When the exception becomes the rule: The disappearance of limiting similarity in the Lotka-Volterra model

We investigate the transition between limiting similarity and coexistence of a continuum in the competitive Lotka-Volterra model. It is known that there exist exceptional cases in which, contrary to the limiting similarity expectation, all phenotypes coexist along a trait axis. Earlier studies established that the distance between surviving phenotypes is in the magnitude of the niche width 2σ provided that the carrying capacity curve differs from the exceptional one significantly enough. In this paper we studied the outcome of competition for small perturbations of the exceptional (Gaussian) carrying capacity. We found that the average distance between the surviving phenotypes goes to zero when the perturbation vanishes. The number of coexisting species in equilibrium is proportional to the negative logarithm of the perturbation. Nevertheless, the niche width provides a good order of magnitude for the distance between survivors if the perturbations are larger than 10%. Therefore, we conclude that limiting similarity is a good framework of biological thinking despite the lack of an absolute lower bound of similarity.     

How Gaussian competition leads to lumpy or uniform species distributions

A central model in theoretical ecology considers the competition of a range of species for a broad spectrum of resources. Recent studies have shown that essentially two different outcomes are possible. Either the species surviving competition are more or less uniformly distributed over the resource spectrum, or their distribution is “lumped” (or “clumped”), consisting of clusters of species with similar resource use that are separated by gaps in resource space. Which of these outcomes will occur crucially depends on the competition kernel, which reflects the shape of the resource utilization pattern of the competing species. Most models considered in the literature assume a Gaussian competition kernel. This is unfortunate, since predictions based on such a Gaussian assumption are not robust. In fact, Gaussian kernels are a border case scenario, and slight deviations from this function can lead to either uniform or lumped species distributions. Here, we illustrate the non-robustness of the Gaussian assumption by simulating different implementations of the standard competition model with constant carrying capacity. In this scenario, lumped species distributions can come about by secondary ecological or evolutionary mechanisms or by details of the numerical implementation of the model. We analyze the origin of this sensitivity and discuss it in the context of recent applications of the model.     

The importance of carrying capacity,fertility, and viability for the competitive ability of populations

Summary A simple mathematical model is analyzed which describes the behaviour of a population under the influence of viability and fertility as well as carrying capacity and competition by another species. One locus with two alleles determining viability, fertility, and carrying capacity is considered. Throughout the analysis absolute rather than relative numbers are used. The following results have been obtained: 1. The species producing more zygotes wins the competition. 2. Absolute numbers must be used for viability and fertility in theoretical considerations, because the outcome of competition may be changed by using different absolute numbers of the same relation. 3. Differences in viability are amplified by limiting the number of individuals. 4. Differences in every parameter can be compensated by any other parameter.     

The shape of the competition and carrying capacity kernels affects the likelihood of disruptive selection

Many quantitative genetic and adaptive dynamic models suggest that disruptive selection can maintain genetic polymorphism and be the driving force causing evolutionary divergence. These models also suggest that disruptive selection arises from frequency-dependent intraspecific competition. For convenience or historical precedence, these models assume that carrying capacity and competition functions follow a Gaussian distribution. Here, we propose a new analytical framework that relaxes the assumption of Gaussian competition and carrying capacity functions, and investigate how alternative shapes affect the likelihood of disruptive selection. We found that the shape of both carrying capacity and competition kernels interact to determine the likelihood of disruptive selection. For certain regions of the parametric space disruptive selection is facilitated, whereas for others it becomes more difficult. Our results suggest that the relationship between the degree of frequency dependence and the likelihood of disruptive selection is more complex than previously thought, depending on how resources are distributed and competition interference takes place. It is now important to describe the empirical patterns of resource distribution and competition in nature as a way to determine the likelihood of disruptive selection in natural populations.     

Species packing in nonsmooth competition models

Despite the potential for competition to generate equilibrium coexistence of infinitely tightly packed species along a trait axis, prior work has shown that the classical expectation of system-specific limits to the similarity of stably coexisting species is sound. A key reason is that known instances of continuous coexistence are fragile, requiring fine-tuning of parameters: A small alteration of the parameters leads back to the classical limiting similarity predictions. Here we present, but then cast aside, a new theoretical challenge to the expectation of limiting similarity. Robust continuous coexistence can arise if competition between species is modeled as a nonsmooth function of their differences—specifically, if the competition kernel (differential response of species’ growth rates to changes in the density of other species along the trait axis) has a nondifferentiable sharp peak at zero trait difference. We will say that these kernels possess a “kink.” The difference in predicted behavior stems from the fact that smooth kernels do not change to a first-order approximation around their maxima, creating strong competitive interactions between similar species. “Kinked” kernels, on the other hand, decrease linearly even for small species differences, reducing interspecific competition compared with intraspecific competition for arbitrarily small species differences. We investigate what mechanisms would lead to kinked kernels in the first place. It turns out that discontinuities in resource utilization generate them. We argue that such sudden jumps in the utilization of resources are unrealistic, and therefore, one should expect kernels to be smooth in reality.     

