Integrating nested spatial scales: implications for the coexistence of competitors on a patchy resource

1. Intraspecific aggregation at a single spatial scale can promote the coexistence of competitors. This paper demonstrates how this same mechanism can be applied to the many systems that are patchy at two scales, with patches nested within 'superpatches'.
2. Data are presented from a field study showing that insects living in rotting fruits have aggregated distributions in the fruits under a single tree, and that the mean density and degree of aggregation varies significantly among trees. Observations in this system motivate the following models.
3. A model of competition has been developed between two species which explicitly represents spatial variation at two scales. By integrating the probability distributions for each scale, the marginal distributions of competitors over all patches can be found and used to calculate coexistence criteria. This model assumes global movement of the competitors.
4. Although spatial variation at a single scale may not be sufficient for coexistence, the total variation over all patches can allow coexistence. Variation in mean densities among superpatches and variation in the degree of aggregation among superpatches both promote coexistence, but act in different ways.
5. A second model of competition between two species is described which incorporates the effects of limited movement among superpatches. Limited movement among superpatches generally promotes coexistence, and also leads to correlations among aggregation and the mean densities of competitors.     

Population size dependence, competitive coexistence and habitat destruction

1. Spatial dynamics can lead to coexistence of competing species even with strong asymmetric competition under the assumption that the inferior competitor is a better colonizer given equal rates of extinction. Patterns of habitat fragmentation may alter competitive coexistence under this assumption.
2. Numerical models were developed to test for the previously ignored effect of population size on competitive exclusion and on extinction rates for coexistence of competing species. These models neglect spatial arrangement.
3. Cellular automata were developed to test the effect of population size on competitive coexistence of two species, given that the inferior competitor is a better colonizer. The cellular automata in the present study were stochastic in that they were based upon colonization and extinction probabilities rather than deterministic rules.
4. The effect of population size on competitive exclusion at the local scale was found to have little consequence for the coexistence of competitors at the metapopulation (or landscape) scale. In contrast, population size effects on extinction at the local scale led to much reduced landscape scale coexistence compared to simulations not including localized population size effects on extinction, especially in the cellular automata models. Spatially explicit dynamics of the cellular automata vs. deterministic rates of the numerical model resulted in decreased survival of both species. One important finding is that superior competitors that are widespread can become extinct before less common inferior competitors because of limited colonization.
5. These results suggest that population size–extinction relationships may play a large role in competitive coexistence. These results and differences are used in a model structure to help reconcile previous spatially explicit studies which provided apparently different results concerning coexistence of competing species.     

Starve a competitor: evolution of luxury consumption as a competitive strategy

Organisms are often observed to acquire an excess of non-limiting resources, a process known as luxury consumption. Luxury consumption has been largely treated as a bet hedging strategy for temporal variation in resource supply, but may also function as a competitive strategy. We incorporate luxury resource consumption into a derivation of the classic resource ratio model for competition between terrestrial plant, and explore its consequences for population dynamics and competition. We show that luxury consumption reduces the potential for coexistence between two species competing for two resources. Furthermore, we demonstrate that luxury consumption can be selected for because of the competitive advantage that luxury consumers gain. Luxury consumption evolves when competition for resources is local rather than global, there is potential for coexistence between the two species and the competitive environment remains stable over a sufficient period of time to allow selection to act. The evolutionary outcome can be either extinction of one of the competing species or coexistence of the two species with maximum luxury consumption. The potential for selection to favor luxury consumption is well predicted by the competitive outcome between individuals of the two species with and without luxury consumption.     

Fusing spatial resource heterogeneity with a competition–colonization trade-off in model communities

Two commonly cited mechanisms of multispecies coexistence in patchy environments are spatial heterogeneity in competitive abilities caused by variation in resources and a competition–colonization trade-off. In this paper, a model that fuses these mechanisms together is presented and analyzed. The model suggests that spatial variation in resource ratios can lead to multispecies coexistence, but this mechanism by itself is weak when the number of resources for which species compete is small. However, spatial resource heterogeneity is a powerful mechanism for multispecies coexistence when it acts synergistically with a competition–colonization trade-off. The model also shows how resource supply can control the competitive balance between species that are weak competitors but superior colonizers and strong competitors/inferior colonizers. This provides additional theoretical support for a possible explanation of empirically observed hump-shaped relationships between species diversity and ecological productivity.     

