Minimum requirements for modelling bivalve carrying capacity

The concept of carrying capacity of an ecosystem fornatural populations is derived from the logisticgrowth curve in population ecology, and defined as themaximum standing stock that can be supported by agiven ecosystem for a given time. This definitionneeds to be modified for the exploitation ofecosystems. Carrying capacity for exploitation isdefined as the standing stock at which the annualproduction of a marketable cohort is maximized. Forbivalve suspension feeders, the dominant factordetermining the exploitation carrying capacity at theecosystem scale is primary production. At a localscale carrying capacity depends on physicalconstraints such as substrate, shelter and food supplyby tidal currents.We evaluate critically some existing models ofexploited ecosystems for shellfish cultivation inorder to formulate the minimum requirements of ageneric carrying capacity model. Generic models canbe developed for both the ecosystem scale and thelocal scale, depending on the aim of the modelling.Transport processes, sediment dynamics and submodelsfor organism and population level processes areminimum requirements for carrying capacity modelling. This revised version was published online in August 2006 with corrections to the Cover Date.     

Impacts of Biotic Resource Enrichment on a Predator–Prey Population

The environmental carrying capacity is usually assumed to be fixed quantity in the classical predator–prey population growth models. However, this assumption is not realistic as the environment generally varies with time. In a bid for greater realism, functional forms of carrying capacities have been widely applied to describe varying environments. Modelling carrying capacity as a state variable serves as another approach to capture the dynamical behavior between population and its environment. The proposed modified predator–prey model is based on the ratio-dependent models that have been utilized in the study of food chains. Using a simple non-linear system, the proposed model can be linked to an intra-guild predation model in which predator and prey share the same resource. Distinct from other models, we formulate the carrying capacity proportional to a biotic resource and both predator and prey species can directly alter the amount of resource available by interacting with it. Bifurcation and numerical analyses are presented to illustrate the system’s dynamical behavior. Taking the enrichment parameter of the resource as the bifurcation parameter, a Hopf bifurcation is found for some parameter ranges, which generate solutions that posses limit cycle behavior.     

Estimating demographic models for the range dynamics of plant species

Aims To better understand how demographic processes shape the range dynamics of woody plants (in this case, Proteaceae), we introduce a likelihood framework for fitting process‐based models of range dynamics to spatial abundance data. Location The fire‐prone Fynbos biome (Cape Floristic Region, South Africa). Methods Our process‐based models have a spatially explicit demographic submodel (describing dispersal, reproduction, mortality and local extinction) as well as an observation submodel (describing imperfect detection of individuals), and are constrained by species‐specific predictions of habitat distribution models and process‐based models for seed dispersal by wind. Free model parameters were varied to find parameter sets with the highest likelihood. After testing this approach with simulated data, we applied it to eight Proteaceae species that differ in breeding system (monoecy versus dioecy) and adult fire survival. We assess the importance of Allee effects and negative density dependence for range dynamics, by using the Akaike information criterion to select between alternative models fitted for the same species. Results The best model for all dioecious study species included Allee effects, whereas this was true for only one of four monoecious species. As expected, sprouters (in which adults survive fire) were estimated to have lower rates of reproduction and catastrophic population extinction than related non‐sprouters. Overcompensatory population dynamics seem important for three of four non‐sprouters. We also found good quantitative agreement between independent data and most estimates of reproduction, carrying capacity and extinction probability. Main conclusions This study shows that process‐based models can quantitatively describe how large‐scale abundance distributions arise from the movement and interaction of individuals. It stresses links between the life history, demography and range dynamics of Proteaceae: dioecious species seem more susceptible to Allee effects which reduce migration ability and increase local extinction risk, and sprouters seem to have high persistence of established populations, but their low reproduction limits habitat colonization and migration.     

