Discrete time models for two-species competition.

A discrete (difference) single age-class model for two-species competition is presented and its stability properties discussed. It resembles the Lotka-Volterra model in having linear zero growth isoclines, and thus, also in its general requirements for the coexistence of competing species. It differs in allowing the populations to show damped oscillations, stable cycles or even apparent “chaos” if competition is sufficiently severe. A similar two age-class model is discussed where there is both intra- and interspecific competition in one of the developmental stages, but only intraspecific competition in the other. Even this slight increase in complexity leads to markedly different properties. The zero growth curves become nonlinear and up to three equilibria between two competing species are possible.     

Predicting stability of mixed microbial cultures from single species experiments: 1. Phenomenological model

The growth of mixed microbial cultures on mixtures of substrates is a fundamental problem of both theoretical and practical interest. On the one hand, the literature is abundant with experimental studies of mixed-substrate phenomena [T. Egli, The ecological and physiological significance of the growth of heterotrophic microorganisms with mixtures of substrates, Adv. Microbiol. Ecol. 14 (1995) 305-386]. On the other hand, a number of mathematical models of mixed-substrate growth have been analyzed in the last three decades. These models typically assume specific kinetic expressions for substrate uptake and biomass growth rates and their predictions are formulated in terms of parameters of the model. In this work, we formulate and analyze a general mathematical model of mixed microbial growth on mixtures of substitutable substrates. Using this model, we study the effect of mutual inhibition of substrate uptake rates on the stability of the equilibria of the model. Specifically, we address the following question: How much of the dynamics exhibited by two competing species can be inferred from single species data? We provide geometric criteria for stability of various types of equilibria corresponding to non-competitive exclusion, competitive exclusion, and coexistence of two competing species in terms of growth isoclines and consumption curves. A growth isocline is a curve in the plane of substrate concentrations corresponding to the zero net growth of a given species. In [G.T. Reeves, A. Narang, S.S. Pilyugin, Growth of mixed cultures on mixtures of substitutable substrates: The operating diagram for a structured model, J. Theor. Biol. 226 (2004) 143-157], we introduced consumption curves as sets of all possible combinations of substrate concentrations corresponding to balanced growth of a single microbial species. Both types of curves can be obtained in single species experiments.     

Competitive exclusion through reproductive interference

A simple differential equation model was developed to describe the competitive interaction that may occur between species through reproductive interference. The model has the form comparable to Volterra's competition equations, and the graphical analysis of the outcome of the two-species interaction based on its zero-growth isoclines proved that: (1) The possible outcome in this model, as in usual models of resource competition, is either stable coexistence of both species or gradual exclusion of one species by the other, depending critically upon the values of the activity overlapping coefficient c_ij; (2) but, for the same c_ij-values, competitive exclusion is much more ready to occur here than in resource competition; (3) and moreover, the final result of the competition is always dependent on the initial-condition due to its non-linear isoclines, i.e., even under the parameter condition that generally allows both species to coexist, an extreme bias in intial density to one species can readily cause subsequent complete exclusion of its counterparts. Thus, it may follow that the reproductive interference is likely to be working in nature as an efficient mechanism to bring about habitat partitioning in either time or space between some closely related species in insect communities, even though they inhabit heterogeneous habitats where resource competition rarely occurs so that they could otherwise attain steady coexistence.     

Niche construction in the light of niche theory

Ecological niche construction, the process whereby an organism improves its environment to enhance its growth and persistence, is an important missing element of niche theory. Niche theory has mainly focused on niche-deteriorating processes, such as resource consumption, predation and competition, which have negative effects on population growth. Here, we integrate niche construction explicitly into modern niche theory. We use a graphical approach to analyse how a species' niche-improving impacts interplay with niche-deteriorating impacts to modify its response to the environment. In a model of two consumers that compete for one limiting resource and one predator, we show how niche construction modifies the traditional niche-deteriorating impacts of its agent or of competing species, and hence the potential for species coexistence. By altering the balance between intraspecific and interspecific competitive effects, niche construction can either generate net interspecific facilitation or strengthen interspecific competition. The adaptive benefit derived from niche construction also strongly affects the realized niche of a niche-constructing species.     

