Quasi-equilibrium reduction in a general class of stoichiometric producer–consumer models

This article compares a general closed nutrient, stoichiometric producer–consumer model to a two-dimensional ‘quasi-equilibrium’ approximation. We demonstrate that the quasi-equilibrium system can be rigorously analysed, resulting in nullcline-based criteria for the local stability of system equilibria and for the non-existence of periodic orbits. These results are applied to a study of the dependence of the reduced system on nutrient and energy enrichment. When energy and nutrient enrichment are considered together, the associated bifurcation structures of the two models are seen to share the same essential qualitative characteristics. However, numerical simulations of the three-dimensional parent model show highly complex domains of the persistence and extinction that by Poincare–Bendixson theory are not possible for the two-dimensional reduction. This complexity demonstrates a major difference between the two models, and suggests potential challenges in the use of either model for predicting the long-term behaviour of real-world systems at specific nutrient and energy levels.     

Carrying capacity and demographic stochasticity: scaling behavior of the stochastic logistic model

The stochastic logistic model is the simplest model that combines individual-level demography with density dependence. It explicitly or implicitly underlies many models of biodiversity of competing species, as well as non-spatial or metapopulation models of persistence of individual species. The model has also been used to study persistence in simple disease models. The stochastic logistic model has direct relevance for questions of limiting similarity in ecological systems. This paper uses a biased random walk heuristic to derive a scaling relationship for the persistence of a population under this model, and discusses its implications for models of biodiversity and persistence. Time to extinction of a species under the stochastic logistic model is approximated by the exponential of the scaling quantity U=(R-1)(2) N/R(R+1), where N is the habitat size and R is the basic reproductive number.     

Herbivores, the Functional Diversity of Plants Species, and the Cycling of Nutrients in Ecosystems

Numerous investigators have suggested that herbivores almost always increase rates of nutrient and energy flow through terrestrial ecosystems by returning to the soil fecal material and urine with faster turnover rate than shed plant litter. These previous theories and models always treat the producer compartment as a homogenous pool. Essentially, they assume that consumers feed through a pureed cream of vegetable soup. However, many field observations and experiments have shown that consumers feed selectively (i.e., in a cafeteria) and that consumer choice is made on the same chemical basis that determines decomposition rates. Plants that are preferred food sources often have higher nutrient content, higher growth rates, and faster decomposition rates. As consumption reduces dominance of these species in favor of unpreferred species with slower decomposition, rates of nutrient cycling and energy flow should therefore decline. We analyze a model in which the consumer is given a choice among producers that vary in nutrient uptake rates, rates of nutrient return to decomposers, and consumer preference, and which is parameterized for plants and consumers characteristic of boreal regions. In this model, in an open, well-mixed system with one consumer and two such producers, the nutrient/energy flow will not exceed that of a system without the consumer. If the consumer has a choice between two such producers, it must choose one plant over the other at a greater ratio than that between the two plants in uptake and decay rates. In contrast, in a closed system the consumer must be less selective to coexist with the two plants. The system behavior is determined by the level of nutrient return through the consumer and the differences between the plants in nutrient uptake rates and consumer preference. Species richness affects properties of this model system to the extent that species are functionally distinct (i.e., have different rate constants) in a multivariate space of life history traits (i.e., nutrient uptake and palatability). We suggest that the biochemical variability of plant tissues that simultaneously determines both consumer preference and decomposition rates is an essential feature of food webs that cannot be ignored. Thus, ecosystem models should, at minimum, consider more than one producer type with consumer preference.     

Long‐term nutrient addition alters arthropod community composition but does not increase total biomass or abundance