The impact of sea‐level rise on Snowy Plovers in Florida: integrating geomorphological,habitat, and metapopulation models

Sea‐level rise (SLR) is a projected consequence of global climate change that will result in complex changes in coastal ecosystems. These changes will cause transitions among coastal habitat types, which will be compounded by human‐made barriers to the gradual inland migration of these habitat types. The effect of these changes on the future viability of coastal species will depend on the habitat requirements and population dynamics of these species. Thus, realistic assessments of the impact of SLR require linking geomorphological models with habitat and population models. In this study, we implemented a framework that allows this linkage, and demonstrated its feasibility to assess the effect of SLR on the viability of the Snowy Plover population in Florida. The results indicate that SLR will cause a decline in suitable habitat and carrying capacity for this species, and an increase in the risk of its extinction and decline. The model projected that the population size will decline faster than the area of habitat or carrying capacity, demonstrating the necessity of incorporating population dynamics in assessing the impacts of SLR on coastal species. The results were most sensitive to uncertainties in survival rate and fecundity, and suggested that future studies on this species should focus on the average and variability of these demographic rates and their dependence on population density. The effect of SLR on this species’ viability was qualitatively similar with most alternative models that used the extreme values of each uncertain parameter, indicating that the results are robust to uncertainties in the model.     

A quantitative test of the size efficiency hypothesis by means of a physiologically structured model

According to the size‐efficiency hypothesis (SEH) larger bodied cladocerans are better competitors for food than small bodied species. In environments with fish, however, the higher losses of the large bodied species due to size‐selective predation may shift the balance in favor of the small bodied species. Here we present a theoretical framework for the analysis of the competitive abilities of zooplankton species that takes both competition and predation into account in one coherent analysis. By applying the conceptually well‐understood framework of physiologically structured population models we were able to predict the relative difference in predation rates necessary to cause a shift in dominance of the large‐bodied species (Daphnia pulicaria) to the small‐bodied species (D. galeata). These predictions depend only on seven easily interpretable parameters per species: size at birth, size at maturity and maximum size, age at maturity, maximal clutch size, egg development time and finally the half‐saturation constant for food. The critical equilibrium mortality of D. pulicaria was 0.16 d^?1 at food concentrations close to the critical food concentration of D. galeata, i.e. D. pulicaria will win the competition as long as its mortality rate is below 0.16 d^?1. At higher food concentrations the differential mortality curve (plotting equilibrium mortalities of both species against each other) approached a linear function with a slope of one and an intercept equal to the difference in maximal population birth rates. The prediction of critical predation rates was independent of the ingestion rate of the cladocerans and the algal carrying capacity and food regeneration rate of the environment although the mechanism works through competition for a shared algal food resource. We interpret these findings in terms of the relative predation risk large and small‐bodied cladocerans will face in various freshwater ecosystems.     

Relationships between survival and habitat suitability of semi‐aquatic mammals

Spatial distribution and habitat selection are integral to the study of animal ecology. Habitat selection may optimize the fitness of individuals. Hutchinsonian niche theory posits the fundamental niche of species would support the persistence or growth of populations. Although niche‐based species distribution models (SDMs) and habitat suitability models (HSMs) such as maximum entropy (Maxent) have demonstrated fair to excellent predictive power, few studies have linked the prediction of HSMs to demographic rates. We aimed to test the prediction of Hutchinsonian niche theory that habitat suitability (i.e., likelihood of occurrence) would be positively related to survival of American beaver (Castor canadensis), a North American semi‐aquatic, herbivorous, habitat generalist. We also tested the prediction of ideal free distribution that animal fitness, or its surrogate, is independent of habitat suitability at the equilibrium. We estimated beaver monthly survival probability using the Barker model and radio telemetry data collected in northern Alabama, United States from January 2011 to April 2012. A habitat suitability map was generated with Maxent for the entire study site using landscape variables derived from the 2011 National Land Cover Database (30‐m resolution). We found an inverse relationship between habitat suitability index and beaver survival, contradicting the predictions of niche theory and ideal free distribution. Furthermore, four landscape variables selected by American beaver did not predict survival. The beaver population on our study site has been established for 20 or more years and, subsequently, may be approaching or have reached the carrying capacity. Maxent‐predicted increases in habitat use and subsequent intraspecific competition may have reduced beaver survival. Habitat suitability‐fitness relationships may be complex and, in part, contingent upon local animal abundance. Future studies of mechanistic SDMs incorporating local abundance and demographic rates are needed.     