物种对资源竞争的动力机制及数值模拟

建立了集合种群物种在两个斑块中对资源竞争的数学模型,并进行了数值模拟实验,结果表明：（1）通过R^＊来预测竞争物种的结局,存在几种可能性：一是具有低R^＊值的物种竞争取代高R^＊值的物种;二是具有不同R^＊值的物种,甚至是具有相同R^＊值的物种也存在共存的可能性;三是具有高R^＊值的物种也可以竞争排斥低R^＊值的物种,结论存在不确定性。（2）竞争物种的随机迁移形成了源一汇结构,对物种竞争共存具有促进作用,但弱的资源利用者（较高的R^＊）的迁移率不宜过高。（3）在种群统计率相同的条件下,资源增长率差异越大,越不利于消费者物种的共存;若种群统计率不相同,在资源增长率相同的情况下,物种共存又是不可能的,在自然界中,物种共存需要资源增长率的差异。（4）不同类型的资源增长对竞争物种的稳定性的影响是不同的。     

病原体感染对物种间资源竞争的影响——基于资源竞争理论

病原体感染对种间竞争的影响可能是因为改变了宿主的资源利用过程,然而竞争模型（Lotka-Volterra）由于参数化竞争系数而忽略了资源的动态变化过程,因此基于此类模型的研究无法揭示病原体对宿主资源利用的影响。基于Tilman的资源竞争理论构建了病原体感染一个物种的资源竞争模型,通过分析宿主物种资源利用效率的变化探讨了病原体对种间竞争的影响。结果表明：（1）病原体降低了宿主对资源的消耗率（消费矢量变短）,抬高了对资源的最低需求（零等倾线上移）,这意味着宿主的竞争力减弱;（2）虽然感染影响了竞争物种的密度,但不会改变共存物种的共存状态;（3）病原体可以使宿主物种的竞争对手更容易入侵,形成共存局面,极大地扩大了竞争物种共存的参数范围,本质上促进了物种多样性维持;（4）病原体的传播率和毒性也复杂地影响了竞争物种共存,传播率越大越能促进物种共存,而中等强度毒性最能促进物种共存。研究结果明确了病原体对物种资源利用模式的潜在改变,强调了病原体在物种共存和生物多样性维持中的重要性。     

Habitat-dependent competition and the coexistence of Australian heathland rodents

When competing species depress one another's fitness in the habitats that they occupy, their competitive effects will emerge in each species' pattern of density-dependent habitat choice. Thus, a regression analysis of joint densities, corrected by the habitat effect, should reveal the magnitude of interspecific competition. We tested this idea by 1) demonstrating the connection between removal experiments and regression estimates of competition with those obtained from isodars (regressions that implicitly incorporate evolutionarily stable strategies of habitat selection) and 2) evaluating whether interspecific competition inferred from isodars corresponded with the inferences emerging from regression and field experiments. Previous removal experiments on two herbivorous rodents occupying coastal wet heathlands in eastern Australia documented that competition between Rattus lutreolus and Pseudomys gracilicaudatus is asymmetrically biased in favor of the much larger Rattus . The asymmetry in competition was also revealed by regression estimates of competition. Isodar analyses illustrate a habitat-dependent mechanism for the asymmetry. Rattus compete effectively with Pseudomys in both 'wetter' and 'drier' patches of heath whereas Pseudomys appear to exert a competitive effect in only the drier sites. The magnitude of competition measured by a removal experiment in an area with more-or-less equal amounts of both habitats will be biased in favor of Rattus . More generally, one can use the isodar estimates to draw isolegs and isoclines of competitive coexistence. Isoclines for the two Australian rodents imply dynamic equilibria of stable competitive coexistence that vary with plant succession in fire-dominated heathland ecosystems.     

