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1.
Double Allee Effects and Extinction in the Island Fox   总被引:3,自引:0,他引:3  
Abstract:  An Allee effect (AE) occurs in populations when individuals suffer a decrease in fitness at low densities. If a fitness component is reduced (component AE), per capita population growth rates may decline as a consequence (demographic AE) and extinction risk is increased. The island fox ( Urocyon littoralis ) is endemic to six of the eight California Channel Islands. Population crashes have coincided with an increase in predation by Golden Eagles ( Aquila chrysaetos ). We propose that AEs could render fox populations more sensitive and may be a likely explanation for their sharp decline. We analyzed demographic data collected between 1988 and 2000 to test whether fox density (1) influences survival and reproductive rates; (2) interacts with eagle presence and affects fox fitness parameters; and (3) influences per capita fox population trends. A double component AE simultaneously influenced survival (of adults and pups) and proportion of breeding adult females. The adult survival AE was driven by predation by eagles. These component AEs led to a demographic AE. Multiple-component AEs, a predation-driven AE, and the simultaneous occurrence of both component and demographic AEs in a mammal are all previously unreported processes. Populations below 7 foxes/km2 could have suboptimal population growth rates due to the demographic AE, and AEs may have contributed to the dramatic declines in three fox populations. Because fox densities in critically endangered populations are well below this level, removing Golden Eagles appears necessary to prevent a predation-driven AE. Conservationists should also be aware of AEs when planning the release of captive foxes. More generally, our findings highlight the danger of overlooking AEs in the conservation of populations of rare or threatened species.  相似文献   

2.
Kendall BE  Fox GA  Fujiwara M  Nogeire TM 《Ecology》2011,92(10):1985-1993
Demographic heterogeneity--variation among individuals in survival and reproduction--is ubiquitous in natural populations. Structured population models address heterogeneity due to age, size, or major developmental stages. However, other important sources of demographic heterogeneity, such as genetic variation, spatial heterogeneity in the environment, maternal effects, and differential exposure to stressors, are often not easily measured and hence are modeled as stochasticity. Recent research has elucidated the role of demographic heterogeneity in changing the magnitude of demographic stochasticity in small populations. Here we demonstrate a previously unrecognized effect: heterogeneous survival in long-lived species can increase the long-term growth rate in populations of any size. We illustrate this result using simple models in which each individual's annual survival rate is independent of age but survival may differ among individuals within a cohort. Similar models, but with nonoverlapping generations, have been extensively studied by demographers, who showed that, because the more "frail" individuals are more likely to die at a young age, the average survival rate of the cohort increases with age. Within ecology and evolution, this phenomenon of "cohort selection" is increasingly appreciated as a confounding factor in studies of senescence. We show that, when placed in a population model with overlapping generations, this heterogeneity also causes the asymptotic population growth rate lambda to increase, relative to a homogeneous population with the same mean survival rate at birth. The increase occurs because, even integrating over all the cohorts in the population, the population becomes increasingly dominated by the more robust individuals. The growth rate increases monotonically with the variance in survival rates, and the effect can be substantial, easily doubling the growth rate of slow-growing populations. Correlations between parent and offspring phenotype change the magnitude of the increase in lambda, but the increase occurs even for negative parent-offspring correlations. The effect of heterogeneity in reproductive rate on lambda is quite different: growth rate increases with reproductive heterogeneity for positive parent-offspring correlation but decreases for negative parent-offspring correlation. These effects of demographic heterogeneity on lambda have important implications for population dynamics, population viability analysis, and evolution.  相似文献   

3.
Savannas commonly consist of a discontinuous cover of overstory trees and a groundcover of grasses. Savanna models have previously demonstrated that vegetation feedbacks on fire frequency can limit the density of overstory trees, thereby maintaining savannas. Positive feedbacks of either savanna trees alone or trees and grasses together on fire frequency have been shown to result in a stable savanna equilibrium. Grass feedbacks on fire frequency, in contrast, have resulted in stable equilibria in either a grassland or forest state, but not in a savanna. These results, however, were derived from a system of differential equations that assumes that fire occurrence is strictly deterministic and that vegetation losses due to fire are continuous in time. We develop an alternative formulation of the grass-fire feedback model that assumes that fires are discrete and occur stochastically in time to examine the influence of these assumptions on the predicted state of the system. We show that incorporating fire as a discrete event can produce a recurring temporal refuge in which both grass and trees co-occur in a stable, bounded savanna. In our model, tree abundance is limited without invoking demographic bottlenecks in the transition from fire-sensitive to fire-resistant life history stages. An increasing strength of grass feedback on fire results in regular, predictable fires, which suggests that the system can also be modeled using a set of difference equations. We implement this discrete system using modified Leslie/Gower difference equations and demonstrate the existence of a bounded savanna state in this model framework. Our results confirm the potential for grass feedbacks to result in stable savannas, and indicate the importance of modeling fire as a discrete event rather than as a loss rate that is continuous in time.  相似文献   