Evaluating the “recovery level” of endangered species without prior information before alien invasion

For maintaining social and financial support for eradication programs of invasive species, quantitative assessment of recovery of native species or ecosystems is important because it provides a measurable parameter of success. However, setting a concrete goal for recovery is often difficult owing to lack of information prior to the introduction of invaders. Here, we present a novel approach to evaluate the achievement level of invasive predator management based on the carrying capacity of endangered species estimated using long‐term monitoring data. In Amami‐Oshima Island, Japan, where the eradication project of introduced small Indian mongoose is ongoing since 2000, we surveyed the population densities of four endangered species threatened by the mongoose (Amami rabbit, the Otton frog, Amami tip‐nosed frog, and Amami Ishikawa's frog) at four time points ranging from 2003 to 2011. We estimated the carrying capacities of these species using the logistic growth model combined with the effects of mongoose predation and environmental heterogeneity. All species showed clear tendencies toward increasing their density in line with decreased mongoose density, and they exhibited density‐dependent population growth. The estimated carrying capacities of three endangered species had small confidence intervals enough to measure recovery levels by the mongoose management. The population density of each endangered species has recovered to the level of the carrying capacity at about 20–40% of all sites, whereas no individuals were observed at more than 25% of all sites. We propose that the present approach involving appropriate monitoring data of native organism populations will be widely applicable to various eradication projects and provide unambiguous goals for management of invasive species.     

Stoichiometry and population dynamics

Population dynamics theory forms the quantitative core from which most ecologists have developed their intuition about how species interactions, heterogeneity, and biodiversity play out in time. Throughout its development, theoretical population biology has built on variants of the Lotka–Volterra equations and in nearly all cases has taken a single‐currency approach to understanding population change, abstracting populations as aggregations of individuals or biomass. In this review, we explore how depicting organisms as built of more than one thing (for example, C and an important nutrient, such as P) in stoichiometrically explicit models results in qualitatively different predictions about the resulting dynamics. Fundamentally, stoichiometric models incorporate both food quantity and food quality effects in a single framework, allow key feedbacks such as consumer‐driven nutrient recycling to occur, and generally appear to stabilize predator–prey systems while simultaneously producing rich dynamics with alternative domains of attraction and occasionally counterintuitive outcomes, such as coexistence of more than one predator species on a single‐prey item and decreased herbivore performance in response to increased light intensity experienced by the autotrophs. In addition to the theoretical background, we also review recent laboratory and field studies considering stoichiometric effects on autotroph–herbivore systems, emphasizing algae–Daphnia interactions. These studies support the predictions of stoichiometric theory, providing empirical evidence for alternative stable states under stoichiometric constraints, for negative effects of solar radiation on herbivores via stoichiometric food quality, and for diversity‐enhancing effects of poor food quality. Stoichiometric theory has strong potential for both quantitative and qualitative improvements in the predictive power of population ecology, a major priority in light of the multivariate anthropogenic and natural perturbations experienced by populations. However, full development and testing of stoichiometric population dynamics theory will require greater intellectual tolerance and exchange between researchers working in ecosystem and population ecology.     

Range Expansion and Population Dynamics of an Invasive Species: The Eurasian Collared-Dove (Streptopelia decaocto)

Invasive species offer ecologists the opportunity to study the factors governing species distributions and population growth. The Eurasian Collared-Dove (Streptopelia decaocto) serves as a model organism for invasive spread because of the wealth of abundance records and the recent development of the invasion. We tested whether a set of environmental variables were related to the carrying capacities and growth rates of individual populations by modeling the growth trajectories of individual populations of the Collared-Dove using Breeding Bird Survey (BBS) and Christmas Bird Count (CBC) data. Depending on the fit of our growth models, carrying capacity and growth rate parameters were extracted and modeled using historical, geographical, land cover and climatic predictors. Model averaging and individual variable importance weights were used to assess the strength of these predictors. The specific variables with the greatest support in our models differed between data sets, which may be the result of temporal and spatial differences between the BBS and CBC. However, our results indicate that both carrying capacity and population growth rates are related to developed land cover and temperature, while growth rates may also be influenced by dispersal patterns along the invasion front. Model averaged multivariate models explained 35–48% and 41–46% of the variation in carrying capacities and population growth rates, respectively. Our results suggest that widespread species invasions can be evaluated within a predictable population ecology framework. Land cover and climate both have important effects on population growth rates and carrying capacities of Collared-Dove populations. Efforts to model aspects of population growth of this invasive species were more successful than attempts to model static abundance patterns, pointing to a potentially fruitful avenue for the development of improved invasive distribution models.     

