Necessary Condition For Coexistence Research Articles

Coexistence under Hierarchical Resource Exploitation: The Role of the R*-Preemption Trade-Off.

AbstractResource competition theory predicts coexistence and exclusion patterns based on species' R*s, the minimum resource values required for a species to persist. A central assumption of the theory is that all species have equal access to resources. However, many systems are characterized by preemption exploitation, where some species deplete resources before their competitors can access them (e.g., asymmetric light competition, contest competition among animals). We hypothesized that coexistence under preemption requires an R*-preemption trade-off-that is, the species with the priority access should have a higher R* (lower "efficiency"). Thus, we developed an extension of resource competition theory to investigate partial and total preemption (in the latter, the preemptor is unaffected by species with lower preemption rank). We found that an R*-preemption trade-off is a necessary condition for coexistence in all models. Moreover, under total preemption, the trade-off alone is sufficient for coexistence. In contrast, under partial preemption, more conditions are needed, which restricts the parameter space of coexistence. Finally, we discuss the implications of our finding for seemingly distinct trade-offs, which we view as special cases of the R*-preemption trade-off. These trade-offs include the digger-grazer trade-off, the competition-colonization trade-off, and trade-offs related to light competition between trees and understories.

Resource competition and technological diversity.

The work develops and investigates a mathematical model for evolution of the technological structure of an economic system where different technologies compete for the common essential resources. The model is represented by a system of consumer–resource rate equations. Consumers are technologies formalized as populations of weakly differentiated firms producing a similar commodity with like average output. Firms are characterized by the Leontief–Liebig production function in stock-flow representation. Firms self-replicate with a rate proportional to production output of the respective technology and dissolve with a constant rate of decay. The resources are supplied to the system from outside and consumed by concerned technologies; the unutilized resource amounts are removed elsewhere. The inverse of a per firm break-even resource availability is proposed to serve as a measure for competitiveness towards a given resource. The necessary conditions for coexistence of different technologies are derived, according to which each contender must be a superior competitor for one specific resource and an inferior competitor for the others. The model yields a version of the principle of competitive exclusion: in a steady state, the number of competing technologies cannot exceed the number of limiting resources. Competitive outcomes (either dominance or coexistence) in the general system of multiple technologies feeding on multiple essential resources are shown to be predictable from knowledge of the resource-dependent consumption and growth rates of each technological population taken alone. The proposed model of exploitative competition with explicit resource dynamics enables more profound insight into the patterns of technological change as opposed to conventional mainstream models of innovation diffusion.

Effects of pigment richness and size variation on coexistence, richness and function in light‐limited phytoplankton

Abstract Trait diversity is traditionally seen as promoting species richness and ecosystem function. Species with dissimilar traits would partition available resources, increasing niche differences, facilitating coexistence and increasing ecosystem function. Here we first show, using theory and simulations for light‐limited phytoplankton, that combing photosynthetic pigments is indeed a necessary condition for coexistence and stimulates ecosystem function. However, pigment richness does mostly not permit the coexistence of more than two species, and increases productivity at most 40% compared to single‐pigment communities. That is because blending in more pigments leads to coexistence of species with many pigments and therefore flat absorption spectra, which equalizes their fitness but decreases their niche differences. Similarly, seeding species with more variable size leads to an excess of large‐celled species, which does not only decrease fitness differences but also niche differences. Empirical data and additional simulations suggest that pigment richness effects can be stronger during transient dynamics but inevitably weaken with time, that is, pigment richness effects on species richness and function are likely short‐lived. Synthesis. Our results highlight the need to apply coexistence theory to understand the long‐term effects of trait diversity on biodiversity and ecosystem function.

