The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.     

Effects of experimental population extinction for the spatial population dynamics of the butterfly Parnassius smintheus

While many studies have examined factors potentially impacting the rate of local population extinction, few experimental studies have examined the consequences of extinction for spatial population dynamics. Here we report results from a large‐scale, long‐term experiment examining the effects of local population extinction for the dynamics of surrounding populations. From 2001–2008 we removed all adult butterflies from two large, neighboring populations within a system of 17 subpopulations of the Rocky Mountain Apollo butterfly, Parnassius smintheus. Surrounding populations were monitored using individual, mark–recapture methods. We found that population removal decreased immigration to surrounding populations in proportion to their connectivity to the removed populations. Correspondingly, within‐generation population abundance declined. Despite these effects, we saw little consistent impact between generations. The extinction rates of surrounding populations were unaffected and local population growth was not consistently reduced by the lack of immigration. The broader results show that immigration affects local abundance within generations, but dynamics are mediated by density‐dependence within populations and by broader density‐independent factors acting between generations. The loss of immigrants resulting from extinction has little impact on the persistence of local populations in this system.     

Finite metapopulation models with density-dependent migration and stochastic local dynamics

The effects of small density-dependent migration on the dynamics of a metapopulation are studied in a model with stochastic local dynamics. We use a diffusion approximation to study how changes in the migration rate and habitat occupancy affect the rates of local colonization and extinction. If the emigration rate increases or if the immigration rate decreases with local population size, a positive expected rate of change in habitat occupancy is found for a greater range of habitat occupancies than when the migration is density-independent. In contrast, the reverse patterns of density dependence in respective emigration and immigration reduce the range of habitat occupancies where the metapopulation will be viable. This occurs because density-dependent migration strongly influences both the establishment and rescue effects in the local dynamics of metapopulations.     

Allee effects in stochastic populations

The Allee effect, or inverse density dependence at low population sizes, could seriously impact preservation and management of biological populations. The mounting evidence for widespread Allee effects has lately inspired theoretical studies of how Allee effects alter population dynamics. However, the recent mathematical models of Allee effects have been missing another important force prevalent at low population sizes: stochasticity. In this paper, the combination of Allee effects and stochasticity is studied using diffusion processes, a type of general stochastic population model that accommodates both demographic and environmental stochastic fluctuations. Including an Allee effect in a conventional deterministic population model typically produces an unstable equilibrium at a low population size, a critical population level below which extinction is certain. In a stochastic version of such a model, the probability of reaching a lower size a before reaching an upper size b , when considered as a function of initial population size, has an inflection point at the underlying deterministic unstable equilibrium. The inflection point represents a threshold in the probabilistic prospects for the population and is independent of the type of stochastic fluctuations in the model. In particular, models containing demographic noise alone (absent Allee effects) do not display this threshold behavior, even though demographic noise is considered an "extinction vortex". The results in this paper provide a new understanding of the interplay of stochastic and deterministic forces in ecological populations.     

Overcoming Allee effects through evolutionary,genetic, and demographic rescue

Despite the amplified threats of extinction facing small founder populations, successful colonization sometimes occurs, bringing devastating ecological and economic consequences. One explanation may be rapid evolution, which can increase mean fitness in populations declining towards extinction, permitting persistence and subsequent expansion. Such evolutionary rescue may be particularly important, given Allee effects. When a population is introduced at low density, individuals often experience a reduction in one or more components of fitness due to novel selection pressures that arise from diminished intraspecific interactions and positive density dependence (i.e. component Allee effects). A population can avoid extinction if it can adapt and recover on its own (i.e. evolutionary rescue), or if additional immigration sustains the population (i.e. demographic rescue) or boosts its genetic variation that facilitates adaptation (i.e. genetic rescue). These various forms of rescue have often been invoked as possible mechanisms for specific invasions, but their relative importance to invasion is not generally understood. Within a spatially explicit modelling framework, we consider the relative impact of each type of rescue on the probability of successful colonization, when there is evolution of a multi-locus quantitative trait that influences the strength of component Allee effects. We demonstrate that when Allee effects are important, the effect of demographic rescue via recurrent immigration overall provides the greatest opportunity for success. While highlighting the role of evolution in the invasion process, we underscore the importance of the ecological context influencing the persistence of small founder populations.     

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.     

