Competition and stoichiometry: coexistence of two predators on one prey

The competitive exclusion principle (CEP) states that no equilibrium is possible if n species exploit fewer than n resources. This principle does not appear to hold in nature, where high biodiversity is commonly observed, even in seemingly homogenous habitats. Although various mechanisms, such as spatial heterogeneity or chaotic fluctuations, have been proposed to explain this coexistence, none of them invalidates this principle. Here we evaluate whether principles of ecological stoichiometry can contribute to the stable maintenance of biodiverse communities. Stoichiometric analysis recognizes that each organism is a mixture of multiple chemical elements such as carbon (C), nitrogen (N), and phosphorus (P) that are present in various proportions in organisms. We incorporate these principles into a standard predator-prey model to analyze competition between two predators on one autotrophic prey. The model tracks two essential elements, C and P, in each species. We show that a stable equilibrium is possible with two predators on this single prey. At this equilibrium both predators can be limited by the P content of the prey. The analysis suggests that chemical heterogeneity within and among species provides new mechanisms that can support species coexistence and that may be important in maintaining biodiversity.     

Enrichment in a general class of stoichiometric producer-consumer population growth models

This paper presents the derivation and partial analysis of a general producer-consumer model. The model is stoichiometric in that it includes the growth constraints imposed by species-specific biomass carbon to nutrient ratios. The model unifies the approaches of other studies in recent years, and is calibrated from an extensive review of the algae-Daphnia literature. Numerical simulations and bifurcation analysis are used to examine the impact of energy enrichment under nutrient and stoichiometric constraints. Our results suggest that the variety of system responses previously cited for related models can be attributed to the size of the total system nutrient pool, which is here assumed fixed. New, more complicated bifurcation sequences, such as multiple homoclinic bifurcations, are demonstrated as well. The mechanistic basis of the model permits us to show the robustness of the system’s dynamics subject to alternate approaches to modeling producer and consumer biomass production.     

Scale-free extinction dynamics in spatially structured host-parasitoid systems

Much of the work on extinction events has focused on external perturbations of ecosystems, such as climatic change, or anthropogenic factors. Extinction, however, can also be driven by endogenous factors, such as the ecological interactions between species in an ecosystem. Here we show that endogenously driven extinction events can have a scale-free distribution in simple spatially structured host-parasitoid systems. Due to the properties of this distribution there may be many such simple ecosystems that, although not strictly permanent, persist for arbitrarily long periods of time. We identify a critical phase transition in the parameter space of the host-parasitoid systems, and explain how this is related to the scale-free nature of the extinction process. Based on these results, we conjecture that scale-free extinction processes and critical phase transitions of the type we have found may be a characteristic feature of many spatially structured, multi-species ecosystems in nature. The necessary ingredient appears to be competition between species where the locally inferior type disperses faster in space. If this condition is satisfied then the eventual outcome depends subtly on the strength of local superiority of one species versus the dispersal rate of the other.     

Competition-similarity relationships and the nonlinearity of competitive effects in consumer-resource systems

Much previous ecological and evolutionary theory about exploitative competition for a continuous spectrum of resources has used the Lotka-Volterra model with competition coefficients given by a Gaussian function of niche separation. Using explicit consumer-resource models, we show that the Lotka-Volterra model and the assumption of a Gaussian competition-similarity relationship both fail to reflect the impact of strong resource depletion, which typically reduces the influence of the most heavily used resources on the competitive interaction. Taking proper account of resource depletion reveals that strong exploitative competition between efficient consumers is usually a highly nonlinear interaction, implying that a single measure is no longer sufficient to characterize the process. The nonlinearity usually entails weak coupling of competing species when their abundances are high and equal. Rare invaders are likely to have effects on abundant residents much larger than those of the resident on the invader. Asymmetrical utilization curves often produce asymmetrical competition coefficients. Competition coefficients are typically non-Gaussian and are often nonmonotonic functions of niche separation. Utilization curve shape and resource growth functions can have major effects on competition-similarity relationships. A variety of previous theoretical findings need to be reassessed in light of these results.     

Diversity and extinction in the Cenozoic history of Caribbean reefs

Occurrences of reef corals are examined at Caribbean fossil localities to determine how biodiversity has changed within the region over the past 50 million years. Analyses of 294 species (66 genera) at 58 fossil localities show that Caribbean generic diversity rose to 44 between 50–22 Ma, ranged from 32–39 between 22–2 Ma, and dropped to 25 afterwards. Regional species diversity was high at 40–36 Ma, 28–22 Ma, and 5–2 Ma. Origination rates were elevated throughout each high diversity interval, but extinction was concentrated near the end of each interval. Regional highs of origination and extinction, therefore, differed in timing and duration, causing the observed regional diversity increases during the three remarkably long intervals of turnover. Highs of generic origination decreased in magnitude as immigration from the Mediterranean ceased, but speciation highs increased in association with emergence of the Central American isthmus. Peaks of extinction coincided with regional changes in climate and oceanic circulation. Maximum species diversities within assemblages increased to 40–60 between 50–36 Ma, and have remained relatively constant ever since. Assemblage compositions differed among localities having similar ages and environments, suggesting that the timing and pattern of turnover varied across the region. Stable diversities but variable compositions within assemblages suggest that dispersal and recruitment influenced the pattern of faunal change during turnover. Accepted: 22 August 1999     