Plant coexistence without asymmetry in competitor–colonizer abilities or spatial heterogeneity in resource distribution

AbstractSpecies coexistence is a central issue in plant community ecology. Among processes potentially promoting coexistence is the competition–colonization trade‐off: the idea that inferior competitors can coexist with superior competitors if they are better colonizers. The competition–colonization trade‐off usually assumes superior competitors have an impact on inferior competitors, but not vice versa. In the absence of spatial heterogeneity in resource distribution, this absolute asymmetry often is considered a necessary condition for coexistence. We used a spatially explicit, individual‐based model to demonstrate that coexistence is possible without strict asymmetry in competitor–colonizer abilities in a habitat in which distribution of resources is spatially homogeneous through time. Our results (1) indicated a strong tendency for spatial clustering of similar plant types (better competitor or better colonizer) due primarily to local intraspecific competition, (2) identified characteristics of competition–colonization trade‐offs that permit long‐term coexistence vs. those that lead to relatively rapid extinction of one plant type, and (3) suggested range of trade‐offs permitting coexistence decrease as a power function of length of competition time.

Positive solutions to the unstirred chemostat model with Crowley-Martin functional response

A food-chain model with Crowley-Martin functional response in the unstirred chemostat is considered. First, the global framework of coexistence solutions is discussed by the maximum principle and bifurcation theory. We obtain the sufficient and necessary conditions for coexistence of steady-state. Second, the stability and uniqueness of coexistence solutions are investigated by means of the combination of the perturbation theory and fixed point index theory. Our results indicate that if the magnitude of interference among predator is sufficiently large, the model has only one unique linearly stable coexistence solution when the maximal growth rate of predator belongs to certain range. Finally, some numerical simulations are carried out to verify and complement the theoretical results.

A necessary condition for coexistence of autocatalytic replicators in a prebiotic environment.

A necessary, but not sufficient, mathematical condition for the coexistence of short replicating species is presented here. The mathematical condition is obtained for a prebiotic environment, simulated as a fed-batch reactor, which combines monomer recycling, variable reaction order and a fixed monomer inlet flow with two replicator types and two monomer types. An extensive exploration of the parameter space in the model validates the robustness and efficiency of the mathematical condition, with nearly 1.7% of parameter sets meeting the condition and half of those exhibiting sustained coexistence. The results show that it is possible to generate a condition of coexistence, where two replicators sustain a linear growth simultaneously for a wide variety of chemistries, under an appropriate environment. The presence of multiple monomer types is critical to sustaining the coexistence of multiple replicator types.

Signature of ecological partitioning in the maintenance of damselfly diversity

1. Ecological differences among co-occurring taxa are often invoked as an explanation for the maintenance of biodiversity. Whether these differences facilitate coexistence, which allows unequal competitors to remain in systems and thus maintain biodiversity, is still unclear. 2. Here, we used observational and experimental studies to test for ecological partitioning in ways that would promote coexistence among three co-occurring damselfly genera. We evaluated two necessary conditions for coexistence: (i) that the damselfly genera differ in their abilities to engage in interactions with other damselfly genera and environmental conditions such that their relative abundances covary differently along environmental gradients and (ii) that an increase in intrageneric abundance is more detrimental to performance-related demographic features of each genus than increases in intergeneric abundances. 3. Observational studies across 40 lakes showed that relative abundances of each genus covaried differently along an environmental gradient of lake abiotic and biotic features consistent with ecological partitioning. Field experiments in which we manipulated both intra- and intergeneric densities demonstrated that per capita growth rates of each genus are negatively density-dependent and are only limited by increases in intra- not intergeneric densities. 4. Collectively, these results show a clear signature of ecological partitioning among each genus, which should prevent competitive exclusion and maintain each genus in this system. The results do not guarantee local coexistence among the three genera but are consistent with criteria that should promote their coexistence. Our results also suggest that a food web model coupling keystone predation and apparent competition is likely necessary to explain the ecological dynamics of persistence among these genera.