Demographic stochasticity versus spatial variation in the competition between fast and slow dispersers

Dispersal is an important strategy that allows organisms to locate and exploit favorable habitats. The question arises: given competition in a spatially heterogeneous landscape, what is the optimal rate of dispersal? Continuous population models predict that a species with a lower dispersal rate always drives a competing species to extinction in the presence of spatial variation of resources. However, the introduction of intrinsic demographic stochasticity can reverse this conclusion. We present a simple model in which competition between the exploitation of resources and stochastic fluctuations leads to victory by either the faster or slower of two species depending on the environmental parameters. A simplified limiting case of the model, analyzed by closing the moment and correlation hierarchy, quantitatively predicts which species will win in the complete model under given parameters of spatial variation and average carrying capacity.     

Effect of habitat degradation on competition,carrying capacity,and species assemblage stability

Changes in species’ trophic niches due to habitat degradation can affect intra‐ and interspecific competition, with implications for biodiversity persistence. Difficulties of measuring species’ interactions in the field limit our comprehension of competition outcomes along disturbance gradients. Thus, information on how habitat degradation can destabilize food webs is scarce, hindering predictions regarding responses of multispecies systems to environmental changes. Seagrass ecosystems are undergoing degradation. We address effects of Posidonia oceanica coverage reduction on the trophic organization of a macroinvertebrate community in the Tyrrhenian Sea (Italy), hypothesizing increased trophic generalism, niche overlap among species and thus competition and decreased community stability due to degraded conditions. Census data, isotopic analysis, and Bayesian mixing models were used to quantify the trophic niches of three abundant invertebrate species, and intra‐ and interspecific isotopic and resource‐use similarity across locations differing in seagrass coverage. This allowed the computation of (1) competition strength, with respect to each other and remaining less abundant species and (2) habitat carrying capacity. To explore effects of the spatial scale on the interactions, we considered both individual locations and the entire study area (“‘meadow scale”). We observed that community stability and habitat carrying capacity decreased as P. oceanica coverage declined, whereas niche width, similarity of resource use and interspecific competition strength between species increased. Competition was stronger, and stability lower, at the meadow scale than at the location scale. Indirect effects of competition and the spatial compartmentalization of species interactions increased stability. Results emphasized the importance of trophic niche modifications for understanding effects of habitat loss on biodiversity persistence. Calculation of competition coefficients based on isotopic distances is a promising tool for describing competitive interactions in real communities, potentially extendible to any subset of ecological niche axes for which specimens’ positions and pairwise distances can be obtained.     

Resource partitioning between large herbivores in Hustai National Park, Mongolia

Re-introduced Przewalski horses in Hustai National Park, Mongolia could suffer from food competition with other herbivore species through food resource depletion. Diet composition of the Przewalski horse (Equus ferus przewalskii), red deer (Cervus elaphus) and four livestock species (sheep, goat, cattle and horse) were studied, using micro histological analysis of faecal samples in the summer of 2005 and winter of 2006 – 2007. We expected that herbivores become less selective in food choice in winter regarding to summer, resulting in a larger diet breadth, a larger similarity in diet and a larger dietary overlap in winter, potentially triggering exploitative competition by depletion of shared resources. Vegetation biomass decreased during winter, and the different herbivores species in HNP changed their diet from summer to winter. As expected diet breadth, diet similarity and dietary overlap were significantly larger in winter in comparison to summer. The existence of competition by resource depletion between the different species cannot be ruled out. Vegetation biomass was probably not a limiting factor according to the correlation between annual rainfall and herbivore species biomass, however the forage quality may be limiting, triggering competition.     