Density compensation,isocline shape and single-level competition models

Criticism of the Lotka-Volterra competition model implies that the theory of competition should be based upon more general concepts. It is suggested that the shape of the competitor isoclines can provide this basis.The relationship between the total density of a competitive community and species number depends crucially upon isocline shape. This has immediate relevance to the interpretation of the excess density compensation seen in some island communities, since if isoclines are sufficiently concave (curved towards the origin) then this phenomenon is expected to be the rule rather than the exception.These observations do not depend upon any specific model, but in order to determine the shape of isoclines in natural communities a link must be made between the biological processes of competition and isocline shape. To this end three types of single-level competition model are distinguished (additive, multiplicative and temporal resource models) depending upon how gains from resources interact in determining individual fitness.The models are based upon resource availability functions (RAFs), which are decreasing functions of the level of competition and determine the availability of each resource to each species. Provided that the argument of these functions is always a weighted sum of the number of competitors then in the case of the additive resource model the shape of the RAFs determines directly the shape of the isoclines. For the multiplicative model, the shape of the logarithm of the RAFs adopts this role.Analysis of a special case of the additive resource model suggests that concave isoclines are likely to predominate, and that the degree of concavity is of an order which minimizes the tendency of total numbers to increase with species number. In some circumstances, involving “scramble”-type competition and habitat selection, the expected concavity is sufficient to cause a decrease. In any event the expected occurrence of concave competition isoclines predicts a much higher incidence of excess density compensation (due to carrying capacity differences) than expected from any model having linear isoclines.The effect of shifting from an additive to a multiplicative resource model is to make the existence of purely concave isoclines less probable and to raise the possibility of purely convex isoclines. On the other hand, shifting from an additive resource model to a temporal resource model apparently has no such simple interpretation and specific predictions must await further analysis.     

Competition among plant species that interact with their environment at different spatial scales

Clonal plants that are physiologically integrated might perceive and interact with their environment at a coarser resolution than smaller, non-clonal competitors. We develop models to explore the implications of such scale asymmetries when species compete for multiple depletable resources that are heterogeneously distributed in space across two patches. Species are either 'non-integrators', whose growth in each patch depends on resource levels in that patch alone, or 'integrators', whose growth is equal between patches and depends on average resource levels across patches. Integration carried both benefits and costs. It tended to be advantageous in poorer patches, where the integrators drew resources down further than the non-integrators (more easily excluding competitors) and might persist by using resources from richer adjacent patches. Integration tended to be disadvantageous in richer patches, where integrators did not draw resources down as far (creating an opportunity for competitors) and could be excluded due to the cost of supporting growth in poorer adjacent patches. Complementarity between patches (each rich in a separate resource) favoured integrators. Integration created new opportunities for local coexistence, and for delayed susceptibility of patches to invasion, but eliminated some opportunities for regional coexistence. Implications for the interpretations of species' zero net growth isoclines and Rs are also discussed.     

The volterra competition equations with resource - Independent growth coefficients and discussion on their biological and biophysical implications

Analysis of the biophysical conditions for a correct application of the Volterra Competition Equations with resource-independent coefficients reveals the following:The traditional, mathematical formalism with the two equations representing two straight lines at the condition of zero growth applies.As a directly resource-limited situation does not permit for stable equilibrium (One-line or K-system, Walker [18]; Ceiling model, Pollard [11]), the combined equilibrium density represented by the intersect of the two lines (Two-line system or S-system; Equilibrium model; lit. ref. see above) is by necessity smaller than the carrying capacity of shared resources would permit. The physical determinant is density in space as a result of behavioural/physiological interaction between organisms. The traditional inequality conditions then mean that coexistence of species sharing the same space is possible provided density-dependent reduction of growth relative to the intrinsic growth rate is more effective within species than between species. The distance of the intersect from the line linking the specific population densities of single species on the axes is a measure for the overlap in the use of space by the two species; thus, overlap of the spatial niche results in stability.This biophysical system allows for coexistence of species in identical space/resource niches.Food-jealous behaviour is a direct function of density in space; it is only indirectly and inconsistently influenced by resource availability.     

Competition between juveniles and adults in age-structured populations

The effect of competition between juveniles and adults is examined in a generalized, two-age-class, discrete-time model. Adult fecundity and juvenile survival are functions of both age-class densities. Possible configurations of the zero growth isoclines are examined, giving special attention to the isocline shapes, the number of equilibria, and the manner in which the population approaches these equilibria. It is found that small increases in the density of one age class may have either a positive or a negative effect on recruitment into the other class, depending upon the degree of density dependence in fecundity and survival. Closely allied to this, an increase in the resources for a given age class may result in either an increase or a decrease in its equilibrium density. Strong juvenile-adult competition generally has destabilizing effects on the population's equilibrium, with the system being more sensitive to juveniles competing with adults than to the reverse.     