Exploring community assembly through an individual-based model for trophic interactions

Traditionally, the dynamics of community assembly has been analyzed by means of deterministic models of differential equations. Despite the theoretical advances provided by such models, they are restricted to questions about community-wide features. The individual-based modeling offers an opportunity to link bionomic features to patterns at the community scale, allowing us to understand how trait-based assembly rules can arise by dynamical processes. The present paper introduces an individual-based model of community assembly, and discusses some of the major advantages and drawbacks of this approach. The model was framed to deal with predation among size-structured populations, incorporating allometric constraints to energetic requirements, movement, life-history features and interaction relationships among individuals. A protocol of assembly procedure is proposed, in which a period of intense species introductions is followed by a period without introductions. The resultant communities did not present any pattern of trait over-dispersion, meaning that the multivariate distances of bionomic features among co-occurring species were neither larger nor more regular than expected in a random collection of species. It suggests a weak influence of interspecific interactions in the model environment and individualistic rules of coexistence, driven mainly by the spatial structure. This highlights that trait over-dispersion and resource partitioning should not be considered a necessary condition for coexistence, even in communities entirely structured by internal processes like predation and competition.

The complete classification for dynamics in a homosexually-transmitted disease model

A sexually-transmitted disease model for two strains of pathogen in a one-sex, heterogeneously-mixing population was proposed by Li et al. in (J Math Biol 10:1037-1052, 1986). The sufficient and necessary conditions for coexistence and the sufficient conditions for stability of the boundary equilibria were provided. This paper will present a thorough classification of dynamics for this model in terms of the first and second so called reproductive numbers of infection in strains I and J. This classification not only solves a conjecture proposed in (Li et al., J Math Biol 10:1037-1052, 1986) but also gives the sufficient and necessary conditions for the competitive exclusion.

Qualitative analysis for recurrent neural networks with linear threshold transfer functions

Multistable networks have attracted much interest in recent years, since multistability is of primary importance for some applications of recurrent neural networks where monostability exhibits some restrictions. This paper focuses on the analysis of dynamical property for a class of additive recurrent neural networks with nonsaturating linear threshold transfer functions. A milder condition is derived to guarantee the boundedness and global attractivity of the networks. Dynamical properties of the equilibria of two-dimensional networks are analyzed theoretically, and the relationships between the equilibria features and network parameters (synaptic weights and external inputs) are revealed. In addition, the sufficient and necessary conditions for coexistence of multiple equilibria are obtained, which confirmed the observations in with a cortex-inspired silicon circuit. The results obtained in this paper are applicable to both symmetric and nonsymmetric networks. Simulation examples are used to illustrate the theory developed in this paper.

Competition, Herbivory, and Enrichment: Nutrient-Based Models for Edible and Inedible Plants

Simple models representing nutrient-dependent growth of plant prey are used to study herbivores and edible and "inedible" plants. Three types of inedible plants are considered: those that have no direct interaction with herbivores, those that interfere with the ingestion of edible plants but are not themselves consumed, and those that are consumed but provide no nutritional benefit to herbivores. A necessary condition for coexistence at equilibrium applies for all types of inedible plants, which requires that edible plants be better competitors than inedible plants in the absence of herbivores but worse in their presence. All models predict compensatory changes in relative densities of edible and inedible plants at equilibrium as herbivore loss rate changes. If inedible plants do not interact with herbivores, then only inedible (not edible) plant biomass increases with nutrient supply; otherwise, biomasses of both edible and inedible plants increase with nutrient supply. If inedible plants interfere with ingestion of edible plants but are not consumed, then equilibria are stable in the parameter space examined, for a parameterized model of a planktonic system. In the other cases examined, equilibria are unstable in most of the parameter space examined.

The growth of pure and simple microbial competitors in a moving distributed medium

The dynamics of pure and simple competition between two microbial species are examined for the case of interaction arising in a distributed and nonstagnant environment. The environment is modeled as a tubular reactor. It is shown that for relatively small values of the dispersion coefficient (i.e., for small, but nonzero, backmixing of the medium), the two competing populations can coexist in a stable steady state. It has been assumed that the species grow uninhibited and that if there are maintenance requirements they are satisfied from endogenous sources. From numerical studies it has been found that a necessary condition for coexistence is that the net specific growth rate curves of the two competitors cross each other at a positive value of the concentration of the rate-limiting substrate. The model equations have been numerically solved by using the methods of orthogonal and spline collocation.