Geographical gradients in the population dynamics of North American prairie ducks

1. Geographic gradients in population dynamics may occur because of spatial variation in resources that affect the deterministic components of the dynamics (i.e. carrying capacity, the specific growth rate at small densities or the strength of density regulation) or because of spatial variation in the effects of environmental stochasticity. To evaluate these, we used a hierarchical Bayesian approach to estimate parameters characterizing deterministic components and stochastic influences on population dynamics of eight species of ducks (mallard, northern pintail, blue-winged teal, gadwall, northern shoveler, American wigeon, canvasback and redhead (Anas platyrhynchos, A. acuta, A. discors, A. strepera, A. clypeata, A. americana, Aythya valisineria and Ay. americana, respectively) breeding in the North American prairies, and then tested whether these parameters varied latitudinally. 2. We also examined the influence of temporal variation in the availability of wetlands, spring temperature and winter precipitation on population dynamics to determine whether geographical gradients in population dynamics were related to large-scale variation in environmental effects. Population variability, as measured by the variance of the population fluctuations around the carrying capacity K, decreased with latitude for all species except canvasback. This decrease in population variability was caused by a combination of latitudinal gradients in the strength of density dependence, carrying capacity and process variance, for which details varied by species. 3. The effects of environmental covariates on population dynamics also varied latitudinally, particularly for mallard, northern pintail and northern shoveler. However, the proportion of the process variance explained by environmental covariates, with the exception of mallard, tended to be small. 4. Thus, geographical gradients in population dynamics of prairie ducks resulted from latitudinal gradients in both deterministic and stochastic components, and were likely influenced by spatial differences in the distribution of wetland types and shapes, agricultural practices and dispersal processes. 5. These results suggest that future management of these species could be improved by implementing harvest models that account explicitly for spatial variation in density effects and environmental stochasticity on population abundance.     

The carrying capacity for species richness

The idea that the number of species within an area is limited by a specific capacity of that area to host species is old yet controversial. Here, we show that the concept of carrying capacity for species richness can be as useful as the analogous concept in population biology. Many lines of empirical evidence indicate the existence of limits of species richness, at least at large spatial and phylogenetic scales. However, available evidence does not support the idea of diversity limits based on limited niche space; instead, carrying capacity should be understood as a stable equilibrium of biodiversity dynamics driven by diversity‐dependent processes of extinction, speciation and/or colonization. We argue that such stable equilibria exist even if not all resources are used and if increasing species richness increases the ability of a community to use resources. Evaluating the various theoretical approaches to modelling diversity dynamics, we conclude that a fruitful approach for macroecology and biodiversity science is to develop theory that assumes that the key mechanism leading to stable diversity equilibria is the negative diversity dependence of per‐species extinction rates, driven by the fact that population sizes of species must decrease with an increasing number of species owing to limited energy availability. The recently proposed equilibrium theory of biodiversity dynamics is an example of such a theory, which predicts that equilibrium species richness (i.e., carrying capacity) is determined by the interplay of the total amount of available resources, the ability of communities to use those resources, environmental stability that affects extinction rates, and the factors that affect speciation and colonization rates. We argue that the diversity equilibria resulting from these biodiversity dynamics are first‐order drivers of large‐scale biodiversity patterns, such as the latitudinal diversity gradient.     

An evaluation of behavior inferences from Bayesian state‐space models: A case study with the Pacific walrus

State‐space models offer researchers an objective approach to modeling complex animal location data sets, and state‐space model behavior classifications are often assumed to have a link to animal behavior. In this study, we evaluated the behavioral classification accuracy of a Bayesian state‐space model in Pacific walruses using Argos satellite tags with sensors to detect animal behavior in real time. We fit a two‐state discrete‐time continuous‐space Bayesian state‐space model to data from 306 Pacific walruses tagged in the Chukchi Sea. We matched predicted locations and behaviors from the state‐space model (resident, transient behavior) to true animal behavior (foraging, swimming, hauled out) and evaluated classification accuracy with kappa statistics (κ) and root mean square error (RMSE). In addition, we compared biased random bridge utilization distributions generated with resident behavior locations to true foraging behavior locations to evaluate differences in space use patterns. Results indicated that the two‐state model fairly classified true animal behavior (0.06 ≤ κ ≤ 0.26, 0.49 ≤ RMSE ≤ 0.59). Kernel overlap metrics indicated utilization distributions generated with resident behavior locations were generally smaller than utilization distributions generated with true foraging behavior locations. Consequently, we encourage researchers to carefully examine parameters and priors associated with behaviors in state‐space models, and reconcile these parameters with the study species and its expected behaviors.     

