Modelling conservation management options for a southern range‐margin population of Golden Plover Pluvialis apricaria vulnerable to climate change

There is considerable interest in understanding how management may help species and populations cope with climate change (climate change adaptation). I used a population model describing the demography of a southern range‐margin European Golden Plover Pluvialis apricaria population vulnerable to climate change to assess the potential benefits associated with site‐based adaptation management. Two forms of management were simulated: (1) counteracting management to reduce the severity of the negative climate change impacts, simulated by increasing tipulid (cranefly) abundance, and (2) compensatory management to increase populations through an alternative mechanism, simulated by manipulating nest and chick predation rates. A 1 °C rise was estimated to require a doubling of cranefly abundance, or a 35% increase in nest and chick survival rates, to maintain a stable population. For a 2 °C rise, a four‐fold increase in craneflies or an 80% increase in survival rates would be required for population stability. A model based on likely realistic estimates of the magnitude of benefit associated with both adaptation management options showed that combined, they may significantly reduce the severity of population decline and risk of extinction associated with a relatively large increase in temperature of 5.8 °C above 1960–90 levels. Site‐based adaptation management may therefore increase the resistance of Golden Plovers to some degree of future climate change. This model framework for informing climate change adaptation decisions should be developed for other species and habitats.     

A discrete-time model with vaccination for a measles epidemic.

A discrete-time, age-independent SIR-type epidemic model is formulated and analyzed. The effects of vaccination are also included in the model. Three mathematically important properties are verified for the model: solutions are nonnegative, the population size is time-invariant, and the epidemic concludes with all individuals either remaining susceptible or becoming immune (a property typical of SIR models). The model is applied to a measles epidemic on a university campus. The simulated results are in good agreement with the actual data if it is assumed that the population mixes nonhomogeneously. The results of the simulations indicate that a rate of immunity greater than 98% may be required to prevent an epidemic in a university population. The model has applications to other contagious diseases of SIR type. Furthermore, the simulated results of the model can easily be compared to data, and the effects of a vaccination program can be examined.     

Effects of re‐breeding rates on population size estimation of biennial breeders: results from a model based on albatrosses

Estimating total breeding populations (I) for species that exhibit biennial breeding is generally done from counts of individuals that breed in each year (N), but can be complicated by the fact that the proportion of individuals breeding varies from year to year. Partly, this reflects the proportion of individuals that re‐breed in successive years (re‐breeding rate, p), which is largely, although not exclusively, governed by reproductive failure. Here we show that variation in counts of breeding individuals not only reflects changes in total breeding population but can be sensitive to and powerfully driven by variation in p. A simulation of annual field counts of a bird exhibiting biennial breeding was constructed to explore the effect of re‐breeding attempts on estimations of the total breeding population. The model was used to simulate the consequences of adult mortality and different annual patterns of nesting failures on total breeding population estimates, and to explore the consequences of variation in p on N, when total breeding population remains constant. N is shown to be very sensitive to variations in p, so that even short‐term fluctuations in p can cause changes in N that oscillate for many years ahead. We compare our modelled results with real data for Grey‐headed Albatrosses Thalassarche chrysostoma and demonstrate that, when I is held constant in the model, actual counts may be simulated by variations in p only. Normally, I is unknown and is extrapolated from N on the assumption that N mirrors changes in the size of the total population. Consequently, applying average values of p can result in misleading estimates of total breeding population. We recommend that annual counts of breeding individuals are supplemented with annual estimates of p. Field protocols that aim to estimate annual breeding population size from counts of breeding individuals should be complemented by independent measures of rates of re‐breeding and nest failure.     

Spatially explicit population responses of crayfish Procambarus alleni to potential shifts in vegetation distribution in the marl marshes of Everglades National Park,USA