Community robustness in discrete-time periodic environments

The influence of seasonal environmental variation on species coexistence is an ecologically important factor. Its two aspects are how seasonal variation contributes to coexistence mechanisms, and, given a seasonally varying coexistence pattern, how sensitive that coexistence is to nonstationary external influences (such as climate change). Here we develop a formula for calculating the robustness of discrete-time periodic dynamics. Robustness is defined as the sensitivity of the position of the cycle in phase space to varying model parameters. Though the results are different, the main biological conclusions are in line with those from a similar study concerning continuous-time cycles (Barabás et al., 2012a): species segregation in the timing of resource use or predator avoidance increases community robustness in a way that is analogous to the effects of resource partitioning. We also connect this formalism with the widely used and successful framework of Chesson (1994), demonstrating that the merging of these two perspectives yields simplified expressions for robustness more amenable to analytical treatment. As an example, we apply our results to a two-cycle in a model of two competing annual plants with seedbanks, using our formulas to calculate the range of parameters that allow for the coexistence of the competitors. This helps us understand which components of the environmental variation the coexistence is sensitive to; in our case, the model is fairly robust against changing seed survival, moderately so against changing the variance in seed germination, and quite sensitive to changing the mean seed germination rates.     

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。     

Coexistence under positive frequency dependence

Negative frequency dependence resulting from interspecific interactions is considered a driving force in allowing the coexistence of competitors. While interactions between species and genotypes can also result in positive frequency dependence, positive frequency dependence has usually been credited with hastening the extinction of rare types and is not thought to contribute to coexistence. In the present paper, we develop a stochastic cellular automata model that allows us to vary the scale of frequency dependence and the scale of dispersal. The results of this model indicate that positive frequency dependence will allow the coexistence of two species at a greater rate than would be expected from chance. This coexistence arises from the generation of banding patterns that will be stable over long time-periods. As a result, we found that positive frequency-dependent interactions over local spatial scales promote coexistence over neutral interactions. This result was robust to variation in boundary conditions within the simulation and to variation in levels of disturbance. Under all conditions, coexistence is enhanced as the strength of positive frequency-dependent interactions is increased.     

Positive interactions among competitors can produce species-rich communities

Although positive interactions between species are well documented, most ecological theory for investigating multispecies coexistence remains rooted in antagonistic interactions such as competition and predation. Standard resource-competition models from this theory predict that the number of coexisting species should not exceed the number of factors that limit population growth. Here I show that positive interactions among resource competitors can produce species-rich model communities supported by a single limiting resource. Simulations show that when resource competitors reduce each others' per capita mortality rate (e.g. by ameliorating an abiotic stress), stable multispecies coexistence with a single resource may be common, even while the net interspecific interaction remains negative. These results demonstrate that positive interactions may provide an important mechanism for generating species-rich communities in nature. They also show that focusing on the net interaction between species may conceal important coexistence mechanisms when species simultaneously engage in both antagonistic and positive interactions.     

Predator selectivity alters the effect of dispersal on coexistence among apparent competitors

While the majority of studies on dispersal effects on patterns of coexistence among species in a metacommunity have focused on resource competitors, dispersal in systems with predator–prey interactions may provide very different results. Here, we use an analytical model to study the effect of dispersal rates on coexistence of two prey species sharing a predator (apparent competition), when the traits of that predator vary. Specifically, we explore the range in immigration rates where apparent competitors are able to coexist, and how that range changes with predator selectivity and efficiency. We find that if the inferior apparent competitor has a higher probability of being consumed, it will require less immigration to invade and to exclude the superior prey as the predator becomes more opportunistic. However, if the inferior apparent competitor has a lower probability of being consumed (and lower growth rates), higher immigration is required for the inferior prey to invade and exclude the superior prey as the predator becomes more opportunistic. We further find that the largest range of immigration rates where prey coexist occurs when predator selectivity is intermediate (i.e. they do not show much bias towards consuming one species or the other). Increasing predator efficiency generally reduces the immigration rates necessary for the inferior apparent competitor to invade and exclude the superior apparent competitor, but also reduces the range of immigration rates where the two apparent competitors can coexist. However, when the superior apparent competitor has a higher probability of being consumed, increased predator efficiency can increase the range of parameters where the species can coexist. Our results are consistent with some of the variation observed in the effect of dispersal on prey species richness in empirical systems with top predators.     