A simple bottom–up hypothesis predicts that plant responses to nutrient addition should determine the response of consumers: more productive and less diverse plant communities, the usual result of long‐term nutrient addition, should support greater consumer abundances and biomass and less consumer diversity. We tested this hypothesis for the response of an aboveground arthropod community to an uncommonly long‐term (24‐year) nutrient addition experiment in moist acidic tundra in arctic Alaska. This experiment altered plant community composition, decreased plant diversity and increased plant production and biomass as a deciduous shrub, Betula nana, became dominant. Consistent with strong effects on the plant community, nutrient addition altered arthropod community composition, primarily through changes to herbivore taxa in the canopy‐dwelling arthropod assemblage and detritivore taxa in the ground assemblage. Surprisingly, however, the loss of more than half of plant species was accompanied by negligible changes to diversity (rarefied richness) of arthropod taxa (which were primarily identified to family). Similarly, although long‐term nutrient addition in this system roughly doubles plant production and biomass, arthropod abundance was either unchanged or decreased by nutrient addition, and total arthropod biomass was unaffected. Our findings differ markedly from the handful of terrestrial studies that have found bottom‐up diversity cascades and productivity responses by consumers to nutrient addition. This is probably because unlike grasslands and salt marshes (where such studies have historically been conducted), this arctic tundra community becomes less palatable, rather than more so, after many years of nutrient addition due to increased dominance of B. nana. Additionally, by displacing insulating mosses and increasing the cover of shrubs that cool and shade the canopy microenvironment, fertilization may displace arthropods keenly attuned to microclimate. These results indicate that terrestrial arthropod assemblages may be more constrained by producer traits (i.e. palatability, structure) than they are by total primary production or producer diversity.     

Indirect and direct controls of macroinvertebrates and small fish by abiotic factors and trophic interactions in the Florida Everglades

1. The roles of nutrients, disturbance and predation in regulating consumer densities have long been of interest, but their indirect effects have rarely been quantified in wetland ecosystems. The Florida Everglades contains gradients of hydrological disturbance (marsh drying) and nutrient enrichment (phosphorus), often correlated with densities of macroinvertebrate infauna (macroinvertebrates inhabiting periphyton), small fish and larger invertebrates, such as snails, grass shrimp, insects and crayfish. However, most causal relationships have yet to be quantified. 2. We sampled periphyton (content and community structure) and consumer (small omnivores, carnivores and herbivores, and infaunal macroinvertebrates inhabiting periphyton) density at 28 sites spanning a range of hydrological and nutrient conditions and compared our data to seven a priori structural equation models. 3. The best model included bottom‐up and top‐down effects among trophic groups and supported top‐down control of infauna by omnivores and predators that cascaded to periphyton biomass. The next best model included bottom‐up paths only and allowed direct effects of periphyton on omnivore density. Both models suggested a positive relationship between small herbivores and small omnivores, indicating that predation was unable to limit herbivore numbers. Total effects of time following flooding were negative for all three consumer groups even when both preferred models suggested positive direct effects for some groups. Total effects of nutrient levels (phosphorus) were positive for consumers and generally larger than those of hydrological disturbance and were mediated by changes in periphyton content. 4. Our findings provide quantitative support for indirect effects of nutrient enrichment on consumers, and the importance of both algal community structure and periphyton biomass to Everglades food webs. Evidence for top‐down control of infauna by omnivores was noted, though without substantially greater support than a competing bottom‐up‐only model.     

Modeling persistence and mutual interference among subpopulations of ecological communities

A model is proposed of an ecological community where some (or all) of the subpopulations exhibit mutual interference. Mutual interference introduces sublinearities which makes the persistence analysis of the community more complex since the model is no longer a dynamical system. A transformation is introduced which yields a dynamical system, thereby making a persistence analysis more tractable. The results are applied to determining top-predator persistence of a simple food chain and to the question of invasibility of a stable community by a new subpopulation. Research partially supported by the Natural Sciences and Engineering Research Council of Canada.     

Quasi-equilibrium reduction in a general class of stoichiometric producer-consumer models

This article compares a general closed nutrient, stoichiometric producer-consumer model to a two-dimensional 'quasi-equilibrium' approximation. We demonstrate that the quasi-equilibrium system can be rigorously analysed, resulting in nullcline-based criteria for the local stability of system equilibria and for the non-existence of periodic orbits. These results are applied to a study of the dependence of the reduced system on nutrient and energy enrichment. When energy and nutrient enrichment are considered together, the associated bifurcation structures of the two models are seen to share the same essential qualitative characteristics. However, numerical simulations of the three-dimensional parent model show highly complex domains of the persistence and extinction that by Poincare-Bendixson theory are not possible for the two-dimensional reduction. This complexity demonstrates a major difference between the two models, and suggests potential challenges in the use of either model for predicting the long-term behaviour of real-world systems at specific nutrient and energy levels.     