Stochastic properties of generalised Yule models, with biodiversity applications

The Yule model is a widely used speciation model in evolutionary biology. Despite its simplicity many aspects of the Yule model have not been explored mathematically. In this paper, we formalise two analytic approaches for obtaining probability densities of individual branch lengths of phylogenetic trees generated by the Yule model. These methods are flexible and permit various aspects of the trees produced by Yule models to be investigated. One of our methods is applicable to a broader class of evolutionary processes, namely the Bellman-Harris models. Our methods have many practical applications including biodiversity and conservation related problems. In this setting the methods can be used to characterise the expected rate of biodiversity loss for Yule trees, as well as the expected gain of including the phylogeny in conservation management. We briefly explore these applications.     

Relationships between spatial configuration of tropical forest patches and woody plant diversity in northeastern Puerto Rico

The destruction and fragmentation of tropical forests are major sources of global biodiversity loss. A better understanding of anthropogenically altered landscapes and their relationships with species diversity and composition is needed in order to protect biodiversity in these environments. The spatial patterns of a landscape may control the ecological processes that shape species diversity and composition. However, there is little information about how plant diversity varies with the spatial configuration of forest patches especially in fragmented tropical habitats. The northeastern part of Puerto Rico provides the opportunity to study the relationships between species richness and composition of woody plants (shrubs and trees) and spatial variables [i.e., patch area and shape, patch isolation, connectivity, and distance to the Luquillo Experimental Forest (LEF)] in tropical forest patches that have regenerated from pasturelands. The spatial data were obtained from aerial color photographs from year 2000. Each photo interpretation was digitized into a GIS package, and 12 forest patches (24–34 years old) were selected within a study area of 28 km². The woody plant species composition of the patches was determined by a systematic floristic survey. The species diversity (Shannon index) and species richness of woody plants correlated positively with the area and the shape of the forest patch. Larger patches, and patches with more habitat edge or convolution, provided conditions for a higher diversity of woody plants. Moreover, the distance of the forest patches to the LEF, which is a source of propagules, correlated negatively with species richness. Plant species composition was also related to patch size and shape and distance to the LEF. These results indicate that there is a link between landscape structure and species diversity and composition and that patches that have similar area, shape, and distance to the LEF provide similar conditions for the existence of a particular plant community. In addition, forest patches that were closer together had more similarity in woody plant species composition than patches that were farther apart, suggesting that seed dispersal for some species is limited at the scale of 10 km.     

Taxonomic and conservation status of a newly discovered giant landsnail from Mount Augustus,New Zealand

The Rhytidae (Mollusca; Gastropoda; Pulmonata) are a group of large carnivorous land snails distributed in the southern hemisphere, with a particularly rich fauna in New Zealand. The endemic genus Powelliphanta consists of at least 10 species and many more recognised subspecies, most of which are restricted to the western margin of South Island, New Zealand. Powelliphanta taxa tend to have restricted ecological and spatial ranges among the mountains of this region, with some species being limited to lowland forest and others to habitats at or above the treeline. Among recent discoveries is a population of snails occupying habitat on and around a peak called Mt Augustus, which is situated at the edge of a large and economically important coalfield. Since recognition of the potential biological significance of the Mt Augustus snails in 2004, almost all of their habitat has been destroyed by opencast mining revealing a direct conflict between economic and biodiversity prioritisation. Our analysis of mtDNA sequence data indicate Powelliphanta “Augustus” is a distinctive evolutionary lineage, more closely related to a nearby lowland species Powelliphanta lignaria than the spatial and ecological neighbour Powelliphanta patrickensis. Powelliphanta “Augustus” appears to be a specialised local endemic species. Despite a growing international awareness of the importance of biodiversity conservation, the demand for foreign earnings continues to take priority over the protection of our biota.     