Coexistence of cycling and dispersing consumer species: Armstrong and McGehee in space

Two competing consumer species may coexist using a single homogeneous resource when the more efficient consumer--the one having the lowest equilibrium resource density--has a more nonlinear functional response that generates consumer-resource cycles. We extend this model of nonequilibrium coexistence, as proposed by Armstrong and McGehee, by putting the interaction into a spatial context using two frameworks: a spatially explicit individual-based model and a spatially implicit metapopulation model. We find that Armstrong and McGehee's mechanism of coexistence can operate in a spatial context. However, individual-based simulations suggest that decreased dispersal restricts coexistence in most cases, whereas differential equation models of metapopulations suggest that a low rate of dispersal between subpopulations often increases the coexistence region. This difference arises in part because of two potentially opposing effects on coexistence due to the asynchrony in the temporal dynamics at different locations. Asynchrony implies that the less efficient species is more likely to be favored in some spatial locations at any given time, which broadens the conditions for coexistence. On the other hand, asynchrony and dispersal can also reduce the amplitude of local population cycles, which restricts coexistence. The relative influence of these two effects depends on details of the population dynamics and the representation of space. Our results also demonstrate that coexistence via the Armstrong-McGehee mechanism can occur even when there is little variation in the global densities of either the consumers or the resource, suggesting that empirical studies of the mechanisms should measure densities on several spatial scales.     

Competition-colonization trade-offs in a ciliate model community

There is considerable theoretical evidence that a trade-off between competitive and colonization ability enables species coexistence. However, empirical studies testing for the presence of a competition–colonization (CC) trade-off and its importance for species coexistence have found mixed results. In a microcosm experiment, we looked for a CC trade-off in a community of six benthic ciliate species. For each species, we measured the time needed to actively disperse to and colonize an empty microcosm. By measuring dispersal rates and growth rates of the species, we were able to differentiate between these two important components of colonization ability. Competitive ability was investigated by comparing species’ growth with or without a competitor in all pairwise species combinations. Species significantly differed in their colonization abilities, with good colonizers having either high growth rates or high dispersal rates or both. Although species showed a clear competitive hierarchy, competitive and colonization ability were uncorrelated. The weakest competitors were also the weakest colonizers, and the strongest competitor was an intermediate colonizer. However, some of the inferior competitors had higher colonization abilities than the strongest competitor, indicating that a CC trade-off may enable coexistence for a subset of the species. Absence of a community-wide CC trade-off may be based on the lack of strong relationships between the traits underlying competitive and colonization ability. We show that temporal effects and differential resource use are alternative mechanisms of coexistence for the species that were both slow colonizers and poor competitors.     

Recycling‐mediated facilitation and coexistence based on plant size

We introduce nutrient recycling into a model where competitors differ in the scale at which they perceive their environment. In a two-resource system with both external nutrient inputs and recycling, larger consumers ("integrators") often generate resource distributions that favor their smaller ("nonintegrator") competitors, and vice versa. This occurs because recycling of integrator biomass reduces between-patch resource heterogeneity, whereas recycling of nonintegrator biomass does not. Combined, recycling and throughput can allow coexistence when it is not possible with either alone. With recycling, the presence of an integrator also may facilitate higher biomass of a co-occurring nonintegrator. Our model provides a context where recycling can generate negative feedback between competitors that differ in size and so promote coexistence. This is opposite to the positive recycling-mediated feedback commonly expected on the basis of litter chemistry differences between competitors. Effects of recycling and homogenization on nonintegrators may also be negative in our model, depending on the conformation of the system's resource supply points and the species' relative resource requirements. Our model suggests that the effects of plant size on competitive outcomes may depend critically on the degree of resource recycling found in the system and, reciprocally, that the effects of recycling may depend on plant size.     

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 Lotka-Volterra competition equations

The Lotka-Volterra competition equations with periodic coefficients derived from the MacArthur-Levins theory of a one-dimensional resource niche are studied when the parameters are allowed to oscillate periodically in time. Specifically, niche positions and widths, resource availability and resource consumption rates are allowed small amplitude periodicities around a specified mean value. Two opposite cases are studied both analytically and numerically. First only resource consumption rates are allowed to oscillate while niche dimensions and resource availability are held constant. The resulting oscillations in population densities and the strength of the system stability as they depend upon crucial relative phase and amplitude differences between the species' consumption rates are studied. This leads to a clear notion of "temporal niche" and of the effects that such oscillations can have on competitive coexistence. Secondly, all system parameters are allowed to oscillate, although the oscillatory consumption rates are assumed identical for both species. The effects on the population density oscillations and their averages are studied and the "best" choice of the common, periodic resource consumption rate for these two "identical" species competing for similar (even identical) niches is considered.     

