges faced by managers is the need to consistently allocate management effort among populations. This allocation should in part be based on comparison of extinction risks among populations. Unfortunately, common criteria that use minimum viable population size or count-based population viability analysis (PVA) often do not provide results that are comparable among populations, primarily because they lack consistency in determining population size measures and threshold levels of population size (e.g., minimum viable population size and quasi-extinction threshold). Here I introduce a new index called the "extinction-effective population index," which accounts for differential effects of demographic stochasticity among organisms with different life-history strategies and among individuals in different life stages. This index is expected to become a new way of determining minimum viable population size criteria and also complement the count-based PVA. The index accounts for the difference in life-history strategies of organisms, which are modeled using matrix population models. The extinction-effective population index, sensitivity, and elasticity are demonstrated in three species of Pacific salmonids. The interpretation of the index is also provided by comparing them with existing demographic indices. Finally, a measure of life-history-specific effect of demographic stochasticity is derived.     

A model for behavioral regulation of metapopulation dynamics

Habitat fragmentation can decrease local population persistence by reducing connectivity, which is a function of dispersal of individuals among habitat fragments. Dispersal is often treated as diffusion in population models, even though for many species it is a result of a series of behavioral decisions. We developed a metapopulation model to explore the potential importance of dispersal behaviors in driving metapopulation dynamics. We incorporated types of behavior that affect dispersal—colonization inhibiting, colonization enhancing, extinction inhibiting, extinction enhancing, rescue enhancing, rescue inhibiting—into Levins’ (1969) metapopulation model and projected occupancy rates for a variety of parameter values. Examples from the literature of behaviors associated with each of these parameters are provided. Our model simplifies into previously published metapopulation models that incorporate only a single behavior, and we present a density-dependent rescue function that leads to multiple non-zero equilibria. We found a variety of behavioral effects on metapopulations. Rescue enhancement fills patches faster than does colonization enhancement or extinction inhibition, and declines in patch occupancy are moderate with extinction enhancement, but colonization inhibition causes metapopulation extinction. We also found that with colonization and extinction inhibitions, equilibrium patch occupancy is inversely related to patch turnover rate. With density-dependent rescue, persistence depends not only on the strength of the strong rescue effect, but also on having a sufficient initial fraction of patches occupied; the stronger the rescue effect, the lower this fraction can be. This study suggests that dispersal behavior can have strong influences on metapopulation dynamics. It confirms the importance of understanding the relationship between landscape structure and dispersal behavior in understanding population persistence.     

Consequences for predators of rescue and Allee effects on prey

The size of a population can be augmented by enriching the carrying capacity of its limiting resource, or by subsidising the renewal of the resource. The well known ‘paradox of enrichment’ models the first case, in which enrichment can force consumers and their limiting resource into destabilising limit cycles, whereas impoverishment stabilises the dynamics. In this paper we model the case of resource subsidy, where the resource is a limiting prey to predators. In contrast to enrichment, the system is stabilised by an influx of prey in the form of a rescue effect, and destabilised by an outflux of prey in the form of an Allee effect. Limit cycles are not sustained by the Allee effect; instead both populations collapse to zero over a large region of the predator-prey phase plane. The catastrophic extinction of prey requires the presence of both an Allee effect on prey and a predator with a type II functional response, though neither needs to contribute a large impact to prey dynamics. The novel implication is that consumers exaggerate the impact of Allee effects on a renewing resource. Conversely, an Allee effect in the form of a cull of resource, even of small value, can trigger local extinction of resource-dependent consumers.     

Consistent scaling of persistence time in metapopulations

Recent theory and experimental work in metapopulations and metacommunities demonstrates that long-term persistence is maximized when the rate at which individuals disperse among patches within the system is intermediate; if too low, local extinctions are more frequent than recolonizations, increasing the chance of regional-scale extinctions, and if too high, dynamics exhibit region-wide synchrony, and local extinctions occur in near unison across the region. Although common, little is known about how the size and topology of the metapopulation (metacommunity) affect this bell-shaped relationship between dispersal rate and regional persistence time. Using a suite of mathematical models, we examined the effects of dispersal, patch number, and topology on the regional persistence time when local populations are subject to demographic stochasticity. We found that the form of the relationship between regional persistence time and the number of patches is consistent across all models studied; however, the form of the relationship is distinctly different among low, intermediate, and high dispersal rates. Under low and intermediate dispersal rates, regional persistence times increase logarithmically and exponentially (respectively) with increasing numbers of patches, whereas under high dispersal, the form of the relationship depends on local dynamics. Furthermore, we demonstrate that the forms of these relationships, which give rise to the bell-shaped relationship between dispersal rate and persistence time, are a product of recolonization and the region-wide synchronization (or lack thereof) of population dynamics. Identifying such metapopulation attributes that impact extinction risk is of utmost importance for managing and conserving the earth's evermore fragmented populations.     

Developing Metapopulation Connectivity Criteria from Genetic and Habitat Data to Recover the Endangered Mexican Wolf

Restoring connectivity between fragmented populations is an important tool for alleviating genetic threats to endangered species. Yet recovery plans typically lack quantitative criteria for ensuring such population connectivity. We demonstrate how models that integrate habitat, genetic, and demographic data can be used to develop connectivity criteria for the endangered Mexican wolf (Canis lupus baileyi), which is currently being restored to the wild from a captive population descended from 7 founders. We used population viability analysis that incorporated pedigree data to evaluate the relation between connectivity and persistence for a restored Mexican wolf metapopulation of 3 populations of equal size. Decreasing dispersal rates greatly increased extinction risk for small populations (<150–200), especially as dispersal rates dropped below 0.5 genetically effective migrants per generation. We compared observed migration rates in the Northern Rocky Mountains (NRM) wolf metapopulation to 2 habitat‐based effective distance metrics, least‐cost and resistance distance. We then used effective distance between potential primary core populations in a restored Mexican wolf metapopulation to evaluate potential dispersal rates. Although potential connectivity was lower in the Mexican wolf versus the NRM wolf metapopulation, a connectivity rate of >0.5 genetically effective migrants per generation may be achievable via natural dispersal under current landscape conditions. When sufficient data are available, these methods allow planners to move beyond general aspirational connectivity goals or rules of thumb to develop objective and measurable connectivity criteria that more effectively support species recovery. The shift from simple connectivity rules of thumb to species‐specific analyses parallels the previous shift from general minimum‐viable‐population thresholds to detailed viability modeling in endangered species recovery planning. Desarrollo de Criterios de Conectividad Metapoblacional a Partir de Datos Genéticos y de Hábitat para Recuperar al Lobo Mexicano en Peligro de Extinción