Bottlenecks in large populations: the effect of immigration on population viability

We model a large population that is subject to successive short bottlenecks, in order to investigate the impact of different extents of immigration on the change in genetic load and on viability. A first simple genetic model uncovers the opposite effects of immigration on fitness according to the type of deleterious mutations considered: immigration increases fitness if the genetic load is comprised of mildly deleterious mutations, whereas it decreases fitness if it is comprised of lethals. When considering both types of mutations and adding explicit stochastic demographic considerations, in which bottlenecks are engendered by random catastrophes, the global impact of immigration on viability is dependent upon a balance between its opposite effects on the two components of the genetic load and on demographic stochasticity. In this context, immigration tends to increase the probability of extinction if occurring preferentially when population density is high, while it decreases extinction if occurring preferentially towards low-density populations.     

Fish species richness in spring‐fed ponds: effects of habitat size versus isolation in temporally variable environments

1. Species richness in a habitat patch is determined by immigration (regional) and extinction (local) processes, and understanding their relative importance is crucial for conservation of biodiversity. In this study, we applied the Island Biogeography concept to spring ponds connected to a river in southwestern Japan to examine how immigration and extinction processes interact to determine fish species richness in temporally variable environments. 2. Fish censuses were conducted 15 times in 13 study ponds at 1–4 month intervals from August 1998 through October 2000. Effects of habitat size (pond area), isolation (distance from the river) and temporal environmental variability (water level fluctuation) on (i) species richness, (ii) immigration and extinction rates and (iii) population size and persistence of each fish species were assessed. 3. The results revealed predominant effects of distance on species richness, immigration/extinction rates and population size and persistence. Species richness decreased with increasing distance but was not related to either pond area or water level fluctuation. A negative effect of distance on immigration rate was detected, while neither pond area nor water level fluctuation had significant effects on extinction rate. Further, population size and persistence of four species increased with decreasing distance, suggesting that, in ponds close to the river, immigrants from the river reduce the probability of extinction (i.e. provide a rescue effect), contributing to the maintenance of high species richness. 4. Overall results emphasise the importance of immigration processes, rather than extinction, in shaping patterns of species richness in our system. The predominant importance of immigration was probably because of (i) high temporal variability that negates habitat‐size effects and (ii) continuous immigration that easily compensates for local extinctions. Our results suggest that consideration of regional factors (e.g. connectivity, locations of source populations and barriers to colonisation) is crucial for conservation and restoration of local habitats.     

Modelling the impacts of two exotic invasive species on a native butterfly: top-down vs. bottom-up effects

1. Exotic invasive species can influence population dynamics of native species through top-down or bottom-up forces. The present study examined separate and interactive effects of multiple exotic species invasions on the native mustard white butterfly, Pieris napi oleracea Harris (Lepidoptera: Pieridae), using a stochastic simulation model. 2. P. n. oleracea populations in North America have decreased regionally since the 1860s. Competition with an exotic congener (P. rapae L.), loss of native host plants and parasitism by the introduced broconid wasp (Cotesia glomerata L.), have been suggested to be independently responsible for its decline. The present study examined these hypotheses, as well as an alternative, invasion by an exotic crucifer, garlic mustard (Alliaria petiolata[Bieb.] Cavara & Grande). 3. A stochastic simulation model of P. n. oleracea population dynamics revealed that decreasing the number of host plants available for oviposition and larval development (i.e. habitat loss), sharply reduced the probability of populations persistence and decreased population size for those that persisted. 4. Simulated invasion by garlic mustard also substantially decreased both probability of persistence (= 0 at approximately 50% cover) and mean population size. Persistence probability never reached zero under any C. glomerata scenarios, even when larval mortality in the second generation due to parasitism was 100%. The impact of garlic mustard was intensified by the addition of C. glomerata parasitism. 5. Results suggest that bottom-up forces, loss of host plants through forest understorey loss and/or garlic mustard invasion are the most important forces driving P. n. oleracea population decline. Parasitism by C. glomerata may interact to reduce P. n. oleracea populations more rapidly, but appears insufficient alone to cause local extinction.     