Community robustness and limiting similarity in periodic environments

Temporal environmental variation has long been considered as one of the potential factors that could promote species coexistence. A question of particular interest is how the ecology of fluctuating environments relates to that of equilibrium systems. Equilibrium theory says that the more similar two species are in their modes of regulation, the less robust their coexistence will be; that is, the volume of external parameters for which all populations persist shrinks with increasing similarity. In this study, we will attempt to generalize these results to temporally varying situations and establish the precise mathematical relationship between the two. Our treatment considers unstructured populations in continuous time with periodic attractors of fixed period length, where the periodic behavior is due to external forcing. Within these conditions, our treatment is general. We provide a coherent theoretical framework for defining measures of species similarity and niche. Our main conclusion is that all factors that function to regulate population growth may be considered as separate regulating factors for each moment of time. In particular, a single resource becomes a resource continuum, along which species may segregate in the same manner as along classical resource continua. Therefore, we provide a mathematical underpinning for considering fluctuation-mediated coexistence as temporal niche segregation.     

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.     

Effects of competition, predation, and dispersal on species richness at local and regional scales

This study explores the consequences of predator-mediated coexistence among competitors for patterns of incidence and diversity at local and regional scales. We develop a model that draws on elements of metapopulation models of competitors and food chains by allowing competitors to coexist locally in the presence of predators but not in their absence. The model predicts that predators promote regional coexistence by greatly expanding the range of conditions under which two competitors persist at equilibrium. Predators could have positive or negative effects on mean local diversity within the region depending on their dispersal rates, those of the prey, and their effects on prey extinction rates. The presence of predators increased the abundance of inferior competitors, thereby expanding the conditions for positive relationships between local and regional diversity. The model also predicted positive correlations between local diversity of predators and prey. These predictions were supported by patterns of phytoplankton, zooplankton, and fish species richness among lakes. The model may help to resolve the apparent contrast between linear patterns of local and regional richness and experimental evidence for strong invasion resistance and rapid dispersal in zooplankton.     

Spatiotemporal population distributions and their implications for species coexistence in a variable environment

A population experiences environmental variation both directly, through effects on life history parameters such as fecundity, and indirectly, through effects on the population distributions of competitors and thus on the distribution of competition. Which spatial and temporal scales of environmental variation most influence the coexistence of two species thus depends in part on the degree to which the resident population responds to different scales of variation. In this paper, I calculate an approximation for a spatiotemporal population distribution as the result of a filter function convolved with the environmental variation. I find that there is no straightforward connection between spatial or temporal scales inherent to an organism's life history, such as mean lifetime or dispersal distance, and the population's sensitivity to variation at different scales. Rather, life history traits interact sensitively with the way environmental variation affects the organism. I comment on the implications for variation-mediated coexistence.     

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.     

Aggregation, intraguild interactions and the coexistence of competitors on small ephemeral patches

It is well established that intraspecific aggregation has the potential to promote coexistence in communities of species competing for patchy ephemeral resources. We developed a simulation model to explore the influence of aggregation on coexistence in such communities when an important assumption of previous studies – that interspecific interactions have only negative effects on the species involved – is relaxed. The model describes a community of competing insect larvae in which an interaction that is equivalent to intraguild predation (IGP) can occur, and is unusual in that it considers species exploiting very small resource patches (carrying capacity=1). Model simulations show that, in the absence of any intraspecific aggregation, variation between species in the way that resource heterogeneity affects survival increases the likelihood of species coexistence. Simulations also show that intraspecific aggregation of the dominant competitor's eggs across resource patches can promote coexistence by reducing the importance of interspecific competition relative to that of intraspecific competition. Crucially, however, this effect is altered if one competitor indulges in IGP. In general, coexistence is only possible when the species that is capable of IGP is less effective at exploiting the shared resource than its competitor. Because it reduces the relative importance of interspecific interactions, intraspecific aggregation of the eggs of a species that is the victim of IGP actually reduces the likelihood of coexistence in parts of parameter space in which the persistence of the other species is dependent on its ability to exploit its competitor. Since resource heterogeneity, intraspecific aggregation and IGP are all common phenomena, these findings shed light on mechanisms that are likely to influence diversity in communities exploiting patchy resources.