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.     

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.     

Multiple limit cycles in the standard model of three species competition for three essential resources

We consider the dynamics of the standard model of 3 species competing for 3 essential (non-substitutable) resources in a chemostat using Liebig's law of the minimum functional response. A subset of these systems which possess cyclic symmetry such that its three single-population equilibria are part of a heteroclinic cycle bounding the two-dimensional carrying simplex is examined. We show that a subcritical Hopf bifurcation from the coexistence equilibrium together with a repelling heteroclinic cycle leads to the existence of at least two limit cycles enclosing the coexistence equilibrium on the carrying simplex- the ``inside' one is an unstable separatrix and the ``outside' one is at least semi-stable relative to the carrying simplex. Numerical simulations suggest that there are exactly two limit cycles and that almost every positive solution approaches either the stable limit cycle or the stable coexistence equilibrium, depending on initial conditions. Bifurcation diagrams confirm this picture and show additional features. In an alternative scenario, we show that the subcritical Hopf together with an attracting heteroclinic cycle leads to an unstable periodic orbit separatrix. This research was partially supported by NSF grant DMS 0211614. KY 40292, USA. This author's research was supported in part by NSF grant DMS 0107160     

Cyclic competition of three species in the time periodic and diffusive case

We consider a semilinear reaction diffusion system with time periodic coefficients. The system models the competitive interaction of three species, which inhabit a bounded domain. We make assumptions that may be loosely stated (cyclically for i {1, 2, 3}) as: Species i outcompetes species i + 1 in the absence of species i + 2. Under those assumptions we prove the existence of a time periodic solution, which is strictly positive in each of its three components. Moreover, we obtain a new result on the nonexistence of positive time-periodic solutions and the extinction of one species for the related two-species subsystem. Our discussion includes a situation, known as cyclic competition in the autonomous ODE-case, in a more general framework that includes temporal and spatial heterogeneity.     

Interspecific competition and coexistence of the two chrysomelids,Gastrophysa atrocyanea motschulsky andGalerucella vittaticollis baly (Coleoptera: Chrysomelidae), under limited food resource conditions

Field experiments were conducted in order to investigate the mode of exploitation of food resources and the mechanism of coexistence of mixed larval populations of the two chrysomelids,Gastrophysa atrocyanea andGalerucella vittaticollis, under limited food resource conditions. The larval survival rates seemed high enough to assure coexistence when hatchlings of the two species were released in 1∶1 and 1∶3 ratios on a host plant. However, the survival rate became almost nil for both species when a 3∶1 ratio was employed, suggesting asymmetrical interspecific competition. Wasted food consumption was much higher inG. atrocyanea larvae. The population ofG. atrocyanea seemed to be regulated more by intraspecific competition, while on the other hand, the population ofG. vittaticollis was considered to be more likely affected by the interspecific competition withG. atrocyanea, depending on the initial ratio and density of the two species.     

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.     

On the coexistence of three predatory stonefly species in a central Japanese stream

Three large-bodied stonefly species (Paragnetina tinctipennis, Oyamia lugubris, and Kamimuria tibialis) coexist in a central Japanese stream. These species have been classified as predators. Here we study their microhabitat use while focusing on the physical environments, physiological activity, and food resources. We show that Paragnetina uses a niche with faster currents than other species throughout the year. Oyamia has seasonal flexibility in microhabitat preference, physiological activity, and food resources. Kamimuria is a rather stable species, independent of seasonal patterns.     