Synergism analysis of biochemical systems. II. Tensor formulation and treatment of stoichiometric constraints

The previous paper outlined a conceptual and mathematical framework for synergism analysis of kinetic models. Though the formalism presented there is adequate for studying simple models, the analysis of large-scale models benefits from the more effective formulation achieved in this work. The present formulation is based on simple tensor operations and takes advantage of the analogy between the formalisms for synergism and log-synergism analysis presented before. Well-known relationships of first-order sensitivity analysis and new relationships for (log-)synergism coefficients of various steady-state properties are cast in the new formal setting. The formalism is then extended to models that are subject to constraints between variables, fluxes and/or parameters. This treatment, which generalizes Reder's concept of link matrices, is applied to networks that include moiety conservation cycles [C. Reder, Metabolic control theory: a structural approach, J. Theor. Biol. 135 (1988) 175]. It is also used to take advantage of flux conservation at steady-state to simplify synergism analysis. Issues of numerical effectiveness are briefly discussed, and the theory illustrated with the study of synergistic behaviour in the metabolism of reactive oxygen species and of a scheme of dynamic channelling.     

Extinction risk assessments at the population and species level: implications for amphibian conservation

Amphibian populations are declining worldwide and this is causing growing concern. High levels of population declines followed by the expansion of red lists are creating demands for effective strategies to maximize conservation efforts for amphibians. Ideally, integrated and comprehensive strategies should be based on complementary information of population and species extinction risk. Here we evaluate the congruence between amphibian extinction risk assessments at the population level (Declining Amphibian Database––DAPTF) and at species level (GAA––IUCN Red List). We used the Declining Amphibian Database––DAPTF that covers 967 time-series records of amphibian population declines assigned into four levels of declines. We assigned each of its corresponding species into GAA––IUCN red list status, discriminated each species developmental mode, and obtained their geographic range size as well. Extinction risk assessments at the population and species level do not fully coincide across geographic realms or countries. In Paleartic, Neartic and Indo-Malayan realms less than 25% of species with reported population declines are formally classified as threatened. In contrast, more than 60% of all species with reported population declines that occur in Australasia and the Neotropics are indeed threatened according to the GAA––IUCN Red List. Species with aquatic development presented proportionally higher extinction risks at both population and species level than those with terrestrial development, being this pattern more prominent at Australasia, Paleartic, and Neartic realms. Central American countries, Venezuela, Mexico and Australia presented the highest congruence between both population and species risk. We address that amphibian conservation strategies could be improved by using complementary information on time-series population trends and species threat. Whenever feasible, conservation assessments should also include life-history traits in order to improve its effectiveness.     

Bio-economic resource management under threats of environmental catastrophes

We combine ecological and economic dynamics to study the management of a natural resource that supports both ecosystem and human needs. Shrinking the resource base introduces a threat of occurrence of catastrophic ecological events, such as sudden ecosystem collapse. The occurrence conditions involve uncertainty of various types, and the distinction among these types is important for optimal resource management. When uncertainty is due to our ignorance of some aspects of the underlying ecology, the isolated equilibrium states characterizing optimal exploitation for many renewable resource problems become equilibrium intervals. Genuinely stochastic events shift the optimal equilibrium states, but maintain the structure of isolated equilibria.     

Habitat destruction and the extinction debt revisited: The Allee effect

Habitat destruction, often caused by anthropogenic disturbance, can lead to the extinction of species at an unprecedented rate. It is important, therefore, to consider habitat destruction when assessing population viability. Another factor often ignored in population viability analysis, is the Allee effect that adds to the risk of populations already on the verge of extinction. Understanding the Allee effect on species dynamics and response to habitat destruction has intrinsic value in conservation prioritization. Here, the Allee effect was considered in a multi-species hierarchical competition model. Results showed that species persistence declines dramatically due to the Allee effect, and certain species become more susceptible to habitat destruction than others. Two extinction orders emerged under habitat destruction: either the best competitor becomes extinct first or the best colonizer first. The extinction debt and order, as well as the time lag between habitat destruction and species extinction, were found to be determined by species abundance and the intensity of the Allee effect.     

Modeling the bacterial contribution to planktonic community respiration in the regulation of solar energy and nutrient availability

In planktonic ecosystems, algae and bacteria exhibit complex interrelationships, as algae provide an important organic matter source for microbial growth while microbial metabolism recycles limiting nutrients for algae in a loose commensalism. However, algae and bacteria can also compete for available nutrients if supplies of organic matter are sufficient to satisfy bacterial demand. We developed a stoichiometrically explicit model of bacteria–algae interactions that incorporated realistic assumptions about algal light and nutrient utilization, algal exudation of organic matter, and bacterial processing of organic matter and nutrients. The model makes specific predictions about how the relative balance of algae and bacteria should change in response to varied nutrient and light availability seen in lakes and oceans. The model successfully reproduces published empirical data and indicates that, under moderate nutrient supply, the bacterial percentage of total respiration should be maximal at intermediate light intensity.     

