Consumer-resource Model Research Articles

Permanence of general stage-structured consumer–resource models

Much of the existing stage-structured consumer–resource models ignore the permanence. In this paper, we consider the permanence for a series of staged-structured consumer–resource models with the function response of so-called “prey-dependence”(resource-dependence) type. We show that the systems are permanent, if and only if the adult consumer's recruitment rate at the peak of resource abundance is more than its death rate. Our results indicate that the large consumer's maturation time delay will directly lead to its extinction. Furthermore, our arguments for the main results give a light for permanence in the general stage-structured consumer–resource systems.

Evolutionary regime shifts in age and size at maturation of exploited fish stocks

Worldwide declines of fish stocks raise concerns about deleterious consequences of harvesting for stock abundances and individual life histories, and call for appropriate recovery strategies. Fishes in exploited stocks mature earlier at either larger or smaller sizes due to both genetic and plastic responses. The latter occur commonly when reduced competition for food leads to faster growth. Using a size-structured consumer-resource model, which accounts for both genetic and plastic responses, we show that fisheries-induced evolutionary changes in individual life history and stock properties can easily become irreversible. As a result of annual spawning, early maturation at small sizes and late maturation at large sizes can become alternative, evolutionarily and ecologically stable states under otherwise identical environmental conditions. Exploitation of late-maturing populations can then induce an evolutionary regime shift to smaller maturation sizes associated with stepwise, 1-year decreases in age at first reproduction. Complete and early fishing moratoria slowly reverse this process, but belated or partial closure of fisheries may accelerate or even instigate further evolution to smaller sizes at maturation. We suggest that stepwise decreases in maturation age can be used as early warnings of upcoming evolutionary changes, and should inspire timely restrictions of fisheries.

Effects of Multi-chain Omnivory on the Strength of Trophic Control in Lakes

Omnivory has been implicated in both diffusing and intensifying the effects of consumer control in food chains. Some have postulated that the strong, community level, top-down control apparent in lakes is not expressed in terrestrial systems because terrestrial food webs are reticulate, with high degrees of omnivory and diverse plant communities. In contrast, lake food webs are depicted as simple linear chains based on phytoplankton-derived energy. Here, we explore the dynamic implications of recent evidence showing that attached algal (periphyton) carbon contributes substantially to lake primary and secondary productivity, including fish production. Periphyton production represents a cryptic energy source in oligotrophic and mesotrophic lakes that is overlooked by previous theoretical treatment of trophic control in lakes. Literature data demonstrate that many fish are multi-chain omnivores, exploiting food chains based on both littoral and pelagic primary producers. Using consumer-resource models, we examine how multiple food chains affect fourth-level trophic control across nutrient gradients in lakes. The models predict that the stabilizing effects of linked food chains are strongest in lakes where both phytoplankton and periphyton contribute substantially to production of higher trophic levels. This stabilization enables a strong and persistent top down control on the pelagic food chain in mesotrophic lakes. The extension of classical trophic cascade theory to incorporate more complex food web structures driven by multi-chain predators provides a conceptual framework for analysis of reticulate food webs in ecosystems.

MODELING CANCER AS AN EVOLUTIONARY GAME

We present two different mathematical models that examine the role of cellular evolution in the development and treatment of cancer. The first is a Lotka-Volterra model of an invasive cancer subjected to chemotherapy that demonstrates rapid evolution of drug-resistant phenotypes. The second model represents a major refinement of the first one with the objective of producing a simple, but realistic model of carcinogenesis. It is a consumer-resource model (consumers = normal + mutuant cells, resource = glucose). We conclude that if mutant cells are not allowed to evolve, they will not ordinarily progress to cancer. However, if accumulating mutations allow cellular evolution and successful adaptation to proliferation constraints, normal cells will develop into invasive cancer. We find this progression is possible because normal cellular populations are not at an evolutionarily stable state and so are subject to invasion by more fit phenotypes.

Resource–consumer models and the biomass conversion principle

In the search for a unified basis for constructing food-web models, the long-standing discrepancy between biomass conversion (BC) and individual survival (IS) modeling has been revitalized by Ginzburg (J. Anim. Ecol. 67 (1998) 325) and Berryman (J. Anim. Ecol. 68 (1999) 1263) in the context of resource–consumer interactions. In this work, the principles underlying the confronting approaches are summarized and the criticisms addressed against each. Also, it is argued that the achievement of a single theory of resource–consumer ecology could benefit from this debate by incorporating key elements of both approaches. A logical procedure is suggested to build simple continuous resource–consumer models that follow the principle of biomass conversion, possess structural homogeneity, and distinguish the effects of depletable and fixed resource availability. Additionally, a new conversion function and a general Holling type extraction function (functional response) are introduced. Finally, it is shown that some well known IS models can be obtained as special cases of a general BC model.

