Competition of two microbial populations for a single resource in a chemostat when one of them exhibits wall attachment

It is known that two microbial populations competing for a single resource in a homogeneous environment with time-invariant inputs cannot coexist in a steady state. The case where two microbial populations compete for a single resource in a chemostat but one of them exhibits attachment to the chemostat walls is studied theoretically. Because of the cells' attachment to the walls, the environment is no longer homogeneous. The present article considers the case where the attached cells form no more than a monolayer. Other situations occur, often frequently, but we do not consider them here. Two models are used to represent the attachment to the walls: the Topiwala-Hamer model and a model which assumes that the attachment of microbial cells to the solid surfaces is a reversible process. The first model does not allow the population that exhibits wall attachment to wash out from the chemostat, in contrast to the second model (which nevertheless reduces to the first one in the limit). It has been found that in most of the possible cases for both models, the two competitors can coexist in a stable steady state for a wide range of the operating parameters space. The results of the stability analysis are discussed and analytical expressions for the conditions and the boundaries of the domains of stable coexistence are given for all the possible situations that may arise.     

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.     

Optimal cell size for resource uptake in fluids: a new facet of resource competition

Planktonic microorganisms are affected by various size-dependent processes both from the bottom up and from the top down. We developed a simple resource-consumer model to explore how size-dependent resource uptake and resource loss influence the growth of, and competition between, planktonic microorganisms. We considered three steps of resource uptake: diffusive transport of resource molecules, uptake by membrane transporters, and cellular enzymatic catalysis, and we investigated optimal cell size when one, two, or three of those steps limit resource uptake. Optimal cell size depends negatively on the size of resource molecules when resource uptake is limited by diffusive transport and membrane uptake. When competing for two resources of different molecular sizes, two different-sized consumers can coexist if the inputs of resources and sizes of consumers are correctly chosen. The model suggests that mixtures of various-sized resources can promote coexistence and size diversity of microorganisms even if the availability of one element, such as carbon, nitrogen, or phosphorus, limits the whole community. Model predictions include that bacteria grown on maltose or polysaccharides should be smaller compared with those grown on glucose under carbon limitation. Our results suggest that size of resource molecules can be an important factor in microbial resource competition in aquatic environments.     

Spatial self-organization in a cyclic resource-species model

Biological communities are remarkable in their ability to form cooperative ensembles that lead to coexistence through various types of niche partitioning, usually intimately tied to spatial structure. This is especially true in microbial settings where differential expression and regulation of genes allows members of a given species to alter their lifestyle so as to fill a functional role within the community. The resulting species interactions can involve feedback, as in the case of some bacterial consortia that participate in the cooperative degradation of a given resource in a succession of steps and in such a way that certain "later" species provide catalytic support for the primary degrader. We seek to capture the essential features of such spatially extended biological systems by introducing a lattice-based stochastic spatial model (interacting particle system) with cyclic local dynamics. Here, a given site progresses through a sequence of resource and species states in a prescribed order. Furthermore, this succession of states (at a site) is assumed to form a cyclic pattern due to a natural feedback mechanism. We explore conditions under which all the species are able to coexist and consider the extent to which this coexistence requires the development of spatio-temporal patterns, including spiral waves. This self-organization, if it occurs, results when synchronization of the dynamics at the microscopic level leads to macroscopic patterns. These patterns result in consumer-driven resource fluctuations that generate a form of spatio-temporal niche partitioning. As with most models of this complexity, we employ a mixture of mathematical analysis and simulations to develop an understanding of the resulting dynamics.     

Global positive coexistence of a nonlinear elliptic biological interacting model

The purpose of this note is to give a necessary and sufficient condition for the coexistence of positive solutions to a rather general type of elliptic predator-prey system of the Dirichlet problem on the bounded domain omega when omega is a subset of Rn is large. The result is that the partial differential equation system possesses positive coexistence if and only if the corresponding ordinary differential equation system has positive equilibrium, the positive constant states. This result thus yields an algebraically computable criterion for the positive coexistence of predator and prey in many biological models.     

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.     

