Periodic coexistence of four species competing for three essential resources

We show the existence of a periodic solution in which four species coexist in competition for three essential resources in the standard model of resource competition. By assuming that species i is limited by resource i for each i near the positive equilibrium, and that the matrix of contents of resources in species is a combination of cyclic matrix and a symmetric matrix, we obtain an asymptotically stable periodic solution of three species on three resources via Hopf bifurcation. A simple bifurcation argument is then employed which allows us to add a fourth species. In principle, the argument can be continued to obtain a periodic solution adding one new species at a time so long as asymptotic stability can be assured at each step. Numerical simulations are provided to illustrate our analytical results. The results of this paper suggest that competition can generate coexistence of species in the form of periodic cycles, and that the number of coexisting species can exceed the number of resources in a constant and homogeneous environment.     

The impact of consumer-resource cycles on the coexistence of competing consumers

This article seeks to determine the extent to which endogenous consumer-resource cycles can contribute to the coexistence of competing consumer species. It begins with a numerical analysis of a simple model proposed by Armstrong and McGehee. This model has a single resource and two consumers, one with a linear functional response and one with a saturating response. Coexistence of the two consumer species can occur when the species with a saturating response generates population cycles of the resource, and also has a lower resource requirement for zero population growth. Coexistence can be achieved over a wide range of relative efficiencies of the two consumers provided that the functional response of the saturating consumer reaches its half-saturation value when the resource population is a small fraction of its carrying capacity. In this case, the range of efficiencies allowing coexistence is comparable to that when two competitors have stable dynamics and a high degree of resource partitioning. A variety of modifications of this basic model are analyzed to investigate the consequences for coexistence of different resource growth equations, different functional and numerical response shapes, and other factors. Large differences in functional response shape appear to be the most important factor in producing robust coexistence via resource cycles. If the unstable species has a concave numerical response, this greatly expands the conditions allowing coexistence. If the stable consumer species has a convex (accelerating) functional and/or numerical response, the range of conditions allowing coexistence is also expanded. We argue that large between-species differences in functional response form can often be produced by between-consumer differences in the adaptive adjustments of foraging effort to food density. Consumer-resource cycles can also expand the conditions allowing coexistence when there is resource partitioning, but do so primarily when resource partitioning is relatively slight; this makes the ease of coexistence relatively independent of consumer similarity.     

Periodic coexistence in the chemostat with three species competing for three essential resources

A chemostat model of three species of microorganisms competing for three essential, growth-limiting nutrients is considered. J. Husiman and F.J. Weissing [Nature 402 (1999) 407] show numerically that this model can generate periodic oscillations. The present contribution is concerned with rigorous analysis regarding the existence of periodic oscillations in this model. Our analysis is based on the following observation made by Huisman and Weissing: there is a cyclic replacement of species, if each species becomes limited by the resource for which it is the intermediate competitor. Using a permanence theory, an index theory, and a Poincaré-Bendixson theory for three-dimensional competitive systems, we analytically succeed to give sufficient conditions for the existence of periodic orbits in the limit sets in this model. The results in this paper suggest that with a wide range of parameter values, sustained periodic oscillations of species abundances for the model are possible, without involving external disturbances. Our results also suggest that competition is not necessarily destructive, i.e., in the case of existence of sustained periodic oscillations, if one of three competitors is absent, one of the other two rivals cannot survive.     

Dispersal-mediated coexistence of competing predators

Models of metapopulations have often ignored local community dynamics and spatial heterogeneity among patches. However, persistence of a community as a whole depends both on the local interactions and the rates of dispersal between patches. We study a mathematical model of a metacommunity with two consumers exploiting a resource in a habitat of two different patches. They are the exploitative competitors or the competing predators indirectly competing through depletion of the shared resource. We show that they can potentially coexist, even if one species is sufficiently inferior to be driven extinct in both patches in isolation, when these patches are connected through diffusive dispersal. Thus, dispersal can mediate coexistence of competitors, even if both patches are local sinks for one species because of the interactions with the other species. The spatial asynchrony and the competition-colonization trade-off are usual mechanisms to facilitate regional coexistence. However, in our case, two consumers can coexist either in synchronous oscillation between patches or in equilibrium. The higher dispersal rate of the superior prompts rather than suppresses the inferior. Since differences in the carrying capacity between two patches generate flows from the more productive patch to the less productive, loss of the superior by emigration relaxes competition in the former, and depletion of the resource by subsidized consumers decouples the local community in the latter.     