4.
A central challenge in ecology is to understand the interplay of internal and external controls on the growth of populations. We examined the effects of temporal variation in weather and spatial variation in vegetation on the strength of density dependence in populations of large herbivores. We fit three subsets of the model ln(Nt) = a + (1 + b) x ln(N(t-1)) + c x ln(N(t-2)) to five time series of estimates (Nt) of abundance of ungulates in the Rocky Mountains, USA. The strength of density dependence was estimated by the magnitude of the coefficient b. We regressed the estimates of b on indices of temporal heterogeneity in weather and spatial heterogeneity in resources. The 95% posterior intervals of the slopes of these regressions showed that temporal heterogeneity strengthened density-dependent feedbacks to population growth, whereas spatial heterogeneity weakened them. This finding offers the first empirical evidence that density dependence responds in different ways to spatial heterogeneity and temporal heterogeneity.  相似文献   

5.
Effective population size (N(e)) determines the strength of genetic drift and can influence the level of genetic diversity a population can maintain. Assessing how changes in demographic rates associated with environmental variables and management actions affect N(e) thus can be crucial to the conservation of endangered species. Calculation of N(e) through demographic models makes it possible to use elasticity analyses to study this issue. The elasticity of N(e) to a given vital rate is the proportional change in N(e) associated with a proportional increase in that vital rate. In addition, demographic models can be used to study N(e) and population growth rate (λ) simultaneously. Simultaneous examination is important because some vital rates differ diametrically in their associations with λ and N(e). For example, in some cases increasing these vital rates increases λ and decreases N(e). We used elasticity analysis to study the effect of stage-specific survival and flowering rates on N(e), annual effective population size (N(a)), and λ in seven populations of the endangered plant Austrian dragonhead (Dracocephalum austriacum). In populations with λ ≥ 1, the elasticities of N(e) and N(a) were similar to those of λ. Survival rates of adults were associated with greater elasticities than survival rates of juveniles, flowering rates, or fecundity. In populations with λ < 1, N(e) and N(a) exhibited greater elasticities to juvenile than to adult vital rates. These patterns are similar to those observed in other species with similar life histories. We did not observe contrasting effects of any vital rate on λ and N(e); thus, management actions that increase the λ of populations of Austrian dragonhead will not increase genetic drift. Our results show that elasticity analyses of N(e) and N(a) can complement elasticity analysis of λ. Moreover, such analyses do not require more data than standard matrix models of population dynamics.  相似文献   

6.
Shefferson RP  Roach DA 《Ecology》2012,93(4):793-802
The theory of evolution via natural selection predicts that the genetic composition of wild populations changes over time in response to the environment. Different genotypes should exhibit different demographic patterns, but genetic variation in demography is often impossible to separate from environmental variation. Here, we asked if genetic variation is important in determining demographic patterns. We answer this question using a long-term field experiment combined with general linear modeling of deterministic population growth rates (lambda), deterministic life table response experiment (LTRE) analysis, and stochastic simulation of demography by paternal lineage in a short-lived perennial plant, Plantago lanceolata, in which we replicated genotypes across four cohorts using a standard breeding design. General linear modeling showed that growth rate varied significantly with year, spatial block, and sire. In LTRE analysis of all cohorts, the strongest influences on growth rate were from year x spatial block, and cohort x year x spatial block interactions. In analysis of genetics vs. temporal environmental variation, the strongest impacts on growth rate were from year and year x sire. Finally, stochastic simulation suggested different genetic composition among cohorts after 100 years, and different population growth rates when genetic differences were accounted for than when they were not. We argue that genetic variation, genotype x environment interactions, natural selection, and cohort effects should be better integrated into population ecological studies, as these processes should result in deviations from projected deterministic and stochastic population parameters.  相似文献   