Modeling ecological traps for the control of feral pigs

Ecological traps are habitat sinks that are preferred by dispersing animals but have higher mortality or reduced fecundity compared to source habitats. Theory suggests that if mortality rates are sufficiently high, then ecological traps can result in extinction. An ecological trap may be created when pest animals are controlled in one area, but not in another area of equal habitat quality, and when there is density‐dependent immigration from the high‐density uncontrolled area to the low‐density controlled area. We used a logistic population model to explore how varying the proportion of habitat controlled, control mortality rate, and strength of density‐dependent immigration for feral pigs could affect the long‐term population abundance and time to extinction. Increasing control mortality, the proportion of habitat controlled and the strength of density‐dependent immigration decreased abundance both within and outside the area controlled. At higher levels of these parameters, extinction was achieved for feral pigs. We extended the analysis with a more complex stochastic, interactive model of feral pig dynamics in the Australian rangelands to examine how the same variables as the logistic model affected long‐term abundance in the controlled and uncontrolled area and time to extinction. Compared to the logistic model of feral pig dynamics, the stochastic interactive model predicted lower abundances and extinction at lower control mortalities and proportions of habitat controlled. To improve the realism of the stochastic interactive model, we substituted fixed mortality rates with a density‐dependent control mortality function, empirically derived from helicopter shooting exercises in Australia. Compared to the stochastic interactive model with fixed mortality rates, the model with the density‐dependent control mortality function did not predict as substantial decline in abundance in controlled or uncontrolled areas or extinction for any combination of variables. These models demonstrate that pest eradication is theoretically possible without the pest being controlled throughout its range because of density‐dependent immigration into the area controlled. The stronger the density‐dependent immigration, the better the overall control in controlled and uncontrolled habitat combined. However, the stronger the density‐dependent immigration, the poorer the control in the area controlled. For feral pigs, incorporating environmental stochasticity improves the prospects for eradication, but adding a realistic density‐dependent control function eliminates these prospects.     

Complex dynamics of survival and extinction in simple population models with harvesting

We study the effects of constant harvesting in a discrete population model that includes density-independent survivorship of adults in a population with overcompensating density dependence. The interaction between the survival parameter and other parameters of the model (harvesting rate, natural growth rate) reveal new phenomena of survival and extinction. The main differences with the dynamics of survival and extinction reported for semelparous populations with overcompensatory density dependence are that there can be multiple windows of extinction and conditional persistence as harvesting increases or the intrinsic growth rate is increased, and that, in case of bistability, the basin of attraction of the nontrivial attractor may consist of an arbitrary number of disjoint connected components.     

Comparative population dynamics of large and small mammals in the Northern Hemisphere: deterministic and stochastic forces

Deterministic feedbacks within populations interact with extrinsic, stochastic processes to generate complex patterns of animal abundance over time and space. Animals inherently differ in their responses to fluctuating environments due to differences in body sizes and life history traits. However, controversy remains about the relative importance of deterministic and stochastic forces in shaping population dynamics of large and small mammals. We hypothesized that effects of environmental stochasticity and density dependence are stronger in small mammal populations relative to their effects in large mammal populations and thus differentiate the patterns of population dynamics between them. We conducted an extensive, comparative analysis of population dynamics in large and small mammals to test our hypothesis, using seven population parameters to describe general dynamic patterns for 23 (14 species) time series of observations of abundance of large mammals and 38 (21 species) time series for small mammals. We used state‐space models to estimate the strength of direct and delayed density dependence as well as the strength of environmental stochasticity. We further used phylogenetic comparative analysis to detect differences in population dynamic patterns and individual population parameters, respectively, between large and small mammals. General population dynamic patterns differed between large and small mammals. However, the strength of direct and delayed density dependence was comparable between large and small mammals. Moreover, the variances of population growth rates and environmental stochasticity were greater in small mammals than in large mammals. Therefore, differences in population response to stochastic forces and strength of environmental stochasticity are the primary factor that differentiates population dynamic patterns between large and small mammal species.     