Limiting similarity and Darwin’s naturalization hypothesis: understanding the drivers of biotic resistance against invasive plant species

Several hypotheses have been proposed to explain biotic resistance of a recipient plant community based on reduced niche opportunities for invasive alien plant species. The limiting similarity hypothesis predicts that invasive species are less likely to establish in communities of species holding similar functional traits. Likewise, Darwin’s naturalization hypothesis states that invasive species closely related to the native community would be less successful. We tested both using the invasive alien Ambrosia artemisiifolia L. and Solidago gigantea Aiton, and grassland species used for ecological restoration in central Europe. We classified all plant species into groups based on functional traits obtained from trait databases and calculated the phylogenetic distance among them. In a greenhouse experiment, we submitted the two invasive species at two propagule pressures to competition with communities of ten native species from the same functional group. In another experiment, they were submitted to pairwise competition with native species selected from each functional group. At the community level, highest suppression for both invasive species was observed at low propagule pressure and not explained by similarity in functional traits. Moreover, suppression decreased asymptotically with increasing phylogenetic distance to species of the native community. When submitted to pairwise competition, suppression for both invasive species was also better explained by phylogenetic distance. Overall, our results support Darwin’s naturalization hypothesis but not the limiting similarity hypothesis based on the selected traits. Biotic resistance of native communities against invasive species at an early stage of establishment is enhanced by competitive traits and phylogenetic relatedness.     

Limiting similarity and the intensity of competitive effects on the mottled sculpin,Cottus bairdi,in experimental stream communities

The identification of potential competitors has been driven by the concept of limiting similarity. Lacking are explicit tests of interaction strength among morphologically similar and dissimilar species. I used the mottled sculpin, Cottus bairdi, as a focal species in an artificial stream experiment designed to compare the effect of intraspecific competition to interspecific cometition from two very different species: a congener, the Kanawha sculpin (C. carolinae ssp.), and an unrelated species, the fantail darter (Etheostoma flabellare). The differences in morphology between these two species generate specific predictions under limiting similarity regarding the likelihood of competition and its relative strength: the congener should be a more important potential competitor. Increased fish density had a strong effect on the multivariate response of survival and growth, and on the relative condition of C. bairdi, indicating competition. The effect of additional C. bairdi or Kanawha sculpins were roughly equal, but the effect of E. flabellare was significantly greater. The most important potential impact on C. bairdi came from interspecific competition by a species that is smaller and very different in morphology, contrary to predictions based on limiting similarity.     

Can limiting similarity increase invasion resistance? A meta‐analysis of experimental studies

Synthesis We used meta‐analyses to examine experimental evidence that functional similarity between invaders and resident communities reduces invasion. We synthesized evidence from studies that experimentally added seed to resident communities in which the functional group composition had been manipulated. We found communities containing functionally similar resident species reduced invasion of forb but not grass invaders. However, experimental design dramatically influenced the results – with evidence for limiting similarity only found in artificially assembled communities, and not when studies used functional group removal from more ‘natural communities’. We suggest that functional group similarity plays a limited role in biotic resistance in established communities. The principle of limiting similarity suggests that species must be functionally different to coexist; based on the assumption that inter‐specific competition should be greatest between functionally similar species. There has been controversy over the generality of this assembly rule for plant communities with some studies finding evidence for limiting similarity and others not. One approach to testing this is to examine the ‘invasion’ success of species into communities in which the functional group composition has been manipulated. Using a meta‐analysis approach, we examined the generality of limiting similarity for plant communities based on published experimental studies. We asked – is establishment of an invading species less successful if it belongs to a functional group that is already present in the community compared to a community in which that functional group is absent? We explored separately colonisation (i.e. germination, establishment or seedling survival) and performance (i.e. biomass, cover or growth) of different functional groups (forbs and grasses) and experimental designs (removal experiments of more or less natural communities and synthetic‐assemblage experiments). We found that communities containing functionally similar resident species did reduce invader colonisation and performance of forb invaders, but did not reduce colonisation or performance of grass invaders. Evidence in support of limiting similarity was only detected in synthetic‐assemblage experiments and not when studies used functional group removal from ‘natural’ communities. Functional similarity is an important aspect of biotic resistance for forb invaders, but was only found in artificial communities. This has implications for restoration ecology especially when communities are built de novo. However, we suggest that limiting similarity plays a limited role in biotic resistance, because no evidence was detected in established communities.     

Biologic interactions determining geographic range size: a one species response to phylogenetic community structure

Range size variation in closely related species suggests different responses to biotic and abiotic heterogeneity across large geographic regions. Species turnover generates a wide spectrum of species assemblages, resulting in different competition intensities among taxa, creating restrictions as important as environmental constraints. We chose to adopt the widely used phylogenetic relatedness (NRI) measurement to define a metric that depicts competition strength (via phylogenetic similarity), which one focal species confronts in its environment. This new approach (NRI_focal) measures the potential of the community structure effect over performance of a single species. We chose two ecologically similar Peucaea sparrows, which co‐occur and have highly dissimilar range size to test whether the population response to competition intensity is different between species. We analyzed the correlation between both Peucaea species population sizes and NRI_focal using data from point counts. Results indicated that the widespread species population size was not associated with NRI_focal, whereas the population of restricted‐sized species exhibited a negative relationship with competition intensity. Consequently, a species' sensitivity to competition might be a limiting factor to range expansion, which provides new insights into geographic range analysis and community ecology.     