Spatial dynamics of specialist seed predators on synchronized and intermittent seed production of host plants.

Masting, the synchronized and intermittent seed production by plant populations, provides highly variable food resources for specialist seed predators. Such a reproductive mode helps minimize seed losses through predator satiation and extinction of seed predator populations. The seed predators can buffer the resource variation through dispersal or extended diapause. We developed a spatially explicit resource-consumer model to understand the effect of masting on specialist seed predators. The masting dynamics were assumed to follow a resource-based model for plant reproduction, and the population dynamics of the predator were represented by a spatially extended Nicholson-Bailey model. The resultant model demonstrated that when host plants reproduce intermittently, seed predator populations go locally extinct, but global persistence of the predator is facilitated by dispersal or extended diapause. Global extinction of the predator resulted when the intermittent reproduction is highly synchronized among plants. An approximate invasion criterion for the predators showed that negative lag-1 autocorrelation in seeding reduces invasibility, and positive lag-1 cross-correlation enhances invasibility. Spatial synchronization in seeding at local scale caused by pollen coupling (or climate forcing) further prevented invasion of the predators. If the predators employed extended diapause, extremely high temporal variability in reproduction was required for plants to evade the predators.

Stabilization of chaotic and non-permanent food-web dynamics

Several decades of dynamical analyses of food-web networks (1-6) have led to important insights into the effects of complexity, omnivory and interaction strength on food-web stability (6-8). Several re- cent insights (7, 8) are based on nonlinear bioenergetic consumer-resource models (9) that display chaotic behavior in three species food chains (10, 11) which can be stabilized by omnivory (7) and weak interaction of a fourth species (8). We slightly relax feeding on low-density prey in these models by modifying stan- dard food-web interactions known as "type II" functional responses (12). This change drastically alters the dynamics of realistic systems containing up to ten species. Our modification stabilizes chaotic dynamics in three species systems and reduces or eliminates extinctions and non-persistent chaos (11) in ten species systems. This increased stability allows analysis of systems with greater biodiversity than in earlier work and suggests that dynamic stability is not as severe a constraint on the structure of large food webs as previously thought. The sensitivity of dynamical models to small changes in the predator-prey functional response well within the range of what is empirically observed suggests that functional response is a crucial aspect of species interactions that must be more precisely addressed in empirical studies.

Probabilistic risk assessment of abalone Haliotis diversicolor supertexta exposed to waterborne zinc

This paper describes a risk assessment approach that integrates predicted tissue concentrations of zinc (Zn) with a concentration–response relationship and leads to predictions of survival risk for pond abalone Haliotis diversicolor supertexta as well as to the uncertainties associated with these predictions. The models implemented include a probabilistic bioaccumulation model, which linking biokinetic and consumer-resource models, accounts for Zn exposure profile and a modified Hill model for reconstructing a dose–response profile for abalone exposed to waterborne Zn. The growth risk is assessed by hazard quotients characterized by measured water level and chronic no-observed effect concentration. Our risk analyses for H. diversicolor supertexta reared near Toucheng, Kouhu, and Anping, respectively, in north, central, and south Taiwan region indicate a relatively low likelihood that survival is being affected by waterborne Zn. Expected risks of mortality for abalone were estimated as 0.46 (Toucheng), 0.36 (Kouhu), and 0.29 (Anping). The predicted 90th-percentiles of hazard quotient for potential growth risk were estimated as 1.94 (Toucheng), 0.47 (Kouhu), and 0.51 (Anping). These findings indicate that waterborne Zn exposure poses no significant risk to pond abalone in Kouhu and Anping, yet a relative high growth risk in Toucheng is alarming. Because of a scarcity of toxicity and exposure data, the probabilistic risk assessment was based on very conservative assumptions.

Population dynamics of prey exhibiting inducible defenses: the role of associated costs and density-dependence

The effect of antipredator behavior on the dynamics of a resource–consumer model was analyzed in relation to the magnitude of associated costs, and the strength of density-dependence. For this purpose, I present a deterministic continuous resource–consumer model that exhibits biomass conversion, structural homogeneity, and competition for renewable and fixed resources as separate processes. Antipredator behavior is incorporated as an inducible response to consumer density, and has metabolic and feeding costs. By means of numerical methods, I show: (1) that antipredator behavior is stabilizing for certain parameter ranges, where other stabilizing forces do not dominate the dynamics; (2) intraspecific competition for both fixed and renewable resources have a stabilizing role; (3) metabolic cost is always stabilizing, and feeding cost can be stabilizing or destabilizing, depending on the relative strength of the two competition forces.