A Simple, Stable Model of Mutualism Incorporating Handling Time

Description d'un modele simple de mutualisme faisant intervenir 2 especes, qui prend en compte le temps de «manipulation» d'une ressource alimentaire (depuis sa rencontre jusqu'au debit de la recherche de nouvelles ressources)

Competition of two suspension-feeding protozoan populations for a growing bacterial population in continuous culture

Mathematical studies for ecosystems involving 2 predators competing for a growing prey population have shown that the 2 competitors can coexist in a state of sustained oscillations for a range of values of the system parameters. For the case of 1 suspension-feeding protozoan population, recent experimental observations suggest that the predator-prey interaction is complicated by the ability of the bacteria to grow on products produced by the lysis of protozoan cells. This situation is studied here for the case where 2 suspension-feeding protozoan populations compete for a growing bacterial population in a chemostat. Computer simulations show that the 2 protozoan populations can coexist over a range of the operating parameters. Some necessary conditions for coexistence are presented as are some speculations regarding the possible physical explanations of results.

Coexistence of two microbial populations competing for a renewable resource in a non-predator-prey system

If two microbial populations compete for a single resource in a homogeneous environment with time invariant inputs they cannot coexist indefinitely if the resource competed for is not renewed by biological activity within the system. Mathematical studies have shown that in a predator-prey system, where the resource (prey) is self-renewing, the two competitors (predators) can coexist in a limit cycle. This suggests that if the resource competed for is renewed by biological activity within the system coexistence can occur in any microbial system provided that it exhibits the same features as, but without being, a predator-prey one. A food chain involving commensalism, competition and amensalism is presented here. Two subcases are considered. It is only when maintenance effects are taken into account that coexistence, in limit cycles, can occur for this system. Limit cycle solutions for the system are demonstrated with the help of computer simulations. Some necessary conditions for coexistence are presented, as are some speculations regarding the possible physical explanations of the results.

Controllability of populations: The competitive exclusion principle

Necessary conditions for coexistence of competitive species described by a simple model of population growth and resource exploitation are derived using controllability and observability conditions for time-varying linear systems. The "competitive exclusion principle" of ecological theory follows from these conditions.

Bill Dimensions of the Three Species of Zosterops on Norfolk Island

Species pairs of birds on islands, originating from double invasions of single stocks from other islands or mainland, differ greatly in bill size where they are sympatric. Differences in bill length have been found to range from 15.6 to 50% (Grant 1968). This has been interpreted to indicate that substantial differences in diet, presumed to be associated with the differences in bill length, are necessary for the sustained coexistence of two species derived in the above manner. One of the species pairs used in the analysis was Zosterops albogularis and Z. tenuirostris on Norfolk Island. Mean measurements given by Lack (1944) were found to yield a difference of 16.7%. Mees (1970) has recently pointed out that these measurements are not correctly representative of the populations. They are from single individuals of each species, and were made much earlier by G. Matthews. Mees's own more extensive, and hence more representative, measurements yield a difference in mean bill length of no more than 4.5%. Such a small difference renders these two Zosterops species an exception to the empirical rule that sympatric, double-invasion, species pairs of birds differ in bill length by at least 15%.1 Further, it raises a doubt that appreciable differences in diet are a necessary condition for coexistence. However, dimensions other than length are as important or more so in contributing to the efficiency of the bill in dealing with foods (Grant 1968). Differences in other dimensions may reflect differences in diet more accurately for some species than differences in length. Only length was used

Invalidation of principle of competitive exclusion defended.

Gause and Witt1 give as necessary conditions for the coexistence of two species living in the same microcosm α K1/K2 and β>K2/K1. The two species of Drosophila coexist even though there is a “strong mutual depression”. For example, the sizes of D. pseudoobscura AR and of D. serrata are 252 and 278 flies respectively in the competing populations; in populations of single species their respective sizes are 664 and 1,251 flies. This result warrants the rejection of the principle of competitive exclusion, “two species competing for limiting resources can only coexist if they inhibit the growth of the competing species less than they inhibit their own growth”. Similarly this experimental result warrants the rejection of the competition equations from which the necessary conditions for coexistence stated above are derived.