Hydropattern disturbance has had wide-ranging impacts on wetland communities of the Florida Everglades, especially on the habitats and the aquatic biota of the seasonally flooded marl marshes. We used the Everglades crayfish Procambarus alleni as a model to study the associations among hydrology, vegetation distribution, and population dynamics to assess the potential impacts of hydrological changes on the aquatic faunal community in Everglades National Park. To classify benthic habitats as sources or sinks for the crayfish population, we quantified vegetation community structure using GIS maps in which dominant vegetation types were weighted by local hydroperiod (length of inundation). Regression analysis showed that this habitat classification was associated with crayfish density distribution. We then used a spatially explicit, stage-structured population model to describe crayfish population fluctuations under current environmental conditions and to simulate the potential population-level responses to habitat changes that might occur following hydrological restoration. In habitat that was initially saturated with crayfish, the crayfish population size declined under current environmental conditions and then stabilized at about 13% of the initial density over a 50-year period. A 4-month increase in hydroperiod was then simulated by converting shorter-hydroperiod Muhlenbergia-dominated marsh habitat to longer-hydroperiod Cladium-dominated marshes. The model predicted a rapid 7-fold increase in crayfish density following the simulated habitat restoration. This indicated that several functional effects may result from the restoration of historical hydropatterns in marl marshes: (1) the areal extent of habitat sinks will be reduced to isolated patches, whereas the spatial distribution of aquatic source habitats will expand; (2) crayfish population size will increase and persist over time; (3) the minimum threshold needed to increase secondary aquatic productivity may be a 7-month hydroperiod over 90% of the marl marsh landscape. Restoration of historical hydropatterns could thus have cascading positive effects throughout the Everglades aquatic food web.     

Evaluating methods to visualize patterns of genetic differentiation on a landscape

With advances in sequencing technology, research in the field of landscape genetics can now be conducted at unprecedented spatial and genomic scales. This has been especially evident when using sequence data to visualize patterns of genetic differentiation across a landscape due to demographic history, including changes in migration. Two recent model‐based visualization methods that can highlight unusual patterns of genetic differentiation across a landscape, SpaceMix and EEMS, are increasingly used. While SpaceMix's model can infer long‐distance migration, EEMS’ model is more sensitive to short‐distance changes in genetic differentiation, and it is unclear how these differences may affect their results in various situations. Here, we compare SpaceMix and EEMS side by side using landscape genetics simulations representing different migration scenarios. While both methods excel when patterns of simulated migration closely match their underlying models, they can produce either un‐intuitive or misleading results when the simulated migration patterns match their models less well, and this may be difficult to assess in empirical data sets. We also introduce unbundled principal components (un‐PC), a fast, model‐free method to visualize patterns of genetic differentiation by combining principal components analysis (PCA), which is already used in many landscape genetics studies, with the locations of sampled individuals. Un‐PC has characteristics of both SpaceMix and EEMS and works well with simulated and empirical data. Finally, we introduce msLandscape, a collection of tools that streamline the creation of customizable landscape‐scale simulations using the popular coalescent simulator ms and conversion of the simulated data for use with un‐PC, SpaceMix and EEMS.     

Patterns and processes of population dynamics with fluctuating habitat size: a case study of a marine copepod inhabiting tide pools

The logistic model is a fundamental population model often used as the basis for analyzing wildlife population dynamics. In the classic logistic model, however, population dynamics may be difficult to characterize if habitat size is temporally variable because population density can vary at a constant abundance, which results in variable strength of density‐dependent feedback for a given population size. To incorporate habitat size variability, we developed a general population model in which changes in population abundance, density, and habitat size are taken into account. From this model, we deduced several predictions for patterns and processes of population dynamics: 1) patterns of fluctuation in population abundance and density can diverge, with respect of their correlation and relative variability; and 2) along with density dependence, habitat size fluctuation can affect population growth with a time lag because changes in habitat size result in changes in population density. In order to test these predictions, we applied our model to population dynamics data of 36 populations of Tigriopus japonicus, a marine copepod inhabiting tide pools of variable sizes caused by weather processes. As expected, we found a significant difference in the fluctuation patterns of population abundance and density of T. japonicus populations with respect to the correlation between abundance and density and their relative variability, which correlates positively with the variability of habitat size. In addition, we found direct and lagged‐indirect effects of weather processes on population growth, which were associated with density dependence and impose regulatory forces on local and regional population dynamics. These results illustrate how changes in habitat size can have an impact on patterns and processes of wildlife population dynamics. We suggest that without knowledge of habitat size fluctuation, measures of population size and its variability as well as inferences about the processes of population dynamics may be misleading.     