Herbivore-induced coexistence of competing plant species

We study a series of spatially implicit lottery models in which two competing plant species, with and without defensive traits, are grazed by a herbivore in a homogeneous habitat. One species (palatable) has no defensive traits, while the other (defended) has defensive traits but suffers reduced reproduction as the result of an assumed trade-off. Not surprisingly, coexistence of these plants cannot occur when the herbivore density is very low (the palatable plant always wins) or very high (the defended plant wins). At intermediate densities, however, herbivory can mediate plant coexistence, even in a homogeneous environment. If the herbivore eats several plants per bite, and its forage-selection depends on the average palatability of the plants it eats, then palatable species in the immediate neighbourhood of defended plants may be more likely to persist (associational resistance) even at higher grazing pressure. If the herbivore shows a positive numerical response to the average palatability of the habitat as a whole, then both plant populations are stabilized and coexistence is promoted, because both species obtain a minority advantage through the negative feedback caused by herbivory. If the herbivore exhibits both of these traits, the system may have at most two non-trivial equilibria, one of which is stable and the other unstable. This means that coexistence in such a system is vulnerable to large fluctuations in herbivore density and identity, and this has implications for conservation in systems where large herbivores are managed to promote plant diversity.     

Disturbances allow coexistence of competing species

We give the first mathematically rigorous proof that disturbances allow competing species to coexist. This work provides a mathematical framework to explain the existence of fugitive species and the role played by disturbances in increasing or decreasing the biodiversity of ecosystems. We study modifications of the metapopulation model for patchy environments proposed by Caswell and Cohen (1990, 1991). For the one- and two-species models we give necessary and sufficient conditions on the parameters for the existence of a non-trivial equilibrium solution, which is shown to be always globally stable.     

Body size variation in aquatic consumers causes pervasive community effects,independent of mean body size

Intraspecific phenotypic variation is a significant component of biodiversity. Body size, for example, is variable and critical for structuring communities. We need to understand how homogenous and variably sized populations differ in their ecological responses or effects if we are to have a robust understanding of communities. We manipulated body size variation in consumer (tadpole) populations in mesocosms (both with and without predators), keeping mean size and density of these consumers constant. Size‐variable consumer populations exhibited stronger antipredator responses (reduced activity), which had a cascading effect of increasing the biomass of the consumer's resources. Predators foraged less when consumers were variable in size, and this may have mediated the differential effects of predators on the community composition of alternative prey (zooplankton). All trophic levels responded to differences in consumer size variation, demonstrating that intrapopulation phenotypic variability can significantly alter interspecific ecological interactions. Furthermore, we identify a key mechanism (size thresholds for predation risk) that may mediate impacts of size variation in natural communities. Together, our results suggest that phenotypic variability plays a significant role in structuring ecological communities.     

Coexistence of species competing for shared resources

In this paper we develop a mathematical model in which any number of competing species can coexist on four resources which regenerate according to an algebraic relationship. We show that previous attempts to prove that n species cannot coexist on fewer that n resources (the “competitive exclusion principle”) all make use of the very restrictive assumption that the specific growth rates of all competing species are linear functions of resource densities. When this restriction is relaxed, it becomes possible to find situations in which n species can coexist on fewer than n resources. On the basis of this and other observations we conclude that the competitive exclusion principle should be considered to apply only to coexistence at fixed densities.     

The invasion and coexistence of competing Wolbachia strains

Cytoplasmic incompatibility between arthropods infected with different strains of Wolbachia has been proposed as an important mechanism for speciation. However, a basic requirement for this mechanism is the coexistence of different strains in neighbouring populations. Here we test whether this required coexistence is possible in a spatial context. Continuous-time models for the behaviour of one and two strains of Wolbachia within a single well-mixed population demonstrate the Allee effect and founder control, such that one strain is always driven extinct. In contrast, discretised spatial models show patchy persistence of the two strains although coexistence within the same habitat is rare. A simplified model of such founder control suggests that it is fragmentation of (or barriers within) the habitat rather than space itself that leads to persistence.     

Epitope down-modulation as a mechanism for the coexistence of competing T-cells

Efficient immune responses against pathogens are frequently characterized by the simultaneous targeting of multiple epitopes. However, it remains unclear how the targeting of multiple epitopes is maintained in the face of competition for antigenic stimulation. Here, we investigate this question by using mathematical models of the population dynamics of a viral pathogen, antigen presentation sites and T-cells. We first show that direct competition for access to antigen presenting sites and indirect competition through killing of the pathogen select for dominance of the T-cell response with the highest affinity for its epitope. We then incorporate in our model that epitopes can become down-modulated following interaction with epitope specific T-cells. We demonstrate that epitope down-modulation leads to differentiation of epitope presentation on antigen presenting sites. This differentiation promotes the coexistence of multiple epitope specific responses. Hence, we propose that the functional relevance of epitope down-modulation may be to enable the persistence of a broad immune response despite competition for antigenic stimulation.     