7.
Plant-soil feedbacks have been implicated in several successful plant invasions. However, simple identification of a feedback alone may not be enough to establish feedbacks as a mechanism behind plant invasion. I suggest that the relationship between soil community density and plant growth is an important unknown that strongly influences the impact of plant-soil feedbacks. I developed a mathematical model of two-plant species competition with plant-soil feedbacks. Each plant species obligately generates its own soil community. Each soil community then influences both plant species’ growth. The model allows for every possible combination of positive and negative effects of the soil community on plant growth. I model the relationship between soil community density and plant growth with non-linear functional responses. I use a range of plant competitive abilities and feedback scenarios from the literature to explore how different functional responses influence the outcome of plant competition. Sensitivity analysis of the model reveals that altering the relationship between feedback strength and soil community development can reverse the outcome of plant competition. Analysis of the model also shows how the importance of different feedback scenarios depends on the strength of plant competition.  相似文献   

8.
Abstract: Effective conservation metrics are needed to evaluate the success of management in a rapidly changing world. Reproductive rates and densities of breeding birds (as a surrogate for reproductive rate) have been used to indicate the quality of avian breeding habitat, but the underlying assumptions of these metrics rarely have been examined. When birds are attracted to breeding areas in part by the presence of conspecifics and when breeding in groups influences predation rates, the effectiveness of density and reproductive rate as indicators of habitat quality is reduced. It is beneficial to clearly distinguish between individual‐ and population‐level processes when evaluating habitat quality. We use the term reproductive rate to refer to both levels and further distinguish among levels by using the terms per capita fecundity (number of female offspring per female per year, individual level) and population growth rate (the product of density and per capita fecundity, population level). We predicted how density and reproductive rate interact over time under density‐independent and density‐dependent scenarios, assuming the ideal free distribution model of how birds settle in breeding habitats. We predicted population density of small populations would be correlated positively with both per capita fecundity and population growth rate due to the Allee effect. For populations in the density‐dependent growth phase, we predicted no relation between density and per capita fecundity (because individuals in all patches will equilibrate to the same success rate) and a positive relation between density and population growth rate. Several ecological theories collectively suggest that positive correlations between density and per capita fecundity would be difficult to detect. We constructed a decision tree to guide interpretation of positive, neutral, nonlinear, and negative relations between density and reproductive rates at individual and population levels.  相似文献   

9.
Demographic Side Effects of Selective Hunting in Ungulates and Carnivores   总被引:2,自引:0,他引:2  
Abstract:  Selective harvesting regimes are often implemented because age and sex classes contribute differently to population dynamics and hunters show preferences associated with body size and trophy value. We reviewed the literature on how such cropping regimes affect the demography of the remaining population (here termed demographic side effects ). First, we examined the implications of removing a large proportion of a specific age or sex class. Such harvesting strategies often bias the population sex ratio toward females and reduce the mean age of males, which may consequently delay birth dates, reduce birth synchrony, delay body mass development, and alter offspring sex ratios. Second, we reviewed the side effects associated with the selective removal of relatively few specific individuals, often large trophy males. Such selective harvesting can destabilize social structures and the dominance hierarchy and may cause loss of social knowledge, sexually selected infanticide, habitat changes among reproductive females, and changes in offspring sex ratio. A common feature of many of the reported mechanisms is that they ultimately depress recruitment and in some extreme cases even cause total reproductive collapse. These effects could act additively and destabilize the dynamics of populations, thus having a stronger effect on population growth rate than first anticipated. Although more experimental than observational studies reported demographic side effects, we argue that this may reflect the quite subtle mechanisms involved, which are unlikely to be detected in observational studies without rigorous monitoring regimes. We call for more detailed studies of hunted populations with marked individuals that address how the expression of these effects varies across mating systems, habitats, and with population density. Theoretical models investigating how strongly these effects influence population growth rates are also required.  相似文献   