Competition in size-structured populations: mechanisms inducing cohort formation and population cycles

In this paper we investigate the consequences of size-dependent competition among the individuals of a consumer population by analyzing the dynamic properties of a physiologically structured population model. Only 2 size-classes of individuals are distinguished: juveniles and adults. Juveniles and adults both feed on one and the same resource and hence interact by means of exploitative competition. Juvenile individuals allocate all assimilated energy into development and mature on reaching a fixed developmental threshold. The combination of this fixed threshold and the resource-dependent developmental rate, implies that the juvenile delay between birth and the onset of reproduction may vary in time. Adult individuals allocate all assimilated energy to reproduction. Mortality of both juveniles and adults is assumed to be inversely proportional to the amount of energy assimilated. In this setting we study how the dynamics of the population are influenced by the relative foraging capabilities of juveniles and adults. In line with results that we previously obtained in size-structured consumer–resource models with pulsed reproduction, population cycles primarily occur when either juveniles or adults have a distinct competitive advantage. When adults have a larger per capita feeding rate and are hence competitively superior to juveniles, population oscillations occur that are primarily induced by the fact that the duration of the juvenile period changes with changing food conditions. These cycles do not occur when the juvenile delay is a fixed parameter. When juveniles are competitively superior, two different types of population fluctuations can occur: (1) rapid, low-amplitude fluctuations having a period of half the juvenile delay and (2) slow, large-amplitude fluctuations characterized by a period, which is roughly equal to the juvenile delay. The analysis of simplified versions of the structured model indicates that these two types of oscillations also occur if mortality and/or development is independent of food density, i.e. in a situation with a constant juvenile developmental delay and a constant, food-independent background mortality. Thus, the oscillations that occur when juveniles are more competitive are induced by the juvenile delay per se. When juveniles exert a larger foraging pressure on the shared resource, maturation implies an increase not only in adult density, but also in food density and consequently fecundity. Our analysis suggests that this correlation in time between adult density and fecundity is crucial for the occurrence of population cycles when juveniles are competitively superior.

Will Small Population Sizes Warn Us of Impending Extinctions?

Several models are used to show that population sizes are often relatively insensitive to deteriorating environmental conditions over most of the range of environments that allow population persistence. As conditions continue to worsen in these cases, equilibrium population sizes ultimately decline rapidly toward extinction from sizes similar to or larger than those before environmental decline began. Consumer-resource models predict that equilibrium or average population size can increase with deteriorating environmental conditions over a large part of the range of the environmental parameter that allows persistence. The initial insensitivity or increase in the population of the focal species occurs because changes in the populations of other components of the food web compensate for the decline in one or more fitness components of the focal population. However, the compensatory processes are generally nonlinear and often approach their limits abruptly rather than gradually. When there is steady directional change in the environment, populations lag behind the equilibrium population size specified by current environmental conditions. The environmental variable can then decline below the level required for population persistence while the population size is still close to or greater than its original value. Efficient consumers and self-reproducing resources are especially likely to produce this outcome. More complex models with adaptive behavior, additional consumers, or additional resources often exhibit similar trajectories of population size under environmental deterioration.

Coexistence in MacArthur-Style Consumer–Resource Models

The nature of and conditions for permanent coexistence of consumers and resources are characterized in a family of models that generalize MacArthur's consumer–resource model. The generalization is of the resource dynamics, which need not be of Lotka–Volterra form but are subject only to certain restrictions loose enough to admit many resource dynamics of biological interest. For any such model, (1) if there is an interior equilibrium, then it is globally attracting, else some boundary equilibrium is globally attracting—thus permanent coexistence is coexistence at a globally attracting equilibrium; (2) there is an interior equilibrium if and only if for any species, the equilibrium approached in the absence of that species and the presence of the others is invasible by that species—thus permanent coexistence is equivalent to mutual invasibility; (3) for resources without direct interactions, the conditions for permanent coexistence of the consumers admit an instructive formulation in terms of regression statistics. The significance and limitations of the models and results are discussed.

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.

Physiologically structured models – from versatile technique to ecological theory

