The effect of density-independent mortality on the coexistence of exploitative competitors for renewing resources

Many ecologists believe that higher mortality imposed on competing species increases the probability that they will coexist. This belief has persisted in spite of many theoretical counterarguments. However, few of those counterarguments have been based on models having explicit representation of the resources for which competition is occurring. This article analyzes a series of consumer-resource models of competition for nutritionally substitutable renewable resources and determines the range of relative resource requirements that allow coexistence. In most cases, if consumers are initially efficient at reducing resource densities, increasing density-independent mortality widens the range of resource requirements of the consumers that allow coexistence, provided the increase in mortality is not too great. The coexistence-promoting effects of mortality occur because a very efficient consumer species usually reduces the diversity of the set of resources it consumes. This lessens the extent to which resource utilization differences between consumer species can be expressed. Mortality, in this case, increases the diversity of resource types, widening the conditions for coexistence. However, sufficiently high mortality will usually reduce the range of parameters allowing coexistence, in agreement with much previous theory. The results presented here also predict maximal diversity at intermediate levels of productivity. Previous empirical studies and theory are reviewed in light of the theory developed here.     

Competition between two species for two complementary or substitutable resources

The classical models of interspecific competition are phenomenological, that is, they purport to describe the trajectories followed by the abundances of both competitors, without specifying either mechanisms or the dynamics of what the competition is for, i.e. resources. Yet the conditions for the different outcomes of competition inferred from these models are most often interpreted in terms of resources. Here it is contended that the dynamics of resource supply and exploitation must be explicitly taken into account if these conditions are to be obtained in any accurate form. First, the distinction between perfectly substitutable, imperfectly substitutable and perfectly complementary resources is examined. This leads to an array of models intended to mirror interspecific exploitative competition for two resources for each category considered. Then, the conditions for the coexistence of the two competitors in each case are derived and presented with direct reference to the modes of resource use.     

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.     

A stage-structured predator–prey model with distributed maturation delay and harvesting

ABSTRACT

A stage-structured predator–prey system with distributed maturation delay and harvesting is investigated. General birth and death functions are used. The local stability of each feasible equilibria is discussed. By using the persistence theory, it is proven that the system is permanent if the coexistence equilibrium exists. By using Lyapunov functional and LaSalle invariant principle, it is shown that the trivial equilibrium is globally stable when the other equilibria are not feasible, and that the boundary equilibrium is globally stable if the coexistence equilibrium does not exist. Finally, sufficient conditions are derived for the global stability of the coexistence equilibrium.     

Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210-219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine "interior" and "exterior" species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Kolmogorov-type competition model with finitely supported allocation profiles and its applications to plant competition for sunlight

A Kolmogorov-type competition model featuring allocation profiles, gain functions, and cost parameters is examined. For plant species that compete for sunlight according to the canopy partitioning model [R.R. Vance and A.L. Nevai, Plant population growth and competition in a light gradient: a mathematical model of canopy partitioning, J. Theor. Biol. 245 (2007), pp. 210–219] the allocation profiles describe vertical leaf placement, the gain functions represent rates of leaf photosynthesis at different heights, and the cost parameters signify the energetic expense of maintaining tall stems necessary for gaining a competitive advantage in the light gradient. The allocation profiles studied here, being supported on three alternating intervals, determine “interior” and “exterior” species. When the allocation profile of the interior species is a delta function (a big leaf) then either competitive exclusion or coexistence at a single globally attracting equilibrium point occurs. However, if the allocation profile of the interior species is piecewise continuous or a weighted sum of delta functions (multiple big leaves) then multiple coexistence states may also occur.     

Adaptive dynamics of competition for nutritionally complementary resources: character convergence, displacement, and parallelism

Consumers acquire essential nutrients by ingesting the tissues of resource species. When these tissues contain essential nutrients in a suboptimal ratio, consumers may benefit from ingesting a mixture of nutritionally complementary resource species. We investigate the joint ecological and evolutionary consequences of competition for complementary resources, using an adaptive dynamics model of two consumers and two resources that differ in their relative content of two essential nutrients. In the absence of competition, a nutritionally balanced diet rarely maximizes fitness because of the dynamic feedbacks between uptake rate and resource density, whereas in sympatry, nutritionally balanced diets maximize fitness because competing consumers with different nutritional requirements tend to equalize the relative abundances of the two resources. Adaptation from allopatric to sympatric fitness optima can generate character convergence, divergence, and parallel shifts, depending not on the degree of diet overlap but on the match between resource nutrient content and consumer nutrient requirements. Contrary to previous verbal arguments that suggest that character convergence leads to neutral stability, coadaptation of competing consumers always leads to stable coexistence. Furthermore, we show that incorporating costs of consuming or excreting excess nonlimiting nutrients selects for nutritionally balanced diets and so promotes character convergence. This article demonstrates that resource-use overlap has little bearing on coexistence when resources are nutritionally complementary, and it highlights the importance of using mathematical models to infer the stability of ecoevolutionary dynamics.     