京郊密云县农业生态系统蛋白人口承载力模型分析

农业生态系统蛋白人口承载力不仅是农业生态学系统生产力研究的重要方面,而且是制定我国人口政策和食物战略的重要依据。中央曾指出:要把我国人民的膳物结构问题作为战略问题来考虑。分析我国的膳食特点可概括为“一够两缺”(即热量够,蛋白质和脂肪缺)温饱型营养水平。食物蛋白质的改善是我国膳食结     

Flexibility in phenology and habitat use act as buffers to long‐term population declines in UK passerines

Ecological responses to environmental change are wide‐ranging, from alterations in the timing of life‐history events to range and population changes. Explaining the variation across species in these responses is essential for identifying vulnerable species and for developing adequate conservation or mitigation strategies. Using population trend data from the UK Breeding Bird Survey, this study examined the association between long‐term population trends (1994–2007) and phenological, life‐history and resource‐use traits of UK passerine species. Phenology, as well as productivity and resource use were significantly associated with long‐term population trends. Average laying date and first clutch laying period were key predictors, with higher population growth rates associated with earlier laying dates and longer laying periods. This suggests that flexibility in the duration of reproductive periods buffers species against environmental changes. Average laying period was particularly important for migrant species. Flexibility in laying dates for these species is constrained by their arrival dates; mean change in arrival date over a twenty‐five year period strongly predicted population trends amongst migrant species. Besides the key role phenological flexibility plays in buffering population declines, we also showed that more productive, generalist species were less likely to have declining populations than species with specialized habitat requirements, particularly those associated with farmland and urban areas and those reliant on highly seasonal food items (i.e. invertebrate eaters). These results underscore the need for a multi‐faceted approach to understanding the mechanisms governing population trends. Additionally, species’ sensitivity to environmental change is likely to depend on interactions between species‐specific phenology, habitat and resource‐use traits.     

The role of fish life histories in allometrically scaled food‐web dynamics

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Inferring dispersal: a Bayesian approach to phylogeny-based island biogeography, with special reference to the Canary Islands

Aim Oceanic islands represent a special challenge to historical biogeographers because dispersal is typically the dominant process while most existing methods are based on vicariance. Here, we describe a new Bayesian approach to island biogeography that estimates island carrying capacities and dispersal rates based on simple Markov models of biogeographical processes. This is done in the context of simultaneous analysis of phylogenetic and distributional data across groups, accommodating phylogenetic uncertainty and making parameter estimates more robust. We test our models on an empirical data set of published phylogenies of Canary Island organisms to examine overall dispersal rates and correlation of rates with explanatory factors such as geographic proximity and area size. Location Oceanic archipelagos with special reference to the Atlantic Canary Islands. Methods The Canary Islands were divided into three island‐groups, corresponding to the main magmatism periods in the formation of the archipelago, while non‐Canarian distributions were grouped into a fourth ‘mainland‐island’. Dispersal between island groups, which were assumed constant through time, was modelled as a homogeneous, time‐reversible Markov process, analogous to the standard models of DNA evolution. The stationary state frequencies in these models reflect the relative carrying capacity of the islands, while the exchangeability (rate) parameters reflect the relative dispersal rates between islands. We examined models of increasing complexity: Jukes–Cantor (JC), Equal‐in, and General Time Reversible (GTR), with or without the assumption of stepping‐stone dispersal. The data consisted of 13 Canarian phylogenies: 954 individuals representing 393 taxonomic (morphological) entities. Each group was allowed to evolve under its own DNA model, with the island‐model shared across groups. Posterior distributions on island model parameters were estimated using Markov Chain Monte Carlo (MCMC) sampling, as implemented in MrBayes 4.0, and Bayes Factors were used to compare models. Results The Equal‐in step, the GTR, and the GTR step dispersal models showed the best fit to the data. In the Equal‐in and GTR models, the largest carrying capacity was estimated for the mainland, followed by the central islands and the western islands, with the eastern islands having the smallest carrying capacity. The relative dispersal rate was highest between the central and eastern islands, and between the central and western islands. The exchange with the mainland was rare in comparison. Main conclusions Our results confirm those of earlier studies suggesting that inter‐island dispersal within the Canary Island archipelago has been more important in explaining diversification within lineages than dispersal between the continent and the islands, despite the close proximity to North Africa. The low carrying capacity of the eastern islands, uncorrelated with their size or age, fits well with the idea of a historically depauperate biota in these islands but more sophisticated models are needed to address the possible influence of major recent extinction events. The island models explored here can easily be extended to address other problems in historical biogeography, such as dispersal among areas in continental settings or reticulate area relationships.     