Biodiversity in model ecosystems, I: coexistence conditions for competing species

This is the first of two papers where we discuss the limits imposed by competition to the biodiversity of species communities. In this first paper, we study the coexistence of competing species at the fixed point of population dynamic equations. For many simple models, this imposes a limit on the width of the productivity distribution, which is more severe the more diverse the ecosystem is (1994, Theor. Popul. Biol. 45, 227-276). Here we review and generalize this analysis, beyond the "mean-field"-like approximation of the competition matrix used in previous works, and extend it to structured food webs. In all cases analysed, we obtain qualitatively similar relations between biodiversity and competition: the narrower the productivity distribution is, the more species can stably coexist. We discuss how this result, considered together with environmental fluctuations, limits the maximal biodiversity that a trophic level can host.     

Spatial patterns of coexistence of competing species in patchy habitat

The single-species spatially realistic patch occupancy metapopulation model is, in this study, extended to a metacommunity of many competing species. Competition is assumed to reduce the local carrying capacity (effective patch area), which in turn increases local extinction rates and reduces colonization rates because of smaller population sizes. Each species is described by three parameters: pre-competitive abundance (equilibrium incidence of patch occupancy, which reflects the rate of colonization in relation to extinction rate), the spatial range of migration, and competitive ability. The model ignores spatio–temporal correlations caused by interspecific interactions, because in metacommunities of unequal competitors inhabiting heterogeneous landscapes, correlations in the occurrence of species are driven more by patch heterogeneity than by competition. The model allows the calculation of multispecies equilibria in patchy habitats without simulations. In general, the number of coexisting species in the metacommunity increases with decreasing strength of competition, increasing rate of colonization, and decreasing range of migration. Habitat heterogeneity in the form of spatial variation in patch areas tends to facilitate coexistence. Poor competitors may coexist with superior competitors in the patch network if the former have higher colonization rates (competition–colonization trade-off). When migration distances are short, competition leads to spatial pattern formation: Species tend to have restricted spatial distributions in the network, but contrary to intuitive expectations, often the distributions of many species are nested. Having more dispersive species enhances both local and global diversity, whereas more local migration decreases local but increases global diversity.     

Testing competing measures of profitability for mobile resources

Optimal diet theory often fails to predict a forager’s diet choice when prey are mobile. Because they escape or defend themselves, mobile prey are likely to increase the forager’s handling time, thereby decreasing its fitness gain rate. Many animals have been shown to select their prey so as to maximize either their fitness gain or their fitness gain rate. However, no study has yet compared directly these two measures of profitability by generating testable predictions about the choice of the forager. Under laboratory conditions, we compared these two measures of profitability, using the aphid parasitoid Aphidius colemani and its host, Myzus persicae. Fitness gain was calculated for parasitoids developing in each host instar by measuring life-history traits such as developmental time, sex ratio and fecundity. Fitness gain rate was estimated by dividing fitness gain by handling time, the time required to subdue the host. Fourth instar aphids provided the best fitness gain to parasitoids, whereas second instar aphids were the most profitable in terms of fitness gain rate. Host choice tests showed that A. colemani females preferred second instar hosts, suggesting that their decision maximizes fitness gain rate over fitness gain. Our results indicate that fitness gain rate is a reliable predictor of animal’s choice for foragers exploiting resources that impose additional time cost due to their mobility. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Coevolution of competing Callosobruchus species does not stabilize coexistence

Interspecific resource competition is expected to select for divergence in resource use, weakening interspecific relative to intraspecific competition, thus promoting stable coexistence. More broadly, because interspecific competition reduces fitness, any mechanism of interspecific competition should generate selection favoring traits that weaken interspecific competition. However, species also can adapt to competition by increasing their competitive ability, potentially destabilizing coexistence. We reared two species of bean beetles, the specialist Callosobruchus maculatus and the generalist C. chinensis, in allopatry and sympatry on a mixture of adzuki beans and lentils, and assayed mutual invasibility after four, eight, and twelve generations of evolution. Contrary to the expectation that coevolution of competitors will weaken interspecific competition, the rate of mutual invasibility did not differ between sympatry and allopatry. Rather, the invasion rate of C. chinensis, but not C. maculatus, increased with duration of evolution, as C. chinensis adapted to lentils without experiencing reduced adaptation to adzuki beans, and regardless of the presence or absence of C. maculatus. Our results highlight that evolutionary responses to interspecific competition promote stable coexistence only under specific conditions that can be difficult to produce in practice.     