10.
11.
Population viability analysis programs are being used increasingly in research and management applications, but there has not been a systematic study of the congruence of different program predictions based on a single data set. We performed such an analysis using four population viability analysis computer programs: GAPPS, INMAT, RAMAS/AGE, and VORTEX. The standardized demographic rates used in all programs were generalized from hypothetical increasing and decreasing grizzly bear ( Ursus arctos horribilis ) populations. Idiosyncracies of input format for each program led to minor differences in intrinsic growth rates that translated into striking differences in estimates of extinction rates and expected population size. In contrast, the addition of demographic stochasticity, environmental stochasticity, and inbreeding costs caused only a small divergence in viability predictions. But, the addition of density dependence caused large deviations between the programs despite our best attempts to use the same density-dependent functions. Population viability programs differ in how density dependence is incorporated, and the necessary functions are difficult to parameterize accurately. Thus, we recommend that unless data clearly suggest a particular density-dependent model, predictions based on population viability analysis should include at least one scenario without density dependence. Further, we describe output metrics that may differ between programs; development of future software could benefit from standardized input and output formats across different programs.  相似文献   

12.
Two contrasting approaches to the analysis of population dynamics are currently popular: demographic approaches where the associations between demographic rates and statistics summarizing the population dynamics are identified; and time series approaches where the associations between population dynamics, population density, and environmental covariates are investigated. In this paper, we develop an approach to combine these methods and apply it to detailed data from Soay sheep (Ovis aries). We examine how density dependence and climate contribute to fluctuations in population size via age- and sex-specific demographic rates, and how fluctuations in demographic structure influence population dynamics. Density dependence contributes most, followed by climatic variation, age structure fluctuations and interactions between density and climate. We then simplify the density-dependent, stochastic, age-structured demographic model and derive a new phenomenological time series which captures the dynamics better than previously selected functions. The simple method we develop has potential to provide substantial insight into the relative contributions of population and individual-level processes to the dynamics of populations in stochastic environments.  相似文献   

13.
Abstract:  Although land preservation and promotion of successful regeneration are important conservation actions, their ability to increase population growth rates of slow-growing, long-lived trees is limited. We investigated the demography of Taxus floridana Nutt., a rare understory conifer, in three populations in different ravine forests spanning its entire geographic range along the Apalachicola River Bluffs in northern Florida (U.S.A.). We examined spatial and temporal patterns in demographic parameters and projected population growth rates by using four years of data on the recruitment and survival of seedlings and established stems, and on diameter growth from cross-sections of dead stems. All populations experienced a roughly 10-fold increase in seedling recruitment in 1996 compared with other years. The fates of seedlings and stems between 8 and 16 mm differed among populations. The fates of stems in two other size classes (the 2- to 4-mm class and the 4- to 8-mm class) differed among both populations and years. Individual stems in all populations exhibited similarly slow growth rates. Stochastic matrix models projected declines in all populations. Stochastic matrix analysis revealed the high elasticity of a measure of stochastic population growth rate to perturbations in the stasis of large reproductive stems for all populations. Additional analyses also indicated that occasional episodes of high recruitment do not greatly affect population growth rates. Conservation efforts directed at long-lived, slow-growing rare plants like Taxus floridana should both protect established reproductive individuals and further enhance survival of individuals in other life-history stages, such as juveniles, that often do not appear to contribute greatly to population growth rates.  相似文献   

14.
A pressing need exists to develop new approaches for obtaining information on demographic rates without causing further threats to imperiled animal populations. In this paper, we illustrate and apply a data-fitting technique based on quadratic programming that uses stage-specific abundance data to estimate demographic rates and asymptotic population growth rates (lambda). We used data from seven breeding colonies of California sea lions (Zalophus californianus) in the Gulf of California, Mexico. Estimates of lambda were similar to those from previous studies relying on a diffusion approximation using trends in total abundance. On average, predicted abundances were within 24% of the observed value for the inverse estimation method and within 29% of the observed value for the diffusion approximation. Our results suggest that three of the seven populations are declining (lambda < 1), but as many as six may be at risk. Elasticity and sensitivity analyses suggest that population management in most sites should focus on the protection of adults, whose survival generally contributes the most to lambda. The quadratic programming approach is a promising noninvasive technique for estimating demographic rates and assessing the viability of populations of imperiled species.  相似文献   