Harvesting and predation of a sex- and age-structured population

We study a discrete-time system of equations for a structured ungulate population exploited by human harvesting or a dynamic predator. The population is divided into juveniles, and female and male adults. Harvesting is concentrated on adults (trophy hunting of males or population control measures on females), whereas predation occurs in juveniles. Though the model consists of four nonlinear equations, we find explicit expressions for the steady states. We use these explicit expressions to investigate harvesting rates that allow population persistence, rates that ensure population control, and optimal harvesting efforts. Several reductions of complexity allow for a detailed analysis of the dynamics of the model. Most notably, we find that even compensatory density dependence can lead to a period doubling bifurcation, that the model does not support consumer–resource cycles, and that an Allee effect can emerge from the interplay of stage-specific predation and density-dependent prey reproduction.     

Omnivorous food web,prey preference and allochthonous nutrient input

This work purports to analyze the influence of allochthonous nutrient input into consumer level in the ultimate dynamics of an omnivory food web, where consumption is dictated by non-switching and switching predators. Within this behavioral context, prey consumption structure is shown to have a markedly effect on food web dynamics under a gradient of allochthonous input and primary productivity. A striking feature is that in the non-switching model invasion of consumer and predator occurs sequentially in this order as density of carrying capacity increases, while in the switching model both predators and consumers are able to invade and persist irrespective of the considered carrying capacity levels.     

Natal habitat effects drive density—dependent scaling of dispersal decisions

The dispersal behavior of a species is critical for the stability and persistence of its populations across a landscape. How population density affects dispersal decisions is important for predicting these dynamics, as the form of density‐dependent dispersal influences the stability and persistence of populations. Natal habitat experience often has strong impacts on individual dispersal behavior as well, but its influence on density‐dependent dispersal behaviors remains unexplored. Here we address this conceptual gap in two experiments separately examining habitat selection and emigration from recently colonized patches for two species of flour beetle Tribolium sp. We found that interactions between the quality of habitat experienced during natal development and current habitat for dispersal capable adults can strongly affect the form of density dependence, including reversing the direction of nonlinearities (accelerating to decelerating), or even negating the influence of population density for individual dispersal decisions. Across heterogeneous landscapes, where individuals from different populations may experience different natal habitats, this altering of density‐dependent relationships is predicted by theory to fundamentally influence regional population dynamics. Our results indicate that species which occur across heterogeneous environments, such as during conservation reintroductions, or as invasive species spread, have much potential for natal experience to interact with density dependence and influence local and regional population dynamics.     

Competition in the presence of a virus in an aquatic system: an SIS model in the chemostat

Recent research indicates that viruses are much more prevalent in aquatic environments than previously imagined. We derive a model of competition between two populations of bacteria for a single limiting nutrient in a chemostat where a virus is present. It is assumed that the virus can only infect one of the populations, the population that would be a more efficient consumer of the resource in a virus free environment, in order to determine whether introduction of a virus can result in coexistence of the competing populations. We also analyze the subsystem that results when the resistant competitor is absent. The model takes the form of an SIS epidemic model. Criteria for the global stability of the disease free and endemic steady states are obtained for both the subsystem as well as for the full competition model. However, for certain parameter ranges, bi-stability, and/or multiple periodic orbits is possible and both disease induced oscillations and competition induced oscillations are possible. It is proved that persistence of the vulnerable and resistant populations can occur, but only when the disease is endemic in the population. It is also shown that it is possible to have multiple attracting endemic steady states, oscillatory behavior involving Hopf, saddle-node, and homoclinic bifurcations, and a hysteresis effect. An explicit expression for the basic reproduction number for the epidemic is given in terms of biologically meaningful parameters. Mathematical tools that are used include Lyapunov functions, persistence theory, and bifurcation analysis.     

Seasonal Dynamics of Periphyton in a Large Tropical Lake

Tropical aquatic systems are generally assumed to have little seasonality in productivity patterns. However, this study indicated that there was substantial seasonal variation in epilithic productivity and biomass in tropical Lake Tanganyika, due primarily to seasonal patterns in lake hydrodynamics that influence nutrient availability. Although they support much of the lake’s biological diversity, epilithic algae made a minor contribution to the total energy budget in Lake Tanganyika. A comparison among large, oligotrophic lakes revealed no significant latitudinal trends in periphyton productivity or biomass. However, Lake Tanganyika has relatively low benthic algal biomass and is therefore more efficient at photosynthesis than the temperate lakes. The influence of wave action and consumer density and diversity may be important in moderating productivity of the epilithic community.     