Description of the ratios between metabolite concentrations in a polyenzyme system

It was demonstrated that the relations between substrate and product concentrations for a reaction catalyzed by michaelian enzyme incorporated in a multienzyme system can be graphically represented by a diverging set of straight lines intersecting in one point, the flux velocity being treated as a parameter. A competitive inhibitor shifts the intersection point along the line of equilibrium state. The relations between the concentrations of more than two reagents are represented by a set of equivelocity surfaces. The relations between substrate and product concentrations for a kinetically cooperative reaction conforming to the graphical representation by the second--order curves were analyzed. The stability criterion was obtained for a multienzyme system with the first enzyme allosterically regulated by products of subsequent reactions.     

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.     

Plant interspecies competition for sunlight: a mathematical model of canopy partitioning

We examine the influence of canopy partitioning on the outcome of competition between two plant species that interact only by mutually shading each other. This analysis is based on a Kolmogorov-type canopy partitioning model for plant species with clonal growth form and fixed vertical leaf profiles (Vance and Nevai in J. Theor. Biol., 2007, to appear). We show that canopy partitioning is necessary for the stable coexistence of the two competing plant species. We also use implicit methods to show that, under certain conditions, the species’ nullclines can intersect at most once. We use nullcline endpoint analysis to show that when the nullclines do intersect, and in such a way that they cross, then the resulting equilibrium point is always stable. We also construct surfaces that divide parameter space into regions within which the various outcomes of competition occur, and then study parameter dependence in the locations of these surfaces. The analysis presented here and in a companion paper (Nevai and Vance, The role of leaf height in plant competition for sunlight: analysis of a canopy partitioning model, in review) together shows that canopy partitioning is both necessary and, under appropriate parameter values, sufficient for the stable coexistence of two hypothetical plant species whose structure and growth are described by our model. A. L. Nevai was supported in part by the National Institutes of Health, National Research Service Award (T32-GM008185) from the National Institute of General Medical Sciences (NIGMS).     

Species coexistence under resource competition with intraspecific and interspecific direct competition in a chemostat

Competition theory has developed separately for direct competition and for exploitative competition. However, the combined effects of the two types of competition on species coexistence remain unclear. To examine how intraspecific and interspecific direct competition contributes to the coexistence of species competing for a single resource, we constructed a chemostat-type resource competition model. With general functions for intraspecific and interspecific direct competition, we derived necessary and sufficient conditions (except for a critical case that rarely occurs in a biological sense) that determine the number of stably coexisting species. From these conditions, we found that the number of coexisting species is determined just by the invasibility of each species into subcommunities with a smaller number of species. In addition, using a combination of rigorous mathematical theory and a simple graphical method, we can demonstrate how the stronger intraspecific direct competition facilitates species invasion, leading to a larger number of coexisting species.     

The significance of ratios of detritus types and micro-organism productivity to competitive interactions between aquatic insect detritivores

Investigations of competitive interactions emphasize non-detrital resources, even though detritus is a major component of most food webs. Studies of competing species focus usually on single resource types, although consumers in nature are likely to encounter mixtures of resource types that may affect whether competition results in exclusion or coexistence. The invasive mosquito Aedes albopictus is capable of excluding the native mosquito Ochlerotatus triseriatus in competition for single detritus types in laboratory and field microcosms. In this study, we used nine ratios of two detritus types (animal and leaf) common in natural containers to test whether detritus ratios affect the outcome of competition. Under intraspecific and interspecific competition, A. albopictus attained higher survival and estimated population growth rate than did O. triseriatus. Unlike past studies, both species had positive growth and high adult survival, with little evidence of competitive effects, under one resource ratio (10:1 ratio of leaf : animal detritus) regardless of mosquito densities, suggesting potential coexistence. Path analysis showed that densities of larvae had negative effects on population growth for O. triseriatus but not for A. albopictus, indicating competitive superiority of A. albopictus. Population growth of both species was affected strongly by the direct paths from animal (positive) and leaf (negative) detritus, and the indirect effect of leaf detritus via bacterial production (positive). Field sampling established that detritus entered real tree holes in ratios similar to those in our experiment, suggesting that natural variation in detritus ratios may influence local coexistence of these species. Seasonal variation in ratios of plant and animal detritus indicated that temporal as well as spatial variation in inputs may be important for potential coexistence.     