Effects of density-dependent migrations on stability of a two-patch predator-prey model

We consider a predator-prey model in a two-patch environment and assume that migration between patches is faster than prey growth, predator mortality and predator-prey interactions. Prey (resp. predator) migration rates are considered to be predator (resp. prey) density-dependent. Prey leave a patch at a migration rate proportional to the local predator density. Predators leave a patch at a migration rate inversely proportional to local prey population density. Taking advantage of the two different time scales, we use aggregation methods to obtain a reduced (aggregated) model governing the total prey and predator densities. First, we show that for a large class of density-dependent migration rules for predators and prey there exists a unique and stable equilibrium for migration. Second, a numerical bifurcation analysis is presented. We show that bifurcation diagrams obtained from the complete and aggregated models are consistent with each other for reasonable values of the ratio between the two time scales, fast for migration and slow for local demography. Our results show that, under some particular conditions, the density dependence of migrations can generate a limit cycle. Also a co-dim two Bautin bifurcation point is observed in some range of migration parameters and this implies that bistability of an equilibrium and limit cycle is possible.     

The extinction of slowly evolving dynamical systems

The time evolution of slowly evolving discrete dynamical systems x _{i + 1} = T(r _i ,x _i ), defined on an interval [0, L], where a parameter r _ichanges slowly with respect to i is considered. For certain transformations T, once r _i reaches a critical value the system faces a non-zero probability of extinction because some x _j [0, L]. Recent ergodic theory results of Ruelle, Pianigiani, and Lasota and Yorke are used to derive a simple expression for the probability of survival of these systems. The extinction process is illustrated with two examples. One is the quadratic map, T(r, x) = rx(1 – x), and the second is a simple model for the growth of a cellular population. The survival statistics for chronic myelogenous leukemia patients are discussed in light of these extinction processes. Two other dynamical processes of biological importance, to which our results are applicable, are mentioned.     

Effects of patch scale on density-dependence and species-dependence in two host-parasitoid systems

Summary Data from two host-parasitoid communities were analyzed to ascertain whether patch scale affected the kinds of correlations existing between 1) spatial differences in host density and the intensity of parasitism (density-dependence) and 2) number of species of parasitoids and the intensity of parasitism (species-dependence). We concluded that parasitization rates are usually independent of both host density and number of parasitoid species present regardless of patch scale. Therefore, the responses of parasitoids to host density and the addition of parasitoid species to a community are equally unpredictable in outcome.     

Dynamics of a mechanistically derived stoichiometric producer-grazer model

One of the simplest predator-prey models that tracks the quantity and the quality of prey is the one proposed by [I. Loladze, Y. Kuang, and J.J. Elser, Stoichiometry in producer-grazer systems: Linking energy flow with element cycling, Bull. Math. Biol. 62 (2000) pp. 1137–1162.] (LKE model). In it, the ratio of two essential chemical elements, carbon to phosphorus, C:P, represents prey quality. However, that model does not explicitly track P neither in the prey nor in the media that supports the prey. Here, we extend the LKE model by mechanistically deriving and accounting for P in both the prey and the media. Bifurcation diagrams and simulations show that our model behaves similarly to the LKE model. However, in the intermediate range of the carrying capacity, especially near the homoclinic bifurcation point for the carrying capacity, quantitative behaviour of our model is different. We analyze positive invariant region and stability of boundary steady states. We show that as the uptake rate of P by producer becomes infinite, LKE models become the limiting case of our model. Furthermore, our model can be readily extended to multiple producers and consumers.     

Foraging behavior by <Emphasis Type="Italic">Daphnia</Emphasis> in stoichiometric gradients of food quality

Mismatches in the elemental composition of herbivores and their resources can impact herbivore growth and reproduction. In aquatic systems, the ratio of elements, such as C, P, and N, is used to characterize the food quality of algal prey. For example, large increases in the C:P ratio of edible algae can decrease rates of growth and reproduction in Daphnia. Current theory emphasizes that Daphnia utilize only assimilation and respiration processes to maintain an optimal elemental composition, yet studies of terrestrial herbivores implicate behavioral processes in coping with local variation in food quality. We tested the ability of juvenile and adult Daphnia to locate regions of high-quality food within a spatial gradient of algal prey differing in C:P ratio, while holding food density constant over space. Both juveniles and adults demonstrated similar behavior by quickly locating (i.e., <10 min) the region of high food quality. Foraging paths were centred on regions of high food quality and these differed significantly from paths of individuals exposed to a homogeneous environment of both food density and food quality. Ingestion rate experiments on algal prey of differing stoichiometric ratio show that individuals can adjust their intake rate over fast behavioral time-scales, and we use these data to examine how individuals choose foraging locations when presented with a spatial gradient that trades off food quality and food quantity. Daphnia reared under low food quality conditions chose to forage in regions of high food quality even though they could attain the same C ingestion rate elsewhere along a spatial gradient. We argue that these aspects of foraging behavior by Daphnia have important implications for how these herbivores manage their elemental composition and our understanding of the dynamics of these herbivore–plant systems in lakes and ponds where spatial variation in food quality is present. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.