Evaluating methods for estimating local effective population size with and without migration

Effective population size is a fundamental parameter in population genetics, evolutionary biology, and conservation biology, yet its estimation can be fraught with difficulties. Several methods to estimate N_e from genetic data have been developed that take advantage of various approaches for inferring N_e. The ability of these methods to accurately estimate N_e, however, has not been comprehensively examined. In this study, we employ seven of the most cited methods for estimating N_e from genetic data (Colony2, CoNe, Estim, MLNe, ONeSAMP, TMVP, and NeEstimator including LDNe) across simulated datasets with populations experiencing migration or no migration. The simulated population demographies are an isolated population with no immigration, an island model metapopulation with a sink population receiving immigrants, and an isolation by distance stepping stone model of populations. We find considerable variance in performance of these methods, both within and across demographic scenarios, with some methods performing very poorly. The most accurate estimates of N_e can be obtained by using LDNe, MLNe, or TMVP; however each of these approaches is outperformed by another in a differing demographic scenario. Knowledge of the approximate demography of population as well as the availability of temporal data largely improves N_e estimates.     

From snapshot information to long-term population dynamics of Acacias by a simulation model

The African Acacia species A. raddiana is believed to be endangered in the Negev desert of Israel. The ecology of this species is not well understood. The main idea of our study is to learn more about the long-term population dynamics of these trees using snapshot information in the form of size frequency distributions. These distributions are highly condensed indices of population dynamics acting over many years. In this paper, we analyse field data on recruitment, growth, and mortality and use an existing simulation model of the population dynamics of A. raddiana (SAM) to produce contrasting scenarios of these live history processes that are based on the analysed field evidence. The main properties of simulated as well as observed tree size frequency distributions are characterised with Simpson's index of dominance and a new permutation index. Finally, by running the SAM model under the different scenarios, we study the effect of these different processes on simulated size frequency distributions (pattern) and we compare them to size distributions observed in the field, in order to identify the processes acting in the field. Our study confirms rare recruitment events as a major factor shaping tree size frequency distributions and shows that the paucity of recruitment has been a normal feature of A. raddiana in the Negev over many years. Irregular growth, e.g., due to episodic rainfall, showed a moderate influence on size distributions. Finally, the size frequency distributions observed in the Negev reveal the information that, in this harsh environment, mortality of adult A. raddiana is independent of tree size (age).     

The role of trophic bottlenecks in stunting: a field test of an allocation model of growth and reproduction in yellow perch, Perca flavescens

Synopsis Stunting in freshwater fish populations may be due to low availability of one or more prey components within their diet. If the limiting prey constitute a trophic level (i.e. zooplankton, benthos, or fish), we define the phenomenon as a trophic bottleneck. Growth of a non-stunted population of yellow perch, Perca flavescens, was simulated using an allocation model based on reported ontogenetic shifts in diet (planktivory to benthivory to piscivory). The model was then perturbed by limiting the available ration individually for each of the three feeding types. The resulting simulated growth curves all reflected lower growth rates than the unperturbed state and the shape of the curve differed between the limited food types. Only the reduced benthic ration produced a simulated growth that matched that observed for the stunted yellow perch of Lac Hertel (Québec). To test the hypothesis that benthic ration was limiting growth in Lac Hertel, all the fish species in the lake were sampled for diet composition and total length at age. We predicted that species feeding upon benthic invertebrates should exhibit the slow growth characteristic of stunting (based on the results of the allocation model). Of the seven fish species found in Lac Hertel, four were stunted: yellow perch, pumpkinseed, Lepomis gibbosus, rock bass, Ambloplites rupestris, and brown bullhead, Ictalurus nebulosus. Three species were non-stunted: northern pike, Esox lucius, golden shiner, Notemigonus crysoleucas, and white sucker, Catostomus commersoni. All stunted species fed on benthic invertebrates and all non-stunted species, except the white sucker, did not feed upon benthos. The prediction of the allocation model was thus supported.     