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

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

The Effects of Density Dependence and Immigration on Local Adaptation and Niche Evolution in a Black-Hole Sink Environment

We examine the effects of density dependence and immigration on local adaptation in a "black-hole sink" habitat, i.e., a habitat in which isolated populations of a species would tend to extinction but where a population is demographically maintained by recurrent one-way migration from a separate source habitat in which the species persists. Using a diploid, one-locus model of a discrete-generation sink population maintained by immigration from a fixed source population, we show that a locally favored allele will spread when rare in the sink if the absolute fitness (or, in some cases, the geometric-mean absolute fitness) of heterozygotes with the favored allele is above one in the sink habitat. With density dependence, the criterion for spread can depend on the rate of immigration, because immigration affects local densities and, hence, absolute fitness. Given the successful establishment of a locally favored allele, it will be maintained by a migration-selection balance and the resulting polymorphic population will be sustained deterministically with either stable or unstable dynamics. The densities of stable polymorphic populations tend to exceed densities that would be maintained in the absence of the favored allele. With strong density regulation, spread of the favored allele may destabilize population dynamics. Our analyses show that polymorphic populations which form subsequent to the establishment of favorable alleles have the capacity to persist deterministically without immigration. Finally, we examined the probabilistic rate at which new favored alleles arise and become established in a sink population. Our results suggest that favored alleles are established most readily at intermediate levels of immigration.     

The intrinsic mean time to extinction: a unifying approach to analysing persistence and viability of populations

Analysing the persistence and viability of small populations is a key issue in extinction theory and population viability analysis. However, there is still no consensus on how to quantify persistence and viability. We present an approach to evaluate any simulation model concerned with extinction. The approach is devised from general Markov models of stochastic population dynamics. From these models, we distil insights into the general mathematical structure of the risk of extinction by time t, P₀(t). From this mathematical structure, we devise a simple but effective protocol – the ln(1−P₀)-plot – which is applicable for situations including environmental noise or catastrophes. This plot delivers two quantities which are fundamental to the assessment of persistence and viability: the intrinsic mean time to extinction, T_m, and the probability c₁ of the population reaching the established phase. The established phase is characterized by typical fluctuations of the population's state variable which can be described by quasi-stationary probability distributions. The risk of extinction in the established phase is constant and given by 1/T_m. We show that T_m is the basic currency for the assessment of persistence and viability because T_m is independent of initial conditions and allows the risk of extinction to be calculated for any time horizon. For situations where initial conditions are important, additionally c₁ has to be considered.     

Persistence of structured populations in random environments

Environmental fluctuations often have different impacts on individuals that differ in size, age, or spatial location. To understand how population structure, environmental fluctuations, and density-dependent interactions influence population dynamics, we provide a general theory for persistence for density-dependent matrix models in random environments. For populations with compensating density dependence, exhibiting “bounded” dynamics, and living in a stationary environment, we show that persistence is determined by the stochastic growth rate (alternatively, dominant Lyapunov exponent) when the population is rare. If this stochastic growth rate is negative, then the total population abundance goes to zero with probability one. If this stochastic growth rate is positive, there is a unique positive stationary distribution. Provided there are initially some individuals in the population, the population converges in distribution to this stationary distribution and the empirical measures almost surely converge to the distribution of the stationary distribution. For models with overcompensating density-dependence, weaker results are proven. Methods to estimate stochastic growth rates are presented. To illustrate the utility of these results, applications to unstructured, spatially structured, and stage-structured population models are given. For instance, we show that diffusively coupled sink populations can persist provided that within patch fitness is sufficiently variable in time but not strongly correlated across space.     

Demography_Lab,an educational application to evaluate population growth: Unstructured and matrix models

Training in Population Ecology asks for scalable applications capable of embarking students on a trip from basic concepts to the projection of populations under the various effects of density dependence and stochasticity. Demography_Lab is an educational tool for teaching Population Ecology aspiring to cover such a wide range of objectives. The application uses stochastic models to evaluate the future of populations. Demography_Lab may accommodate a wide range of life cycles and can construct models for populations with and without an age or stage structure. Difference equations are used for unstructured populations and matrix models for structured populations. Both types of models operate in discrete time. Models can be very simple, constructed with very limited demographic information or parameter‐rich, with a complex density‐dependence structure and detailed effects of the different sources of stochasticity. Demography_Lab allows for deterministic projections, asymptotic analysis, the extraction of confidence intervals for demographic parameters, and stochastic projections. Stochastic population growth is evaluated using up to three sources of stochasticity: environmental and demographic stochasticity and sampling error in obtaining the projection matrix. The user has full control on the effect of stochasticity on vital rates. The effect of the three sources of stochasticity may be evaluated independently for each vital rate. The user has also full control on density dependence. It may be included as a ceiling population size controlling the number of individuals in the population or it may be evaluated independently for each vital rate. Sensitivity analysis can be done for the asymptotic population growth rate or for the probability of extinction. Elasticity of the probability of extinction may be evaluated in response to changes in vital rates, and in response to changes in the intensity of density dependence and environmental stochasticity.     