Similarities between invaders and native species: Moving past Darwin's naturalization conundrum

Darwin's naturalization conundrum states that successful invaders must be closely related to native species to possess the traits to tolerate that environment, but distantly related enough to possess traits allowing exploitation of underutilized niches, thereby minimizing competition. Although influential, this hypothesis is based on several simplistic assumptions. In particular, the relationship among phylogenetic relatedness, similarity, and competition is more complex than assumed and changes with spatial and phylogenetic scale. Competitive interactions are determined by limiting similarity and trait hierarchies associated with separate traits. Successful invaders thus need to be similar to native species in some respects, but different in others. This combination of similarities and differences is unlikely to be conserved. Further, many invasive species are represented in their novel range by genotypes with extreme trait values or plasticity relative to the species mean. Selection for these genotypes may alter the similarity between invasive and native species, thus obscuring the relationship between competition and phylogenetic relatedness. As environmental filtering and competition often act on different spatial scales, approaches assessing how individual traits relate to invasion at these scales (species pools vs local community) may improve our understanding of the relationship between similarity and invasion.     

Co‐occurrence and phylogenetic distance in communities of mammalian ectoparasites: limiting similarity versus environmental filtering

Similarity between species plays a key role in the processes governing community assembly. The co‐occurrence of highly similar species may be unlikely if their similar needs lead to intense competition (limiting similarity). On the other hand, persistence in a particular habitat may require certain traits, such that communities end up consisting of species sharing the same traits (environmental filtering). Relatively little information exists on the relative importance of these processes in structuring parasite communities. Assuming that phylogenetic relatedness reflects ecological similarity, we tested whether the co‐occurrence of pairs of flea species (Siphonaptera) on the same host individuals was explained by the phylogenetic distance between them, among 40 different samples of mammalian hosts (rodents and shrews) from different species, areas or seasons. Our results indicate that frequency of co‐occurrence between flea species increased with decreasing phylogenetic distance between them in 37 out of 40 community samples, with 14 of these correlations being statistically significant. A meta‐analysis across all samples confirmed the overall trend for closely related species to co‐occur more frequently on the same individual hosts than expected by chance, independently of the identity of the host species or of environmental conditions. These findings suggest that competition between closely related, and therefore presumably ecologically similar, species is not important in shaping flea communities. Instead, if only fleas with certain behavioural, ecological and physiological properties can encounter and exploit a given host, and if phylogenetic relationships determine trait similarity among flea species, then a process akin to environmental filtering, or host filtering, could favour the co‐occurrence of related species on the same host.     

Density dependent selection incorporating intraspecific competition 1. A haploid model

A haploid model is introduced and analyzed in which intraspecific competition is incorporated within a density dependent framework. It is assumed that each genotype has a unique carrying capacity corresponding to the equilibrium population size when fixed for that type. Each genotypic fitness at a single multi-allelic locus is a function of a distinctive effective population size formed by adding the numbers of each genotype present, weighted by an intraspecific competition coefficient. As a result, the fitnesses depend upon the relative frequencies of the various genotypes as well as the total population size. Intergenotypic interactions can have a profound effect upon the outcome of the population. In particular, when the density effect of one individual upon another depends upon their respective genotypes, a unique stable interior equilibrium is possible in which all alleles are present. This stands in contrast to the purely density dependent haploid system in which the only possible stable state corresponds to fixation for the type with the highest carrying capacity. In the present model selective advantage is determined by a balance between carrying capacity and sensitivity to density pressures from other genotypes. Fixation for the genotype with the highest carrying capacity, for instance, will not be stable if it exerts a sufficiently weak competitive effect upon the other genotypes. In the diallelic case, maintenance of both alleles at a stable equilibrium requires that the net intragenotypic competition between individuals of like genotype be stronger than that between unlike types. As for purely density regulated systems, there may be no stable equilibria and/or regular and chaotic cycling may occur. The results may also be interpreted in terms of a discrete time model of interspecific competition with each haplotype representing a different species.     