Adaptation in a hybrid world with multiple interaction types: a new mechanism for species coexistence

In ecological communities, numerous species coexist and affect each others’ population levels via various types of interspecific interactions. Previous ecological theory explaining multispecies coexistence tended to focus on a single interaction type, such as antagonism, competition, or mutualism, and its consequences on population dynamics. Hence, it remains unclear what, if any, contribution multiple coexisting interaction types have on the multispecies coexistence. Here, we show that the coexistence of multiple interaction types can be essential for multispecies coexistence. We present a simple model in which the exploiter and mutualist adaptively switch between two competing resource species. An adaptive mutualist, which favors the more abundant species, provides a mechanism of majority-advantage and, thus, potentially inhibits the coexistence of resource species. In the absence of an exploiter, an adaptive mutualist leads to competitive exclusion at the resource species level. However, the coexistence of an adaptive exploiter and a mutualist allows the coexistence of all species in the community, because the mutualist-mediated “winner” tends to be suppressed by the adaptive exploiter. The mutualist indirectly increases the abundance of the exploiter through mutualistic interactions, thereby indirectly supporting this coexistence mechanism. In fact, coexistence may occur even if the exploiter or mutualist alone cannot mediate the coexistence of two resources. We conclude that the coexistence of mutualism and antagonism may be the key to the persistence of the four-species module in the presence of adaptive switching.     

Age structure effects in predator-prey interactions

A mathematical model of predator-prey interactions is proposed which incorporates both age structure in the predators and density dependence in the prey. The properties of the model are investigated by a linearized analysis, which enables the conditions for stability to be formulated. The analysis indicates that for a substantial domain of parameter space, a stable equilibrium is possible with the prey well below its carrying capacity. The effect of violating the stability conditions on the behaviour of the model was investigated by computer simulation. Two further types of behaviour are noted in which coexistence is possible. The first is a two point limit cycle in which young and old predators occur in alternate time periods. The second involves a limit cycle in which the component population trajectories lie on closed curves in phase space.     

What Happens When Predators Do Not Completely Consume Their Prey?

A mathematical model is presented for the dynamics of predator-prey interactions when predators do not consume prey (or clumps of prey) in their entirety. Using a combination of analytical and numerical methods, I demonstrate that predator-mediated changes in the distribution of intact and partially consumed prey can affect the outcome of competition between predators in unexpected ways. In some cases, two predators can coexist on a single prey species owing to tradeoffs between the ability to consume prey completely and other competitive abilities. In other cases, predators exhibit frequency-dependent dynamics in which the first predator to occupy the habitat can prevent the other from invading. Conditions for stable coexistence usually expand if the larger predator scatters uneaten prey parts, if prey renewal includes both small and large items, or if the predator with the smaller retrieval capacity is poor at catching intact prey relative to the other predator.     

Effect of resource subsidies on predator-prey population dynamics: a mathematical model

The influence of a resource subsidy on predator-prey interactions is examined using a mathematical model. The model arises from the study of a biological system involving arctic foxes (predator), lemmings (prey), and seal carcasses (subsidy). In one version of the model, the predator, prey and subsidy all occur in the same location; in a second version, the predator moves between two patches, one containing only the prey and the other containing only the subsidy. Criteria for feasibility and stability of the different equilibrium states are studied both analytically and numerically. At small subsidy input rates, there is a minimum prey carrying capacity needed to support both predator and prey. At intermediate subsidy input rates, the predator and prey can always coexist. At high subsidy input rates, the prey cannot persist even at high carrying capacities. As predator movement increases, the dynamic stability of the predator-prey-subsidy interactions also increases.     

Patterns of exclusion in an intraguild predator-prey system depend on initial conditions

1. When intraguild (IG)-prey are superior to IG-predators in competing for a shared resource, theory predicts coexistence of the IG-prey or the IG-predator with the resource depending on the productivity level: (a) resource and IG prey coexist when productivity is low; (b) IG-predator and resource coexist at high productivity; (c) if IG-prey and IG-predators can coexist, it is only at intermediate productivity levels. 2. We tested the existence of productivity-dependent regions of coexistence using an experimental system of two predatory mites and a shared food source (pollen). 3. At high levels of pollen supply (i.e. high productivity), the IG-predator excluded the IG-prey in most, but not all, cases. The same pattern of exclusion was observed at low productivity, at which the IG-prey was expected to exclude the IG-predator. Therefore, species composition could not be predicted by productivity levels. Instead, our results show that initial conditions affected strongly the outcome of the interaction. 4. We emphasize the need for theory on IG-predation that takes the effects of stage structure, initial conditions and transient dynamics into account.     