The use of agent-based modelling of genetics in conservation genetics studies

Animal Landscape and Man Simulation System a genetically explicit agent-based model was used to obtain measures for the genetic and demographic status of simulated populations. This investigation aimed to test the applicability of this approach for assessing the effect of environmental perturbations on populations’ temporal and spatial dynamics. This was achieved by assessing how three simple scenarios with increasing degree of environmental disturbance, simulated by populations bottlenecks repeated at different intervals, affected the genetic and demographic characteristics of the simulated population. Model outputs from a simplified landscape scenario concurred with theoretical expectations validating the model in a qualitative way. Differences in medians, means and coefficient of variation of the observed (H_o) and expected heterozygosity (H_e), population census size (N), effective population size (N_e), inbreeding coefficient (F) and N_e/N ratio were observed for simulated populations. Impacts occurred rapidly after simulated bottleneck events and genetic estimates were less variable, and therefore more reliable, than demographic estimates. Precise genetic consequences of the bottlenecks repeated at different intervals, and resulting population perturbations, are a complex balance between effects on population sub-structure, size and founding events. Agent-based models are appropriate tools to simulate these interactions, being sufficiently flexible to mimic real population processes under a range of environmental conditions. Such models incorporating explicit genetics provide a promising new approach to evaluate the impact of environmental changes on genetic composition of populations.     

Limits to genetic bottlenecks and founder events imposed by the Allee effect

The Allee effect can result in a negative population growth rate at low population density. Consequently, populations below a minimum (critical) density are unlikely to persist. A lower limit on population size should constrain the loss of genetic variability due to genetic drift during population bottlenecks or founder events. We explored this phenomenon by modeling changes in genetic variability and differentiation during simulated bottlenecks of the alpine copepod, Hesperodiaptomus shoshone. Lake surveys, whole-lake re-introduction experiments and model calculations all indicate that H. shoshone should be unlikely to establish or persist at densities less than 0.5–5 individuals m⁻³. We estimated the corresponding range in minimum effective population size using the distribution of habitat (lake) sizes in nature and used these values to model the expected heterozygosity, allelic richness and genetic differentiation resulting from population bottlenecks. We found that during realistic bottlenecks or founder events, >90% of H. shoshone populations in the Sierra Nevada may be resistant to significant changes in heterozygosity or genetic distance, and 70–75% of populations may lose <10% of allelic richness. We suggest that ecological constraints on minimum population size be considered when using genetic markers to estimate historical population dynamics.     

A comparison between computer modeled bioaerosol dispersion and a bioaerosol field spray event

The observed deposition pattern from a field spray ofBacillus subtilus var.niger spores is compared with that of a computer simulated bioaerosol particle dispersion model. Using the same meteorological conditions as the field spray, the model produced a bioaerosol deposition pattern estimated to be reasonably similar (R ²=0.66) to the observed field pattern. Reasons for the differences between the deposition patterns are discussed. The comparison indicates that viable airborne particle deposition models may, with future testing, be useful tools for predicting near source aerial microbial dispersal and deposition. Mention of trade names or commercial products does not constitute endorsement or recommendation for use.     

Common pattern of population decline for freshwater cetacean species in deteriorating habitats

1. Freshwater cetacean species, including the baiji (Lipotes vexillifer), Amazon River dolphin (Inia geoffrensis), Ganges/Indus River dolphins (Platanista spp.) and Yangtze finless porpoise (Neophocaena asiaeorientalis asiaeorientalis), apex predators in megariver ecosystems, face serious challenges owing to the deterioration of habitat quality. 2. We simulated population change of four freshwater cetacean species under increasing habitat deterioration. Carrying capacity (K) was used to represent the habitat quality, and a logistic model was used to describe the rate of habitat deterioration (dK). 3. An individual‐based Leslie matrix model showed that population declines and extinctions in freshwater cetaceans under increasing habitat deterioration exhibit a consistent pattern irrespective of the initial level of K or population size. When dK is low, population abundance fluctuates stochastically around initial K, but a rapid increase in dK is accompanied by a sharp population decline, with a residual population ultimately declining continuously to extinction. 4. Simulations show that traditional census survey techniques used in cetacean species are unlikely to detect early signs of population decline before a critical level is reached. 5. Empirical data of the likely extinction of baiji strongly agree with our simulation exercise, implying that extinction of other freshwater cetacean species may occur sooner than previously considered. Hence, precautionary approaches for habitat restoration and landscape management should be implemented before freshwater cetacean population declines are detected, and ideally, before habitat quality begins to deteriorate.     