Competitive abilities in experimental microcosms are accurately predicted by a demographic index for R*

Resource competition theory predicts that R*, the equilibrium resource amount yielding zero growth of a consumer population, should predict species' competitive abilities for that resource. This concept has been supported for unicellular organisms, but has not been well-tested for metazoans, probably due to the difficulty of raising experimental populations to equilibrium and measuring population growth rates for species with long or complex life cycles. We developed an index (R(index)) of R* based on demography of one insect cohort, growing from egg to adult in a non-equilibrium setting, and tested whether R(index) yielded accurate predictions of competitive abilities using mosquitoes as a model system. We estimated finite rate of increase (λ') from demographic data for cohorts of three mosquito species raised with different detritus amounts, and estimated each species' R(index) using nonlinear regressions of λ' vs. initial detritus amount. All three species' R(index) differed significantly, and accurately predicted competitive hierarchy of the species determined in simultaneous pairwise competition experiments. Our R(index) could provide estimates and rigorous statistical comparisons of competitive ability for organisms for which typical chemostat methods and equilibrium population conditions are impractical.     

Competition in variable environments: experiments with planktonic rotifers

1. In a constant environment, competition often tends to reduce species diversity. However, several theories predict that temporal variation in the environment can slow competitive exclusion and allow competing species to coexist. This study reports on laboratory competition experiments in which two pairs of planktonic rotifer species competed for a phytoplankton resource under different conditions of temporal variability in resource supply.
2. For both species pairs, Keratella cochlearis dominated under all conditions of temporal variability, and the other species ( Brachionus calyciflorus or Synchaeta sp.) almost always went extinct. Increasing temporal variation in resource supply slowed competitive exclusion but did not change competitive outcome or allow coexistence.
3. Rotifers show a gleaner–opportunist trade-off, because gleaner species have low threshold resource levels ( R *) and low maximum population growth rates, while opportunist species have the opposite characteristics. In the competition experiments, the gleaner always won and the opportunists always lost. Thus, a gleaner–opportunist trade-off was not sufficient to facilitate coexistence under conditions of resource variability. Instead, the winning species had both the lowest R * and the greatest ability to store resources and ration their use during times of extreme resource scarcity.     

Interference competition and species coexistence

Interference competition is ubiquitous in nature. Yet its effects on resource exploitation remain largely unexplored for species that compete for dynamic resources. Here, I present a model of exploitative and interference competition with explicit resource dynamics. The model incorporates both biotic and abiotic resources. It considers interference competition both in the classical sense (i.e. each species suffers a net reduction in per capita growth rate via interference from, and interference on, the other species) and in the broad sense (i.e. each species suffers a net reduction in per capita growth rate via interference from, but can experience an increase in growth rate via interference on, the other species). Coexistence cannot occur under classical interference competition even when the species inferior at resource exploitation is superior at interference. Such a trade-off can, however, change the mechanism of competitive exclusion from dominance by the superior resource exploiter to a priority effect. Now the inferior resource exploiter can exclude the superior resource exploiter provided it has a higher initial abundance. By contrast, when interference is beneficial to the interacting species, coexistence is possible via a trade-off between exploitation and interference. These results hold regardless of whether the resource is biotic or abiotic, indicating that the outcome of exploitative and interference competition does not depend on the exact nature of resource dynamics. The model makes two key predictions. First, species that engage in costly interference mechanisms (e.g. territoriality, overgrowth or undercutting, allelopathy and other forms of chemical competition) should not be able to coexist unless they also engage in beneficial interference mechanisms (e.g. predation or parasitism). Second, exotic invasive species that displace native biota should be superior resource exploiters that have strong interference effects on native species with little or negative cost. The first prediction provides a potential explanation for patterns observed in several natural systems, including plants, aquatic invertebrates and insects. The second prediction is supported by data on invasive plants and vertebrates.