Diauxic lags explain unexpected coexistence in multi‐resource environments

How the coexistence of species is affected by the presence of multiple resources is a major question in microbial ecology. We experimentally demonstrate that differences in diauxic lags, which occur as species deplete their own environments and adapt their metabolisms, allow slow‐growing microbes to stably coexist with faster‐growing species in multi‐resource environments despite being excluded in single‐resource environments. In our focal example, an Acinetobacter species (Aci2) competitively excludes Pseudomonas aurantiaca (Pa) on alanine and on glutamate. However, they coexist on the combination of both resources. Experiments reveal that Aci2 grows faster but Pa has shorter diauxic lags. We establish a tradeoff between Aci2’s fast growth and Pa’s short lags as their mechanism for coexistence. We model this tradeoff to accurately predict how environmental changes affect community composition. We extend our work by surveying a large set of competitions and observe coexistence nearly four times as frequently when the slow‐grower is the fast‐switcher. Our work illustrates a simple mechanism, based entirely on supplied‐resource growth dynamics, for the emergence of multi‐resource coexistence.     

Impossibility of coexistence of three pure and simple competitors in configurations of three interconnected chemostats

It is well established that pure and simple microbial competitors cannot coexist at a steady state if their environment is homogeneous. For the case of two microbial populations competing purely and simply in two interconnected chemostats having time-invariant input(s), it is known from the literature that a stable steady state of coexistence arises in domains of the operating parameters space and is attributed to the spatial heterogeneities of the system, which allow a different species to have the competitive advantage in each one of the two sub-environments. This article investigates whether the aforementioned result can be extended to the case of three species competing in three interconnected vessels. By studying all possible alternate configurations of the three-chemostat system, it is shown that coexistence of the three species is impossible, except possibly for some discrete values of the operating parameters. Some potential explanations for the results are discussed.     

Comparing the effects of rapid evolution and phenotypic plasticity on predator-prey dynamics

Ecologists have increasingly focused on how rapid adaptive trait changes can affect population dynamics. Rapid adaptation can result from either rapid evolution or phenotypic plasticity, but their effects on population dynamics are seldom compared directly. Here we examine theoretically the effects of rapid evolution and phenotypic plasticity of antipredatory defense on predator-prey dynamics. Our analyses reveal that phenotypic plasticity tends to stabilize population dynamics more strongly than rapid evolution. It is therefore important to know the mechanism by which phenotypic variation is generated for predicting the dynamics of rapidly adapting populations. We next examine an advantage of a phenotypically plastic prey genotype over the polymorphism of specialist prey genotypes. Numerical analyses reveal that the plastic genotype, if there is a small cost for maintaining it, cannot coexist with the pairs of specialist counterparts unless the system has a limit cycle. Furthermore, for the plastic genotype to replace specialist genotypes, a forced environmental fluctuation is critical in a broad parameter range. When these results are combined, the plastic genotype enjoys an advantage with population oscillations, but plasticity tends to lose its advantage by stabilizing the oscillations. This dilemma leads to an interesting intermittent limit cycle with the changing frequency of phenotypic plasticity.     

Competition and coexistence in flowing habitats with a hydraulic storage zone

This paper examines a model of a flowing water habitat with a hydraulic storage zone in which no flow occurs. In this habitat, one or two microbial populations grow while consuming a single nutrient resource. Conditions for persistence of one population and coexistence of two competing populations are derived from eigenvalue problems, the theory of bifurcation and the theory of monotone dynamical systems. A single population persists if it can invade the trivial steady state of an empty habitat. Under some conditions, persistence occurs in the presence of a hydraulic storage zone when it would not in an otherwise equivalent flowing habitat without such a zone. Coexistence of two competing species occurs if each can invade the semi-trivial steady state established by the other species. Numerical work shows that both coexistence and enhanced persistence due to a storage zone occur for biologically reasonable parameters.     

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.     