Regulation of differentiation in a population of cells interacting through a common pool

We consider a model of a suspension of a cell population in a well-mixed medium. There are two chemical substances, say A and H, reacting in each cell of the population and the substance H can only diffuse from the inside of cell to the medium or vice versa across the cell membrane. The medium is well mixed that the concentration of H is kept uniform over the medium. Cells interact indirectly with each other through the medium. The differential equations governing the dynamics of the suspension are analyzed using standard techniques for differential equations. It is shown that the cell population divides into several groups in respect of the chemical concentrations as time elapses. It is also shown how the fraction of the number of cells belonging to each subgroup to the total number of cells is regulated. The results may be used to explain the mechanism for differentiation of multi-cellular organisms.     

Simulating northern bobwhite population responses to nest predation,nesting habitat,and weather in south Texas

Nest predation is thought to be one of the major factors limiting northern bobwhite (Colinus virginianus) populations. We examined the relative impact of altering nest-predation rate, nesting habitat, and weather (i.e., temp and precipitation) on northern bobwhite population dynamics in a hypothetical 15,000-ha subtropical-rangeland ecosystem in south Texas using a simulation model. The systems model consisted of a 3-stage (i.e., eggs, juv, and ad) bobwhite population with dynamics influenced by variables affecting production, recruitment, nest predation, and mortality. We based model parameters on data collected from a 3-yr nest-predator study employing infrared-camera technology, from ongoing field research using a radio-marked population of wild bobwhites, and from the literature. The baseline simulated bobwhite population dynamics corresponded closely to empirical data, with no difference between medians of simulated (n = 30 yr) and observed bobwhite age ratios over a 28-yr period. Similarly, a time-series comparison of simulated and observed age ratios showed most (89%) observed values fell within the 5th and 95th percentiles of the simulated data over the 28-yr period. We created simulated population scenarios representing 1) baseline historical conditions, 2) predator control, 3) low precipitation, 4) low precipitation with predator control, 5) high temperature, 6) high temperature with predator control, 7) reduced nest-clump availability, and 8) reduced nest-clump availability with predator control that resulted in considerably different median bobwhite densities over 10 yr. For example, under simulated predator control, populations increased by about 55% from the baseline scenario, whereas under simulated reduced nest-clump availability, populations decreased by about 75% from the baseline scenario. Comparisons of time-series for each scenario showed that reduced nest-clump availability, low precipitation, and high temperature reduced bobwhite densities to a larger degree compared to a natural nest predation rate. Reduced nest-clump availability resulted in the most substantial decline of simulated bobwhite densities. Simulations suggested that management efforts should focus on maintaining adequate nest-clump availability and then possibly consider nest predator control as a secondary priority. © 2010 The Wildlife Society     

Choice of model and re-nesting probability function influences behaviour of avian seasonal productivity models and their demographic predictions

Measuring seasonal productivity is difficult in multi-brooded species without labour-intensive ringing studies. Individual-based (IB) models have been used to estimate seasonal productivity with no direct knowledge of number of nesting attempts, but they are often based on simplified re-nesting probability (φ_R) step-functions instead of observed or more biologically plausible ones. We present a new, open-source IB seasonal productivity model parameterized from studies of Black Redstart Phoenicurus ochruros and Yellowhammer Emberiza citrinella. We examined how the φ_R function shape (empirical versus simplified) influenced (1) model performance, (2) re-nesting compensation and (3) population-level predictions of a simulated management intervention. Population-level predictions were made only for Yellowhammer as we had more detailed demographic data, such as survival rates, available. Pattern-oriented modelling revealed that IB models produced realistic within-population distributions of breeding parameters, and those specified with an observed or empirically derived φ_R function generally outperformed those specified with simpler step functions. Strength of re-nesting compensation differed depending on the φ_R function used. For Yellowhammers, type of φ_R function in IB models marginally influenced population-level predictions of a simulated management intervention (potential population growth rate increased between 23% and 29% relative to no management intervention). In contrast, a simple deterministic productivity model, which did not simulate re-nesting compensation, predicted a 41% increase in potential population growth. At a population level, choice of φ_R function may have less influence on IB model predictions, but choice of model itself (IB versus deterministic) may have substantial impact. We discuss how more biologically plausible φ_R functions might either be observed directly, derived from nest data, or estimated from proxy information such as moult or brood patch changes.     