15.
Abstract: The demographic impacts of harvesting nontimber forest products (NTFP) have been increasingly studied because of reports of potentially unsustainable harvest. Nevertheless, our understanding of how plant demographic response to harvest is altered by variation in ecological conditions, which is critical for developing realistic sustainable‐use plans, is limited. We built matrix population models to test whether and how variation in ecological conditions affects population responses to harvest. In particular, we examined the effect of bark and foliage harvest on the demography of populations of African mahogany (Khaya senegalensis) in two contrasting ecological regions of Benin, West Africa. K. senegalensis bark and foliage harvest significantly reduced its stochastic population growth rates, but ecological differences between regions had a greater effect on population growth rates than did harvest. The effect of harvest on population growth rates (Δλ) was slightly stronger in the moist than in the drier region. Life‐table response experiments revealed that the mechanism by which harvesting reduced λ differed between ecological regions. Lowered stasis (persistence) of larger life stages lead to a reduction in λ in the drier region, whereas lowered growth of all life stages lowered λ in moist region. Potential strategies to increase population growth rates should include decreasing the proportion of individuals harvested, promoting harvester‐owned plantations of African mahogany, and increasing survival and growth by promoting no‐fire zones in gallery forests. Our results show how population responses to harvest of NTFP may be altered by ecological differences across sites and emphasize the importance of monitoring populations over the climatic range in which they occur to develop more realistic recommendations for conservation.  相似文献   

16.
Yoo HJ 《Ecology》2006,87(3):634-647
In spatially heterogeneous systems, utilizing population models to integrate the effects of multiple population rates can yield powerful insights into the relative importance of the component rates. The relative importance of demographic rates and dispersal in shaping the distribution of the western tussock moth (Orgyia vetusta) among patches of its host plant was explored using stage-structured population models. Tussock moth dispersal occurs passively in first-instar larvae and is poor or absent in all other life stages. Spatial surveys suggested, however, that moth distribution is not well explained by passive dispersal; moth populations were greater on small patches and on isolated ones. Further analysis showed that several local demographic rates varied significantly with patch characteristics. Two mortality factors in particular may explain the observed patterns. First, crawler mortality both increased with patch size and was density-dependent. A single-patch difference equation model showed mortality related to patch size is strong enough to overcome the homogenizing effect of density dependence; greater equilibrium densities were predicted for smaller patches. Second, although three rates were found to vary with local patch density, only pupal parasitism by a chalcid wasp could potentially account for higher moth abundances on isolated patches. A spatially explicit simulation model of the multiple-patch system showed that spatial variation in pupal parasitism is indeed strong enough to generate such a pattern. These results demonstrate that habitat spatial structure can affect multiple population processes simultaneously, and even relatively low attack rates imposed on a reproductively valuable life stage of the host can have a dominant effect on population distribution among habitat patches.  相似文献   

17.
Goswami VR  Getz LL  Hostetler JA  Ozgul A  Oli MK 《Ecology》2011,92(8):1680-1690
Although ecologists have long recognized that certain mammalian species exhibit high-amplitude, often multiannual, fluctuations in abundance, their causes have remained poorly understood and the subject of intense debate. A key contention has been the relative role of density-dependent and density-independent processes in governing population dynamics. We applied capture-mark-recapture analysis to 25 years of monthly trapping data from a fluctuating prairie vole Microtus ochrogaster population in Illinois, USA, to estimate realized population growth rates and associated vital rates (survival and recruitment) and modeled them as a function of vole density and density-independent climatic variation. We also tested for phase dependence and seasonality in the effects of the above processes. Variation in the realized population growth rate was best explained by phase-specific changes in vole density lagged by one month and mean monthly temperatures with no time lags. The underlying vital rates, survival and recruitment, were influenced by the additive and interactive effects of phase, vole density, and mean monthly temperatures. Our results are consistent with the observation that large-scale population fluctuations are characterized by phase-specific changes in demographic and physiological characteristics. Our findings also support the growing realization that the interaction between climatic variables and density-dependent factors may be a widespread phenomenon, and they suggest that the direction and magnitude of such interactive effects may be phase specific. We conclude that density-dependent and density-independent climatic variables work in tandem during each phase of density fluctuations to drive the dynamics of fluctuating populations.  相似文献   