The Role of Density Regulation in Extinction Processes and Population Viability Analysis

We review the role of density dependence in the stochastic extinction of populations and the role density dependence has played in population viability analysis (PVA) case studies. In total, 32 approaches have been used to model density regulation in theoretical or applied extinction models, 29 of them are mathematical functions of density dependence, and one approach uses empirical relationships between density and survival, reproduction, or growth rates. In addition, quasi-extinction levels are sometimes applied as a substitute for density dependence at low population size. Density dependence further has been modelled via explicit individual spacing behaviour and/or dispersal. We briefly summarise the features of density dependence available in standard PVA software, provide summary statistics about the use of density dependence in PVA case studies, and discuss the effects of density dependence on extinction probability. The introduction of an upper limit for population size has the effect that the probability of ultimate extinction becomes 1. Mean time to extinction increases with carrying capacity if populations start at high density, but carrying capacity often does not have any effect if populations start at low numbers. In contrast, the Allee effect is usually strong when populations start at low densities but has only a limited influence on persistence when populations start at high numbers. Contrary to previous opinions, other forms of density dependence may lead to increased or decreased persistence, depending on the type and strength of density dependence, the degree of environmental variability, and the growth rate. Furthermore, effects may be reversed for different quasi-extinction levels, making the use of arbitrary quasi-extinction levels problematic. Few systematic comparisons of the effects on persistence between different models of density dependence are available. These effects can be strikingly different among models. Our understanding of the effects of density dependence on extinction of metapopulations is rudimentary, but even opposite effects of density dependence can occur when metapopulations and single populations are contrasted. We argue that spatially explicit models hold particular promise for analysing the effects of density dependence on population viability provided a good knowledge of the biology of the species under consideration exists. Since the results of PVAs may critically depend on the way density dependence is modelled, combined efforts to advance statistical methods, field sampling, and modelling are urgently needed to elucidate the relationships between density, vital rates, and extinction probability.     

Predicting coral community recovery using multi‐species population dynamics models

Predicting whether, how, and to what degree communities recover from disturbance remain major challenges in ecology. To predict recovery of coral communities we applied field survey data of early recovery dynamics to a multi‐species integral projection model that captured key demographic processes driving coral population trajectories, notably density‐dependent larval recruitment. After testing model predictions against field observations, we updated the model to generate projections of future coral communities. Our results indicated that communities distributed across an island landscape followed different recovery trajectories but would reassemble to pre‐disturbed levels of coral abundance, composition, and size, thus demonstrating persistence in the provision of reef habitat and other ecosystem services. Our study indicates that coral community dynamics are predictable when accounting for the interplay between species life‐history, environmental conditions, and density‐dependence. We provide a quantitative framework for evaluating the ecological processes underlying community trajectory and characteristics important to ecosystem functioning.     

The evolutionarily optimal rates of body development. The age of onset of sexual maturity and species-specific longevity as the parameters optimized during evolution]

Dynamics of biological community of the "resource--consumer" type considering age structure of consumer population is described by a system of differential equations with special derivatives. On the basis of such a model, a competition model for non-crossing populations with different individual development rates is elaborated. It is shown that only a population with development rates maximizing the Malthusian function (reaching zero value at the equilibrium state of the system) is able to survive under competition for food resources. Equilibrium density of the resources is provided being minimal. Thus maximum energy influx into the population is gained. Search algorithm of evolutionary values of puberty, age and maximal longevity of individuals belonging to the consumer population is proposed. Analytic dependence of maximal longevity on environmental factors and some other parameters are found. Aging is considered to be mechanism leading to death while individual approaches evolutionary optimal longevity.     