Effect of summer weather on internal loading and chlorophyll a in a shallow lake: a modeling approach

The effect of weather on the eutrophication of a shallow lake was estimated by a hydrodynamic lake model coupled with a simple water quality module. The model was applied to Lake Villikkalanjärvi in southern Finland. This shallow, agriculturally loaded lake may stratify during warm and calm periods in summer and as a result oxygen is often consumed from the hypolimnion, causing high internal loading of phosphorus. Vertical mixing and temperature distribution in the lake were simulated by a one-dimensional, horizontally integrated hydrodynamic model. State variables included in the water quality model were dissolved reactive phosphorus, chlorophyll a and dissolved oxygen. The model was first calibrated against observations from 1989 and 1990. Thereafter, simulations were carried out using weather data from the years 1961 to 1988. The results indicated that warm summer periods may cause high chlorophyll a concentrations due to high internal loading. In four years with exceptionally warm summers the model predicted maximum chlorophyll a concentrations almost twice as high as in years without remarkable internal loading. The model simulates accurately temperature and mixing but the reliability of water quality predictions could be improved by adding more factors regulating algal biomass and sediment phosphorus release.     

On parameter estimation in population models

We describe methods for estimating the parameters of Markovian population processes in continuous time, thus increasing their utility in modelling real biological systems. A general approach, applicable to any finite-state continuous-time Markovian model, is presented, and this is specialised to a computationally more efficient method applicable to a class of models called density-dependent Markov population processes. We illustrate the versatility of both approaches by estimating the parameters of the stochastic SIS logistic model from simulated data. This model is also fitted to data from a population of Bay checkerspot butterfly (Euphydryas editha bayensis), allowing us to assess the viability of this population.     

Sperm storage reduces the strength of the mate‐finding Allee effect

Mate searching is a key component of sexual reproduction that can have important implications for population viability, especially for the mate‐finding Allee effect. Interannual sperm storage by females may be an adaptation that potentially attenuates mate limitation, but the demographic consequences of this functional trait have not been studied. Our goal is to assess the effect of female sperm storage durability on the strength of the mate‐finding Allee effect and the viability of populations subject to low population density and habitat alteration. We used an individual‐based simulation model that incorporates realistic representations of the demographic and spatial processes of our model species, the spur‐thighed tortoise (Testudo graeca). This allowed for a detailed assessment of reproductive rates, population growth rates, and extinction probabilities. We also studied the relationship between the number of reproductive males and the reproductive rates for scenarios combining different levels of sperm storage durability, initial population density, and landscape alteration. Our results showed that simulated populations parameterized with the field‐observed demographic rates collapsed for short sperm storage durability, but were viable for a durability of one year or longer. In contrast, the simulated populations with a low initial density were only viable in human‐altered landscapes for sperm storage durability of 4 years. We find that sperm storage is an effective mechanism that can reduce the strength of the mate‐finding Allee effect and contribute to the persistence of low‐density populations. Our study highlights the key role of sperm storage in the dynamics of species with limited movement ability to facilitate reproduction in patchy landscapes or during population expansion. This study represents the first quantification of the effect of sperm storage durability on population dynamics in different landscapes and population scenarios.     

Inferring the origin of populations introduced from a genetically structured native range by approximate Bayesian computation: case study of the invasive ladybird Harmonia axyridis

Correct identification of the source population of an invasive species is a prerequisite for testing hypotheses concerning the factors responsible for biological invasions. The native area of invasive species may be large, poorly known and/or genetically structured. Because the actual source population may not have been sampled, studies based on molecular markers may generate incorrect conclusions about the origin of introduced populations. In this study, we characterized the genetic structure of the invasive ladybird Harmonia axyridis in its native area using various population genetic statistics and methods. We found that native area of H. axyridis most probably consisted of two geographically distinct genetic clusters located in eastern and western Asia. We then performed approximate Bayesian computation (ABC) analyses on controlled simulated microsatellite data sets to evaluate (i) the risk of selecting incorrect introduction scenarios, including admixture between sources, when the populations of the native area are genetically structured and sampling is incomplete and (ii) the ability of ABC analysis to minimize such risks by explicitly including unsampled populations in the scenarios compared. Finally, we performed additional ABC analyses on real microsatellite data sets to retrace the origin of biocontrol and invasive populations of H. axyridis, taking into account the possibility that the structured native area may have been incompletely sampled. We found that the invasive population in eastern North America, which has served as the bridgehead for worldwide invasion by H. axyridis, was probably formed by an admixture between the eastern and western native clusters. This admixture may have facilitated adaptation of the bridgehead population.