The impact of variable stoichiometry on predator-prey interactions: a multinutrient approach

A model for prey and predators is formulated in which three essential nutrients can limit growth of both populations. Prey take up dissolved nutrients, while predators ingest prey, assimilate a fraction of ingested nutrients that depends on their current nutrient status, and recycle the balance. Although individuals are modeled as identical within populations, amounts of nutrients within individuals vary over time in both populations, with reproductive rates increasing with these amounts. Equilibria and their stability depend on nutrient supply conditions. When nutrient supply increases, unusual results can occur, such as a decrease of prey density. This phenomenon occurs if, with increasing nutrient, predators sequester rather than recycle nutrients. Furthermore, despite use of a linear functional response for predators, high nutrient supply can destabilize equilibria. Responses to nutrient supply depend on the balance between assimilation and recycling of nutrients by predators, which differs depending on the identity of the limiting nutrient. Applied to microbial ecosystems, the model predicts that the efficiency of organic carbon mineralization is reduced when supply of mineral nutrients is low and when equilibria are unstable. The extent to which predators recycle or sequester limiting nutrients for their prey is of critical importance for the stability of predator-prey systems and their response to enrichment.     

Stability of a diamond-shaped module with multiple interaction types

Indirect interactions among species emerge from the complexity of ecological networks and can strongly affect the response of communities to disturbances. To determine these indirect interactions and understand better community dynamics, ecologists focused on the interactions within small sets of species or modules. Thanks to their analytical tractability, modules bring insights on the mechanisms occurring in complex interaction networks. So far, most studies have considered modules with a single type of interaction although numerous species are involved in mutualistic and antagonistic interactions simultaneously. In this study, we analyse the dynamics of a diamond-shaped module with multiple interaction types: two resource species sharing a mutualist and a consumer. We describe the different types of indirect interaction occurring between the resource species and the conditions for a stable coexistence of all species. We show that the nature of indirect interactions between resource species (i.e. apparent facilitation, competition or antagonism), as well as stable coexistence, depend on the species generalism and asymmetry of interactions, or in other words, on the distribution of interaction strengths among species. We further unveil that a balance between mutualistic and antagonistic interactions at the level of resource species favours stable coexistence, and that species are more likely to coexist stably if there is apparent facilitation between the two resource species rather than apparent competition. Our results echo existing knowledge on the trophic diamond-shaped module, and confirm that our understanding of communities combining different interaction types can gain from module analyses.     

Growth of mixed cultures on mixtures of substitutable substrates: the operating diagram for a structured model

The growth of mixed microbial cultures on mixtures of substrates is a problem of fundamental biological interest. In the last two decades, several unstructured models of mixed-substrate growth have been studied. It is well known, however, that the growth patterns in mixed-substrate environments are dictated by the enzymes that catalyse the transport of substrates into the cell. We have shown previously that a model taking due account of transport enzymes captures and explains all the observed patterns of growth of a single species on two substitutable substrates (J. Theor. Biol. 190 (1998) 241). Here, we extend the model to study the steady states of growth of two species on two substitutable substrates. The model is analysed to determine the conditions for existence and stability of the various steady states. Simulations are performed to determine the flow rates and feed concentrations at which both species coexist. We show that if the interaction between the two species is purely competitive, then at any given flow rate, coexistence is possible only if the ratio of the two feed concentrations lies within a certain interval; excessive supply of either one of the two substrates leads to annihilation of one of the species. This result simplifies the construction of the operating diagram for purely competing species. This is because the two-dimensional surface that bounds the flow rates and feed concentrations at which both species coexist has a particularly simple geometry: It is completely determined by only two coordinates, the flow rate and the ratio of the two feed concentrations. We also study commensalistic interactions between the two species by assuming that one of the species excretes a product that can support the growth of the other species. We show that such interactions enhance the coexistence region.     

Multiple resource limitation: nonequilibrium coexistence of species in a competition model using a synthesizing unit

During the last two decades, the simple view of resource limitation by a single resource has been changed due to the realization that co-limitation by multiple resources is often an important determinant of species growth. Hence, the multiple resource limitation hypothesis needs to be taken into account, when communities of species competing for resources are considered. We present a multiple species–multiple resource competition model which is based on the concept of synthesizing unit to formulate the growth rates of species competing for interactive essential resources. Using this model, we demonstrate that a more mechanistic explanation of interactive effects of co-limitation may lead to the known complex dynamics including nonequilibrium states as oscillations and chaos. We compare our findings with earlier investigations on biological mechanisms that can predict the outcome of multispecies competition. Moreover, we show that this model yields a periodic state where more species than limiting complementary resources can coexist (supersaturation) in a homogeneous environment. We identify two novel mechanisms, how such a state can emerge: a transcritical bifurcation of a limit cycle and a transition from a heteroclinic cycle. Furthermore, we demonstrate the robustness of the phenomenon of supersaturation when the environmental conditions are varied.