Oscillatory coexistence in a food chain model with competing predators

A food chain model with two predators feeding on a single prey in a chemostat is studied. Using a multiparameter bifurcation analysis, we find parameters values for which there is stable oscillatory coexistence of the predators. It is also shown how these coexistent states provide a transition between two possible states of competitive exclusion. It is shown that the competitive exclusion principle need not hold if one or more of the predators has oscillatory behavior in the absence of other predators.This work was partially supported by National Science Foundation Grant MCS 83-01881     

Costs of jasmonate-induced responses in plants competing for limited resources

We present a design to quantify fitness consequences of jasmonate-induced responses in plants that are competing for limited resources with a conspecific. Under both high and low nitrogen supply rates, uninduced (control) Nicotiana attenuata plants growing next to a plant induced with 250 μg methyl jasmonate (MJ) yielded more seed capsules than control plants competing with another control plant. We conclude that there is a opportunity benefit for control plants growing next to an induced plant. Initially, MJ-induced plants grew more slowly, but by senescence they had produced the same number of seed capsules as control plants that had competed with another control plant. Replacement series showed that the fitness of MJ-induced plants is not influenced by the competitive status of their neighbour plant. We argue that competitive designs are useful tools for evaluating the phenotypic costs of ecologically important traits.     

Variability in resource consumption rates and the coexistence of competing species

A model which incorporates random temporal variation in resource consumption rates is used to investigate the effects that such variation has on the coexistence of competitors. The analysis of the model and several extensions of it suggests that such variation in consumption rates will often allow two or more competitors to coexist while limited by the same resource. For variability to promote coexistence, it is necessary that the time scale of resource population dynamics be fast relative to the time scale of environmental change. Variability is especially likely to promote coexistence if there is a large variance in consumption rates, negative correlation between the consumption rates of different species, and a linear or concave relationship between resource consumption and per capita population growth. Many previous studies which have found coexistence of two or more species on one resource can be interpreted as examples of coexistence due to varying resource consumption rates.     

Temperature fluctuation facilitates coexistence of competing species in experimental microbial communities

1. Temperature fluctuation is a general phenomenon affecting many, if not all, species in nature. While a few studies have shown that temperature fluctuation can promote species coexistence, little is known about the effects of different regimes of temperature fluctuation on coexistence. 2. We experimentally investigated how temperature fluctuation and different regimes of temperature fluctuation ('red' environments in which temperature series exhibited positive temporal autocorrelation vs. 'white' environments in which temperature series showed little autocorrelation) affected the coexistence of two ciliated protists, Colpidium striatum Stein and Paramecium tetraurelia Sonneborn, which competed for bacterial resources. 3. We have previously shown that the two species differed in their growth responses to changes in temperature and in their resource utilization patterns. The two species were not always able to coexist at constant temperatures (22, 24, 26, 28 and 30 degrees C), with Paramecium being competitively excluded at 26 and 28 degrees C. This indicated that resource partitioning was insufficient to maintain coexistence at these temperatures. 4. Here we show that in both red and white environments in which temperature varied between 22 and 32 degrees C, Paramecium coexisted with Colpidium. Consistent with the differential effects of temperature on their intrinsic growth rates, Paramecium population dynamics were largely unaffected by temperature regimes, and Colpidium showed more variable population dynamics in the red environments. 5. Temperature-dependent competitive effects of Colpidium on Paramecium, together with resource partitioning, appeared to be responsible for the coexistence in the white environments; resource partitioning and the storage effect appeared to account for the coexistence in the red environments. 6. These results suggest that temperature fluctuation may play important roles in regulating species coexistence and diversity in ecological communities.     

Modeling of oscillatory scenarios of the coexistence of competing populations

The scenarios of the formation of population distributions have been analyzed for a system of nonlinear reaction–diffusion–advection equations to describe the spatiotemporal distribution of predators and prey. The conditions that must be fulfilled for the model to belong to the class of cosymmetric systems were identified using an analytical approach. Computer simulations of a system with prey and two predators showed that the emergence of families of stationary distributions and oscillatory modes is possible when these conditions are met. The initial distributions of predators were shown to determine the character of the scenario (stationary or non-stationary) at certain combinations of parameters.