Buying time for wild animals with zoos

Zoos and aquariums exhibit many rare species, but sustain few for long periods. Demanding genetic, demographic, and behavioral requirements are a part of the sustainability challenge, and historical zoo goals and limiting animal management objectives are another, but they have been overtaken by worldwide wildlife population contraction and endangerment. New policies are essential for zoo continuance and, if vanishing species are to be helped by zoo propagation, they must be given priority. However, zoos have little animal carrying capacity and propagation must be much more sharply focused. In addition, it is becoming urgent that zoos help to support parks and reserves and, where possible, manage some especially endangered species mutually with parks.     

Can the past predict the future? Experimental tests of historically based population models

A frequently advocated approach for forecasting the population‐level impacts of climate change is to project models based on historical, observational relationships between climate and demographic rates. Despite the potential pitfalls of this approach, few historically based population models have been experimentally validated. We conducted a precipitation manipulation experiment to test population models fit to observational data collected from the 1930s to the 1970s for six prairie forb species. We used the historical population models to predict experimental responses to the precipitation manipulations, and compared these predictions to ones generated by a statistical model fit directly to the experimental data. For three species, a sensitivity analysis of the effects of precipitation and grass cover on forb population growth showed consistent results for the historical population models and the contemporary statistical models. Furthermore, the historical population models predicted population growth rates in the experimental plots as well or better than the statistical models, ignoring variation explained by spatial random effects and local density‐dependence. However, for the remaining three species, the sensitivity analyses showed that the historical and statistical models predicted opposite effects of precipitation on population growth, and the historical models were very poor predictors of experimental responses. For these species, historical observations were not well replicated in space, and for two of them the historical precipitation‐demography correlations were weak. Our results highlight the strengths and weaknesses of observational and experimental approaches, and increase our confidence in extrapolating historical relationships to predict population responses to climate change, at least when the historical correlations are strong and based on well‐replicated observations.     

Density‐dependent and density‐independent drivers of population change in Barton Springs salamanders

Understanding population change is essential for conservation of imperiled species, such as amphibians. Worldwide amphibian declines have provided an impetus for investigating their population dynamics, which can involve both extrinsic (density‐independent) and intrinsic (density‐dependent) drivers acting differentially across multiple life stages or age classes. In this study, we examined the population dynamics of the endangered Barton Springs Salamander (Eurycea sosorum) using data from a long‐term monitoring program. We were interested in understanding both the potential environmental drivers (density‐independent factors) and demographic factors (interactions among size classes, negative density dependence) to better inform conservation and management activities. We used data from three different monitoring regimes and multivariate autoregressive state‐space models to quantify environmental effects (seasonality, discharge, algae, and sediment cover), intraspecific interactions among three size classes, and intra‐class density dependence. Results from our primary data set revealed similar patterns among sites and size classes and were corroborated by our out‐of‐sample data. Cross‐correlation analysis showed juvenile abundance was most strongly correlated with a 9‐month lag in aquifer discharge, which we suspect is related to inputs of organic carbon into the aquifer. However, sedimentation limited juvenile abundance at the surface, emphasizing the importance of continued sediment management. Recruitment from juveniles to the sub‐adult size class was evident, but negative density‐dependent feedback ultimately regulated each size class. Negative density dependence may be an encouraging sign for the conservation of E. sosorum because populations that can reach carrying capacity are less likely to go extinct compared to unregulated populations far below their carrying capacity. However, periodic population declines coupled with apparent migration into the aquifer complicate assessments of species status. Although both density‐dependent and density‐independent drivers of population change are not always apparent in time series of animal populations, both have important implications for conservation and management of E. sosorum.     

Ecological character displacement in quantitative genetic models

We study, both analytically and numerically, models of ecological character displacement for two species that compete for the same set of food sources. These models include quantitative genetics and Lotka-Volterra type competition and are symmetric with respect to the two species. We allow for various shapes of the carrying capacity and the competition function, and we discuss under what general conditions large character displacement can occur. While some of these conditions, like genetic rigidity, or flat and truncated carrying capacity curves, were known before, we also find that slow dynamics of the genetic variance, steep slopes in the interaction function and carrying capacities that are not truncated can lead to large displacements. We interpret these conditions biologically and also give new insights into models which have been previously investigated.