Compartmentalization and niche differentiation: causal patterns of competition and coexistence

The current major models of coexistence of species on the same resources are briefly summarized. It is then shown that analysis of supposedly competitive systems in terms of the physical four dimensions of phase-space is sufficient to understand the causes for coexistence and for competitive exclusion. Thus, the multiple dimensions of niche theory are reduced to factors which define the magnitudes of the phase-spatial system, in particular the boundaries of population spaces and of periods of activity. Excluding possible cooperative interaction between consumers, it appears that coexistence of species on thesame kind of limiting resource is possible only in cases of compartmentalization either in space, or in time, of resource consumption, i.e. if each consumer species disposes of a separate resource supply. Three criteria were found to be decisive for successful compartmentalization (i.e. for coexistence): 1. the vector of the resource flow; 2. relative mobility between consumers and resource units; 3. dependence or independence of resource flow on previous consumption. The traditional terminology of niche theory and of competition theory in general proved to be inadequate for analytical treatment of problems of coexistence.     

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.     

Body size and species coexistence in consumer–resource interactions: A comparison of two alternative theoretical frameworks

Species coexistence involving trophic interactions has been investigated under two theoretical frameworks—partitioning shared resources and accessing exclusive resources. The influence of body size on coexistence is well studied under the exclusive resources framework, but has received less attention under the shared-resources framework. We investigate body-size-dependent allometric extensions of a classical MacArthur-type model where two consumers compete for two shared resources. The equilibrium coexistence criteria are compared against the general predictions of the alternative framework over exclusive resources. From the asymmetry in body size allometry of resource encounter versus demand our model shows, counterintuitively, and contrary to the exclusive resource framework, that a smaller consumer should be competitively superior across a wide range of supplies of the two resource types. Experimental studies are reviewed to resolve this difference among the two frameworks that arise from their respective assumptions over resource distribution. Another prediction is that the smaller consumer may have relatively stronger control over equilibrium resource abundance, and the loss of smaller consumers from a community may induce relatively stronger trophic cascades. Finally, from satiating consumers’ functional response, our model predicts that greater difference among resource sizes can allow a broader range of consumer body sizes to coexist, and this is consistent with the predictions of the alternative framework over exclusive resources. Overall, this analysis provides an objective comparison of the two alternative approaches to understand species coexistence that have heretofore developed in relative isolation. It advances classical consumer–resource theory to show how body size can be an important factor in resource competition and coexistence.     

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     

Coexistence of competitors in metacommunities due to spatial variation in resource growth rates; does R * predict the outcome of competition?

Simple mathematical models are used to investigate the coexistence of two consumers using a single limiting resource that is distributed over distinct patches, and that has unequal growth rates in the different patches. Relatively low movement rates or high demographic rates of an inefficient resource exploiter allow it to coexist at a stable equilibrium with a more efficient species whose ratio of movement to demographic rates is lower. The range of conditions allowing coexistence depends on the between‐patch heterogeneity in resource growth rates, but this range can be quite broad. The between‐patch movement of the more efficient consumer turns patches with high resource growth rates into sources, while low‐growth‐rate patches effectively become sinks. A less efficient species can coexist with or even exclude the more efficient species from the global environment if it is better able to bias its spatial distribution towards the source patches. This can be accomplished with density independent dispersal if the less efficient species has a lower ratio of per capita between‐patch movement rate to demographic rates. Conditions that maximize the range of efficiencies allowing coexistence of two species are: a relatively high level of heterogeneity in resource growth conditions; high dispersal (or low demographic rates) of the superior competitor; and low dispersal (or high demographic rates) of the inferior competitor. Global exclusion of the more efficient competitor requires that the inferior competitor have sufficient movement to also produce a source‐sink environment.     

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。     

Global stability of the coexistence equilibrium for a general class of models of facultative mutualism

Many models of mutualism have been proposed and studied individually. In this paper, we develop a general class of models of facultative mutualism that covers many of such published models. Using mild assumptions on the growth and self-limiting functions, we establish necessary and sufficient conditions on the boundedness of model solutions and prove the global stability of a unique coexistence equilibrium whenever it exists. These results allow for a greater flexibility in the way each mutualist species can be modelled and avoid the need to analyse any single model of mutualism in isolation. Our generalization also allows each of the mutualists to be subject to a weak Allee effect. Moreover, we find that if one of the interacting species is subject to a strong Allee effect, then the mutualism can overcome it and cause a unique coexistence equilibrium to be globally stable.     