Spatial and temporal patterns of coexistence between competing <Emphasis Type="Italic">Aedes</Emphasis> mosquitoes in urban Florida

Understanding mechanisms fostering coexistence between invasive and resident species is important in predicting ecological, economic, or health impacts of invasive species. The mosquito Aedes aegypti coexists at some urban sites in southeastern United States with invasive Aedes albopictus, which is often superior in interspecific competition. We tested predictions for three hypotheses of species coexistence: seasonal condition-specific competition, aggregation among individual water-filled containers, and colonization–competition tradeoff across spatially partitioned habitat patches (cemeteries) that have high densities of containers. We measured spatial and temporal patterns of abundance for both species among water-filled resident cemetery vases and experimentally positioned standard cemetery vases and ovitraps in metropolitan Tampa, Florida. Consistent with the seasonal condition-specific competition hypothesis, abundances of both species in resident and standard cemetery vases were higher early in the wet season (June) versus late in the wet season (September), but the proportional increase of A. albopictus was greater than that of A. aegypti, presumably due to higher dry-season egg mortality and strong wet-season competitive superiority of larval A. albopictus. Spatial partitioning was not evident among cemeteries, a result inconsistent with the colonization-competition tradeoff hypothesis, but both species were highly independently aggregated among standard cemetery vases and ovitraps, which is consistent with the aggregation hypothesis. Densities of A. aegypti but not A. albopictus differed among land use categories, with A. aegypti more abundant in ovitraps in residential areas compared to industrial and commercial areas. Spatial partitioning among land use types probably results from effects of land use on conditions in both terrestrial and aquatic-container environments. These results suggest that both temporal and spatial variation may contribute to local coexistence between these Aedes in urban areas.     

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.     

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.     

An expanded modern coexistence theory for empirical applications

Understanding long‐term coexistence of numerous competing species is a longstanding challenge in ecology. Progress requires determining which processes and species differences are most important for coexistence when multiple processes operate and species differ in many ways. Modern coexistence theory (MCT), formalised by Chesson, holds out the promise of doing that, but empirical applications remain scarce. We argue that MCT's mathematical complexity and subtlety have obscured the simplicity and power of its underlying ideas and hindered applications. We present a general computational approach that extends our previous solution for the storage effect to all of standard MCT's spatial and temporal coexistence mechanisms, and also process‐defined mechanisms amenable to direct study such as resource partitioning, indirect competition, and life history trade‐offs. The main components are a method to partition population growth rates into contributions from different mechanisms and their interactions, and numerical calculations in which some mechanisms are removed and others retained. We illustrate how our approach handles features that have not been analysed in the standard framework through several case studies: competing diatom species under fluctuating temperature, plant–soil feedbacks in grasslands, facilitation in a beach grass community, and niche differences with independent effects on recruitment, survival and growth in sagebrush steppe.     

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.     

Competition for foraging resources and coexistence of two syntopic species of Messor harvester ants in Mediterranean grassland

1. Diet composition of two syntopic species of Messor seed‐harvester ants (M. wasmanni Krausse and M. minor André) was evaluated during different periods over the year (May, July, October), by analysing food type (plant parts and species) and food size (weight, length, width). Morphological traits of foragers (head width and femur length) considered important features promoting diet partitioning were measured. 2. We used two robust randomisation algorithms (RA2 and RA3), adopted in niche overlap studies, to check for random vs non random utilisation of resources at intra‐ and interspecific level for the different periods. 3. Analyses showed high levels of overlap in the diet of the two species and no evidence of interspecific competition during most of the activity season. In particular, there was an aggregated use of resources in summer, whilst niche partitioning and evidence of competition when resources decreased in autumn. Intraspecifically, no evidence of competition was found. 4. Results suggest two different mechanisms for minimising competition: when food resources are abundant (summer), ants collected the same plant species but selected different sizes; when food resource is scarce (autumn), ants foraged on different plants. 5. The importance of different factors (morphological, behavioural, ecological) possibly affecting competition and coexistence are discussed.     

Spatial mechanisms for coexistence of species sharing a common natural enemy

We examine the conditions under which spatial structure can mediate coexistence of apparent competitors. We use a spatially explicit, host-parasitoid metapopulation model incorporating local dynamics of Nicholson-Bailey type and global dispersal. Depending on the model parameters, the resulting system displays a plethora of asynchronous dynamical behaviors for which permanent or transient coexistence is observed. We identify a number of spatially mediated tradeoffs which apparent competitors can utilize and demonstrate that the dynamics of spatial coexistence can typically be understood from consideration of two and three patch systems. The phase relationships of species abundances are different for our model than for some other mechanisms of spatial coexistence. We discuss the implications of our findings relative to issues of community organization and biological conservation.     