Travel costs, oviposition behaviour and the dynamics of insect–plant systems

Insect attack can have major consequences for plant population dynamics. We used individually based simulation models to ask how insect oviposition behaviour influences persistence and potential stability of an herbivore–plant system. We emphasised effects on system dynamics of herbivore travel costs and of two kinds of behaviour that might evolve to mitigate travel costs: insect clutch size behaviour (whether eggs are laid singly or in groups) and female aggregation behaviour (whether females prefer or avoid plants already bearing eggs). Travel costs that increase as plant populations drop lead to inverse density dependence of plant reproduction under herbivore attack. Female clutch size and aggregation behaviours also strongly affect system dynamics. When females lay eggs in large clutches or aggregate their clutches, herbivore damage varies strongly among plants, providing probabilistic refuges that permit plant reproduction and persistence. However, the population dynamics depend strongly on whether insect behaviour is fixed or responds adaptively to plant population size: when (and only when) females increase clutch size or aggregation as plants become rare, refuges from herbivory weaken at high plant density, creating inverse density dependence in plant reproduction. Both herbivore travel costs themselves, and also insect behaviour that might evolve in response to travel costs, can thus create plant density dependence—a basic requirement for regulation of plant populations by their insect herbivores.     

Influence of food availability on demography and local population dynamics in a long-lived seabird

Few studies have addressed the effects of food availability as a proximate factor affecting local adult survival in long-lived organisms and their consequences at local population dynamics. We used capture-recapture analysis of resightings of 10 birth cohorts of ringed Audouin's gulls, Larus audouinii, to estimate adult survival and dispersal (both emigration and immigration). For the first time, permanent emigration (the transient effect in capture-recapture analysis) was modelled for the whole population and not only for the newly marked birds. Gulls exploit to a large extent fishes discarded from trawlers, and a trawling moratorium established since 1991 has decreased food supply for the colony. This was used as a natural experiment of food availability to assess its effects on adult survival and emigration. These and other demographic parameters were used in a projection modelling to assess the probabilities of extinction of the colony under two scenarios of lower and higher food availability. Food availability (together with the age of individuals) influenced emigration probabilities, but not adult survival, which was estimated at 0.91 (s.e. = 0.02). When food was in shorter supply during the chick-rearing period, emigration was very high (ca. 65%) for younger breeders, although this rate decreased sharply with age. Probabilities of extinction were very high when food availability was low, and when environmental stochasticity was introduced, and only stochastic immigration from the outside seemed to prevent extinction. The results highlight the importance of dispersal processes in the population dynamics of long-lived organisms.     

Timescales of population rarity and commonness in random environments

This is a mathematical study of the interactions between non-linear feedback (density dependence) and uncorrelated random noise in the dynamics of unstructured populations. The stochastic non-linear dynamics are generally complex, even when the deterministic skeleton possesses a stable equilibrium. There are three critical factors of the stochastic non-linear dynamics; whether the intrinsic population growth rate (lambda) is smaller than, equal to, or greater than 1; the pattern of density dependence at very low and very high densities; and whether the noise distribution has exponential moments or not. If lambda < 1, the population process is generally transient with escape towards extinction. When lambda > or = 1, our quantitative analysis of stochastic non-linear dynamics focuses on characterizing the time spent by the population at very low density (rarity), or at high abundance (commonness), or in extreme states (rarity or commonness). When lambda >1 and density dependence is strong at high density, the population process is recurrent: any range of density is reached (almost surely) in finite time. The law of time to escape from extremes has a heavy, polynomial tail that we compute precisely, which contrasts with the thin tail of the laws of rarity and commonness. Thus, even when lambda is close to one, the population will persistently experience wide fluctuations between states of rarity and commonness. When lambda = 1 and density dependence is weak at low density, rarity follows a universal power law with exponent -3/2. We provide some mathematical support for the numerical conjecture [Ferriere, R., Cazelles, B., 1999. Universal power laws govern intermittent rarity in communities of interacting species. Ecology 80, 1505-1521.] that the -3/2 power law generally approximates the law of rarity of 'weakly invading' species with lambda values close to one. Some preliminary results for the dynamics of multispecific systems are presented.