On several conjectures from evolution of dispersal

We address several conjectures raised in Cantrell et al. [Evolution of dispersal and ideal free distribution, Math. Biosci. Eng. 7 (2010), pp. 17-36 [ 9 ]] concerning the dynamics of a diffusion-advection-competition model for two competing species. A conditional dispersal strategy, which results in the ideal free distribution of a single population at equilibrium, was found in Cantrell et al. [ 9 ]. It was shown in [ 9 ] that this special dispersal strategy is a local evolutionarily stable strategy (ESS) when the random diffusion rates of the two species are equal, and here we show that it is a global ESS for arbitrary random diffusion rates. The conditions in [ 9 ] for the coexistence of two species are substantially improved. Finally, we show that this special dispersal strategy is not globally convergent stable for certain resource functions, in contrast with the result from [ 9 ], which roughly says that this dispersal strategy is globally convergent stable for any monotone resource function.     

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.     

Character convergence under competition for nutritionally essential resources

Resource competition is thought to drive divergence in resource use traits (character displacement) by generating selection favoring individuals able to use resources unavailable to others. However, this picture assumes nutritionally substitutable resources (e.g., different prey species). When species compete for nutritionally essential resources (e.g., different nutrients), theory predicts that selection drives character convergence. We used models of two species competing for two essential resources to address several issues not considered by existing theory. The models incorporated either slow evolutionary change in resource use traits or fast physiological or behavioral change. We report four major results. First, competition always generates character convergence, but differences in resource requirements prevent competitors from evolving identical resource use traits. Second, character convergence promotes coexistence. Competing species always attain resource use traits that allow coexistence, and adaptive trait change stabilizes the ecological equilibrium. In contrast, adaptation in allopatry never preadapts species to coexist in sympatry. Third, feedbacks between ecological dynamics and trait dynamics lead to surprising dynamical trajectories such as transient divergence in resource use traits followed by subsequent convergence. Fourth, under sufficiently slow trait change, ecological dynamics often drive one of the competitors to near extinction, which would prevent realization of long-term character convergence in practice.     

Grassland-herbivore interactions: how do grazers coexist?

We develop a new approach to modeling grazing systems that links foraging characteristics (intake and digestive constraints) with resource dynamics via the probability of encounter with different grass heights. Three complementary models are presented: the generation of a grass height structure through selective grazing; investigating the conditions for consumer coexistence; and, using a simplified resource structure, the consequences for consumer abundance. The main finding is that coexistence between grazers differing in body size is possible if a single-resource type becomes differentiated in its height structure. Large grazers can facilitate food availability for smaller species but with the latter being competitively dominant. The relative preference given to different resource partitions is important in determining the nature of population interactions. Large-body and small-body grazer populations can interact through competitive, parasitic, commensalist, or amensalist relationships, depending on the way they partition the resource as well as their relative populations and the dynamics of resource renewal. The models provide new concepts of multispecies carrying capacity (stock equilibrium) in grazed systems with implications for conservation and management. We conclude that consumer species are not substitutable; therefore, the use of rangeland management concepts such as "livestock units" may be inappropriate.     

Transient dynamics limit the effectiveness of keystone predation in bringing about coexistence

We analyze the transient dynamics of simple models of keystone predation, in which a predator preferentially consumes the dominant of two (or more) competing prey species. We show that coexistence is unlikely in many systems characterized both by successful invasion of either prey species into the food web that lacks it and by a stable equilibrium with high densities of all species. Invasion of the predator-resistant consumer species often causes the resident, more vulnerable prey to crash to such low densities that extinction would occur for many realistic population sizes. Subsequent transient cycles may entail very low densities of the predator or of the initially successful invader, which may also preclude coexistence of finite populations. Factors causing particularly low minimum densities during the transient cycles include biotic limiting resources for the prey, limited resource partitioning between the prey, a highly efficient predator with relatively slow dynamics, and a vulnerable prey whose population dynamics are rapid relative to the less vulnerable prey. Under these conditions, coexistence of competing prey via keystone predation often requires that the prey's competitive or antipredator characteristics fall within very narrow ranges. Similar transient crashes are likely to occur in other food webs and food web models.     

Dynamics of a intraguild predation model with generalist or specialist predator

Intraguild predation (IGP) is a combination of competition and predation which is the most basic system in food webs that contains three species where two species that are involved in a predator/prey relationship are also competing for a shared resource or prey. We formulate two intraguild predation (IGP: resource, IG prey and IG predator) models: one has generalist predator while the other one has specialist predator. Both models have Holling-Type I functional response between resource-IG prey and resource-IG predator; Holling-Type III functional response between IG prey and IG predator. We provide sufficient conditions of the persistence and extinction of all possible scenarios for these two models, which give us a complete picture on their global dynamics. In addition, we show that both IGP models can have multiple interior equilibria under certain parameters range. These analytical results indicate that IGP model with generalist predator has “top down” regulation by comparing to IGP model with specialist predator. Our analysis and numerical simulations suggest that: (1) Both IGP models can have multiple attractors with complicated dynamical patterns; (2) Only IGP model with specialist predator can have both boundary attractor and interior attractor, i.e., whether the system has the extinction of one species or the coexistence of three species depending on initial conditions; (3) IGP model with generalist predator is prone to have coexistence of three species.