Resource-niche complementarity and autotrophic compensation determines ecosystem-level responses to increased cladoceran species richness

This study examines the relationship between cladoceran species richness and ecosystem functioning. I conducted an experiment in which four cladocerans, Daphnia. magna, D. longispina, D. pulex and Chydorus sphaericus, were cultured in microcosms using different species combinations and levels of species richness. The results demonstrate that even within this closely related group of organisms the effects on ecosystem-level variables, such as total algae and zooplankton biomass, per capita productivity, and nutrient concentrations, as well as phytoplankton community structure, were highly variable between different combinations of these species. Since only four species where involved in this study, species-specific effects dominated the general relationship between species richness and ecosystem functioning. Particular combinations of species resulted in effects that indicated more efficient grazing. These effects, which were most pronounced in combinations including both D. magna and C. sphaericus, were manifested as an indirect effect as the prey community shifted towards grazing-resistant species. As a result, the productivity of the prey community decreased, because phytoplankton species with lower per capita productivity became more dominant. I suggest that the primary mechanism that caused this significant effect was complementarity in prey-size use of D. magna and C. sphaericus. In terms of prey-size range, D. pulex and D. longispina were redundant when D. magna was present and were quickly out-competed by the latter despite higher per capita filtering efficiency. The results show that different mechanisms are important for different combinations of species. Furthermore, the ability of the prey community to respond to changes of consumer species composition is an important factor in experiments in which consumer species richness is experimentally manipulated. Received: 2 November 1998 / Accepted: 8 September 1999     

Nonlinear effects of an aquatic consumer: causes and consequences

The assumption that per capita consumer effects on prey density are independent of consumer and prey density is examined with a large-scale manipulation of an aquatic herbivore (Daphnia). A gradient of consumer removal was maintained long enough to allow the abundances of both consumer and prey (phytoplankton) to equilibrate to the manipulation. Strong and unequivocal nonlinearities were found in the effect of Daphnia on total phytoplankton abundance and the abundance of most of the common phytoplankton species. Daphnia's suppression of phytoplankton was strong between 0 and approximately 400 microg Daphnia L(-1) but essentially nil from approximately 400 to 900 microg Daphnia L(-1). The sharp deceleration in Daphnia's effect was not caused by a shift within the phytoplankton community toward consumption-resistant forms. The most likely explanation for the deceleration was a reduction in Daphnia's filtering effort at low phytoplankton abundance, that is, a Type III functional response. A review of experimental literature suggested that decelerating effects of consumers are the norm in aquatic systems. Nonlinear effects present problems for the estimation of interaction strength and the building of community interaction models from the results of predator manipulations. It is suggested that the role of field experiments in community ecology should be to test rather than to parameterize models.     

Temperature indirectly affects benthic microalgal diversity by altering effects of top–down but not bottom–up control

Understanding the ecological mechanisms that underlie species diversity decline in response to environmental change has become an urgent objective in current ecological research. Not only direct (lethal) effects on single species but also indirect effects altering biotic interactions between species within and across trophic levels comprise the driving force of ecosystem change. In an experimental marine benthic microalgae–grazer system we tested for indirect effects of moderate temperature change on algal diversity by manipulation of temperature, nutrient supply and grazer density. In our model system warming did not exert indirect effects on microalgal diversity via effects on resource competition. However, moderate warming strengthened consumer control and thereby indirectly affected algal community structure which ultimately resulted in decreased diversity. Only in low temperature and low nutrient regimes did the antagonizing mechanisms of bottom–up and top–down regulation establish a balancing effect on algal diversity within 29 days (corresponding to 15–29 algae generations). Effects of thermal habitat change did not appear before 9–18 algae generations, which points to the relevance of longer‐term experiments and ecological monitoring in order to separate transient biotic responses and subtle changes of community dynamics in consequence to global change.     

Critical patch size generated by Allee effect in gypsy moth, Lymantria dispar (L.)

Allee effects are important dynamical mechanisms in small-density populations in which per capita population growth rate increases with density. When positive density dependence is sufficiently severe (a 'strong' Allee effect), a critical density arises below which populations do not persist. For spatially distributed populations subject to dispersal, theory predicts that the occupied area also exhibits a critical threshold for population persistence, but this result has not been confirmed in nature. We tested this prediction in patterns of population persistence across the invasion front of the European gypsy moth (Lymantria dispar) in the United States in data collected between 1996 and 2008. Our analysis consistently provided evidence for effects of both population area and density on persistence, as predicted by the general theory, and confirmed here using a mechanistic model developed for the gypsy moth system. We believe this study to be the first empirical documentation of critical patch size induced by an Allee effect.