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.     

Body size mediated coexistence of consumers competing for resources in space

Body size is a major phenotypic trait of individuals that commonly differentiates co-occurring species. We analyzed inter-specific competitive interactions between a large consumer and smaller competitors, whose energetics, selection and giving-up behaviour on identical resource patches scaled with individual body size. The aim was to investigate whether pure metabolic constraints on patch behaviour of vagile species can determine coexistence conditions consistent with existing theoretical and experimental evidence. We used an individual-based spatially explicit simulation model at a spatial scale defined by the home range of the large consumer, which was assumed to be parthenogenic and semelparous. Under exploitative conditions, competitive coexistence occurred in a range of body size ratios between 2 and 10. Asymmetrical competition and the mechanism underlying asymmetry, determined by the scaling of energetics and patch behaviour with consumer body size, were the proximate determinant of inter-specific coexistence. The small consumer exploited patches more efficiently, but searched for profitable patches less effectively than the larger competitor. Therefore, body-size related constraints induced niche partitioning, allowing competitive coexistence within a set of conditions where the large consumer maintained control over the small consumer and resource dynamics. The model summarises and extends the existing evidence of species coexistence on a limiting resource, and provides a mechanistic explanation for decoding the size-abundance distribution patterns commonly observed at guild and community levels.     

Population biology of multispecies helminth infection: competition and coexistence

The role that interspecific interactions play in shaping parasite communities is uncertain. To date, models of competition between helminth species have assumed that interaction occurs through parasite-induced host death. To our knowledge, there has been no theoretical exploration of other forms of competition. We examine models in which competition acts at the point of establishment within the host, and at the time of egg production by the adult worm. The models used are stochastic and we allow hosts to vary in their rate of exposure to infective larvae. We derive the Lotka-Volterra model of competition when exposure is homogenous and thus demonstrate that two helminth species cannot coexist on a single limiting resource. We show that coexistence of species is promoted by heterogeneity in host exposure provided that the rates of exposure to the two species are not perfectly correlated, and, if they are positively correlated, provided that the degree of heterogeneity in host exposure is similar for the two competing helminth species. These results are robust to the mechanism of competition.     

具有有限时滞的Lotka—Volterra方程出现周期解分歧现象的条件

本文研究了方程组我们给出了该方程组的解出现周期分歧现象的一个条件。     

A positive trait-mediated indirect effect involving the natural enemies of competing herbivores

Trait-mediated indirect effects can have important effects on food web dynamics but are still poorly understood in the field. In a previous population cage study of a small community of aphids and an aphid natural enemy it was found that a trait-mediated indirect effect involving the natural enemy’s behaviour was key to understanding community persistence. Here evidence is presented that a related phenomenon involving some of the same species occurs in the field. Surveys showed that two species of aphid (Acyrthosiphon pisum and Megourella purpurea) tended to share a host plant with a third generally unpalatable species (Megoura viciae) more often than expected by chance. Further evidence suggested this was not due to differential plant suitability or location, but to a positive effect of M. viciae on the performance of the other two species. To test this, field experiments were set up comparing the size and persistence of A. pisum colonies sharing or not sharing a plant individual with M. viciae colonies. A. pisum colonies tended to be larger and persisted for a longer period of time in the presence of M. viciae, an effect that was significant for small colonies exposed to many predators. When protected from predation the presence of M. viciae had no effect on A. pisum colonies. The positive effects of M. viciae on A. pisum is thus likely to be natural-enemy mediated rather than plant mediated. How predation by Syrphidae, the major group observed in the study, is affected by M. viciae is discussed. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.