Effects of sample size and intraspecific variation in phylogenetic comparative studies: a meta‐analytic review

Comparative analyses aim to explain interspecific variation in phenotype among taxa. In this context, phylogenetic approaches are generally applied to control for similarity due to common descent, because such phylogenetic relationships can produce spurious similarity in phenotypes (known as phylogenetic inertia or bias). On the other hand, these analyses largely ignore potential biases due to within‐species variation. Phylogenetic comparative studies inherently assume that species‐specific means from intraspecific samples of modest sample size are biologically meaningful. However, within‐species variation is often significant, because measurement errors, within‐ and between‐individual variation, seasonal fluctuations, and differences among populations can all reduce the repeatability of a trait. Although simulations revealed that low repeatability can increase the type I error in a phylogenetic study, researchers only exercise great care in accounting for similarity in phenotype due to common phylogenetic descent, while problems posed by intraspecific variation are usually neglected. A meta‐analysis of 194 comparative analyses all adjusting for similarity due to common phylogenetic descent revealed that only a few studies reported intraspecific repeatabilities, and hardly any considered or partially dealt with errors arising from intraspecific variation. This is intriguing, because the meta‐analytic data suggest that the effect of heterogeneous sampling can be as important as phylogenetic bias, and thus they should be equally controlled in comparative studies. We provide recommendations about how to handle such effects of heterogeneous sampling.     

Parasite-mediated selection in experimental Daphnia magna populations

Abstract It has been suggested that parasites are a strong selecting force for their hosts and therefore may alter the outcome of competition among host genotypes. We tested the extent to which parasite-mediated selection by different parasite species influenced competition among clones of the cyclic parthenogen Daphnia magna . We monitored clone frequency changes in laboratory microcosm populations consisting of 21 D. magna clones. Parasite treatments (two microsporidians, Glugoides intestinalis and Ordospora colligata ) and a parasite-free control treatment were followed over a nine-month period. A further treatment with the bacterium Pasteuria ramosa failed. We found significant differences in clonal success among the treatments: the two parasite treatments differed from the control treatment and from each other. Additionally, we measured the clone-specific population carrying capacity, competitive ability against tester clones, and reproductive success of infected and uninfected females to test whether they correlate with clonal success in the microcosms. The clone-specific competitive ability was a good predictor of clonal success in the microcosms, but clonal carrying capacity and host reproductive success were not. Our study shows that parasite-mediated selection can strongly alter the outcome of clonal competition. The results suggest that parasites may influence microevolution in Daphnia populations during periods of asexual reproduction.     

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.     

Perturbation analysis of competition and overlap in habitat utilization between Dipodomys ordii and Dipodomys merriami

Summary The populations of two coexisting species of Dipodomys (Heteromyidae, Rodentia) were manipulated on 10, large, unenclosed, trapping grids. These manipulations revealed that, although many kangaroo rats are established residents in an area, a large number are transient individuals who quickly occupy vacated habitats. On plots from which residents had been removed, transients settled at rates of up to 5% of carrying capacity per day. These immigrants were invariably of the same species that was removed, indicating a strong element of intraspecific competition with little or no evidence of competition between the species.Trapping records suggest that these species avoid competition through habitat selection. Dipodomys ordii prefer a grassier habitat, and D. merriami a habitat dominated by creosote bush. Apparent overlap in their utilization of habitats, based on sites of capture, predicts competition coefficients to be higher than those permitted by the theory of limiting similarity and much higher than those actually shown by the perturbation experiments.This study demonstrates the dangers of estimating alpha without experimentation. This is especially true in cases where habitat selection may be important, since organisms may travel in habitats without collecting resources therein. Our results are discussed in light of a theory which examines the optimal (rather than tolerable) amount of overlap in habitat utilization between two potential competitors in a mixed habitat. This theory predicts that the pressure of natural selection should eliminate the interspecific competition entirely.However, the conclusion that the interspecific competitive alpha is zero does not lead to the conclusion that interspecific competition is unimportant in the system. Instead, if our interpretation is correct, such competition has molded the system, and were there not a continual threat of interspecific competition, the habitat specializations would soon disappear.     

A general analysis of resource allocation by competing individuals.

A linear model for population dynamics in a stationary stochastic environment is introduced based on linearizing the N-species Lotka-Volterra competition equations in discrete time. Iteration of the linear model shows the sequence of population sizes to be formed from a simple linear operation on the sequence of carrying capacities. The transfer function for this operation is calculated and the spectral properties of time series data on population size follow directly.The above approach is illustrated with a symmetrical two-species competition system assuming white noise variation in the carrying capacities. The results are interpreted in detail with the following ideas. (1) The intrinsic rate of increase governs the “responsiveness” of the population to changes in the carrying capacity; (2) one effect of competition is to reduce the “effective rate of increase” of the population. Increasing competition can produce effects identical to that of lowering the intrinsic rate of increase; (3) the other effect of competition is to communicate the stochastic variation in one species' carrying capacity to its competitors. The end result of this communication depends critically on the cross-correlation scheme among the carrying capacities of the competing species.