18.
Kramer AM  Sarnelle O  Knapp RA 《Ecology》2008,89(10):2760-2769
Understanding the dynamics of populations at low density and the role of Allee effects is a priority due to concern about the decline of rare species and interest in colonization/invasion dynamics. Despite well-developed theory and observational support, experimental examinations of the Allee effect in natural systems are rare, partly because of logistical difficulties associated with experiments at low population density. We took advantage of fish introduction and removal in alpine lakes to experimentally test for the Allee effect at the whole-ecosystem scale. The large copepod Hesperodiaptomus shoshone is often extirpated from the water column by fish and sometimes fails to recover following fish disappearance, despite the presence of a long-lived egg bank. Population growth rate of this dioecious species may be limited by mate encounter rate, such that below some critical density a colonizing population will fail to establish. We conducted a multi-lake experiment in which H. shoshone was stocked at densities that bracketed our hypothesized critical density of 0.5-5 copoepods/m3. Successful recovery by the copepod was observed only in the lake with the highest initial density (3 copepods/m3). Copepods stocked into small cages at 3000 copepods/m3 survived and reproduced at rates comparable to natural populations, confirming that the lakes were suitable habitat for this species. In support of mate limitation as the mechanism underlying recovery failure, we found a significant positive relationship between mating success and density across experimental and natural H. shoshone populations. Furthermore, a mesocosm experiment provided evidence of increased per capita population growth rate with increasing population density in another diaptomid species, Skistodiaptomus pallidus. Together, these lines of evidence support the importance of the Allee effect to population recovery of H. shoshone in the Sierra Nevada, and to diaptomid copepods in general.  相似文献   

19.
We used population models to explore the effects of the organochlorine contaminant p,p'-DDE and fluctuations in vole availability on the population dynamics of Burrowing Owls (Athene cunicularia). Previous work indicated an interaction between low biomass of voles in the diet and moderate levels of p,p'-DDE in Burrowing Owl eggs that led to reproductive impairment. We constructed periodic and stochastic matrix models that incorporated three vole population states observed in the field: average, peak, and crash years. We modeled varying frequencies of vole crash years and a range of impairment of owl demographic rates in vole crash years. Vole availability had a greater impact on owl population growth rate than did reproductive impairment if vole populations peaked and crashed frequently. However, this difference disappeared as the frequency of vole crash years declined to once per decade. Fecundity, the demographic rate most affected by p,p'-DDE, had less impact on population growth rate than adult or juvenile survival. A life table response experiment of time-invariant matrices for average, peak, and crash vole conditions showed that low population growth under vole crash conditions was due to low adult and juvenile survival rates, whereas the extremely high population growth under vole peak conditions was due to increased fecundity. Our results suggest that even simple models can provide useful insights into complex ecological interactions. This is particularly valuable when temporal or spatial scales preclude manipulative experimental work in the field or laboratory.  相似文献   

20.
Carnivores are widely hunted for both sport and population control, especially where they conflict with human interests. It is widely believed that sport hunting is effective in reducing carnivore populations and related human-carnivore conflicts, while maintaining viable populations. However, the way in which carnivore populations respond to harvest can vary greatly depending on their social structure, reproductive strategies, and dispersal patterns. For example, hunted cougar (Puma concolor) populations have shown a great degree of resiliency. Although hunting cougars on a broad geographic scale (> 2000 km2) has reduced densities, hunting of smaller areas (i.e., game management units, < 1000 km2), could conceivably fail because of increased immigration from adjacent source areas. We monitored a heavily hunted population from 2001 to 2006 to test for the effects of hunting at a small scale (< 1000 km2) and to gauge whether population control was achieved (lambda < or = 1.0) or if hunting losses were negated by increased immigration allowing the population to remain stable or increase (lambda > or = 1.0). The observed growth rate of 1.00 was significantly higher than our predicted survival/fecundity growth rates (using a Leslie matrix) of 0.89 (deterministic) and 0.84 (stochastic), with the difference representing an 11-16% annual immigration rate. We observed no decline in density of the total population or the adult population, but a significant decrease in the average age of independent males. We found that the male component of the population was increasing (observed male population growth rate, lambda(OM) = 1.09), masking a decrease in the female component (lambda(OF) = 0.91). Our data support the compensatory immigration sink hypothesis; cougar removal in small game management areas (< 1000 km2) increased immigration and recruitment of younger animals from adjacent areas, resulting in little or no reduction in local cougar densities and a shift in population structure toward younger animals. Hunting in high-quality habitats may create an attractive sink, leading to misinterpretation of population trends and masking population declines in the sink and surrounding source areas.  相似文献   

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