A ubiquitous feature of natural communities is the variation in size that can be observed between organisms, a variation that to a substantial degree is intraspecific. Size variation within species by necessity implies that ecological interactions vary both in intensity and type over the life cycle of an individual. Physiologically structured population models (PSPMs) constitute a modelling approach especially designed to analyse these size‐dependent interactions as they explicitly link individual level processes such as consumption and growth to population dynamics. We discuss two cases where PSPMs have been used to analyse the dynamics of size‐structured populations. In the first case, a model of a size‐structured consumer population feeding on a non‐structured prey was successful in predicting both qualitative (mechanisms) and quantitative (individual growth, survival, cycle amplitude) aspects of the population dynamics of a planktivorous fish population. We conclude that single generation cycles as a result of intercohort competition is a general outcome of size‐structured consumer–resource interactions. In the second case, involving both cohort competition and cannibalism, we show that PSPMs may predict double asymptotic growth trajectories with individuals ending up as giants. These growth trajectories, which have also been observed in field data, could not be predicted from individual level information, but are emergent properties of the population feedback on individual processes. In contrast to the size‐structured consumer–resource model, the dynamics in this case cannot be reduced to simpler lumped stage‐based models, but can only be analysed within the domain of PSPMs. Parameter values used in PSPMs adhere to the individual level and are derived independently from the system at focus, whereas model predictions involve both population level processes and individual level processes under conditions of population feedback. This leads to an increased ability to test model predictions but also to a larger set of variables that is predicted at both the individual and population level. The results turn out to be relatively robust to specific model assumptions and thus render a higher degree of generality than purely individual‐based models. At the same time, PSPMs offer a much higher degree of realism, precision and testing ability than lumped stage‐based or non‐structured models. The results of our analyses so far suggest that also in more complex species configurations only a limited set of mechanisms determines the dynamics of PSPMs. We therefore conclude that there is a high potential for developing an individual‐based, size‐dependent community theory using PSPMs.

Setting thresholds for pest control: how does pest density affect resource viability?

Conservation in New Zealand is largely focused on reducing the impact introduced mammals have on the abundance of indigenous species. Conservation managers have a range of strategies they can employ to control these pests, but the combination that maximises conservation gains depends on the protection each strategy affords, and the scale at which it can be applied. Given a limited budget, the use of threshold pest densities to initiate pest control can increase control effectiveness by reducing opportunity costs. However, complex trophic relationships between pests and resources mean that thresholds which minimise the costs of controlling pests without reducing the viability of threatened populations to unacceptable levels will often be difficult to identify. Here we review three general consumer–resource models in the context of pest control; (1) the damage function based on the functional response of pests to resource abundance, (2) density dependent predator–prey models, and (3) interactive models. Damage functions can be used to set threshold pest densities that achieve tactical but not strategic conservation outcomes. Density dependent predator–prey models can be used to set threshold pest densities that have strategic consequences for resource conservation, but are limited in their scope where pest or resource abundance is influenced by density independent environmental perturbation. Interactive models can be used to identify thresholds for imposition of pest control that are responsive to pest density, resource abundance and prevailing environmental conditions. We advocate this modelling framework as a basis for setting control thresholds for pests in New Zealand.

INTERACTIONS BETWEEN FOOD CHEMISTRY AND PREDATION RISK IN FOX SQUIRRELS

The role of plant defensive compounds has often been studied within the purview of consumer diet choice. However, consumers are often confronted with foods distributed within depletable patches. To investigate the complication of resource depletion, I merged a consumer–resource model of nutritional relationships between foods with the technique of giving-up densities for measuring foraging behavior in depletable food patches. Theory predicts that foods containing plant defenses that act as digestibility reducers (e.g., lignins and tannins) will be relatively less depleted under higher predation risk than will foods without defenses. In contrast, foods containing defensive toxins (e.g., alkaloids) that affect fitness directly, and not through diminished physiological uptake of energy, will be depleted without bias to predation risk or other foraging costs. I tested the theory using fox squirrels (Sciurus niger) feeding on sunflower seeds that were impregnated with either tannic or oxalic acid. Squirrels had their highest giving-up densities on oxalate-treated seeds and their lowest on water-treated (control) seeds. GUDs were also higher when patches were placed in riskier microhabitats. Moreover, risky microhabitats increased tannin GUDs relative to control GUDs but had no effect on oxalate GUDs. The former observation is consistent with defenses that act as digestibility reducers, and the latter is consistent with defensive toxins. These results indicate that the effects of plant defensive compounds on foraging behavior are complicated and require consideration of not only the potential effects of the compound, but also the environment to which the forager is exposed. Tannins influence squirrel foraging behavior, but the effects may not be seen or realized unless the forager is in a sufficiently risky environment.

Alternative perspectives on consumer–resource dynamics: a reply to Ginzburg

Summary 1. Ginzburg (1998) proposes that models of predator–prey interactions should conform to what he calls the ‘biomass conversion principle’, that consumer reproduction should be directly linked to resource consumption. 2. I argue that consumer–resource models can be built from two different perspectives – biomass conversion or individual survival – and that, as Ginzburg's principle only applies to the first, it should not be used to judge the plausibility of individual survival models. 3. I also suggest that a failure to understand and preserve the distinction between biomass consumption and individual survival may be responsible for much of the confusion and controversy in predator–prey theory.

Dependencies Between Competition and Predation---and Their Consequences for Initial Value Sensitivity

We commence with a discussion of dependencies between various processes in discrete consumer-resource models. Two basic models are formulated. These models are closely related and differ only in terms of which dependencies are assumed. The models can be regarded as hybrids of the Lotka--Volterra model, the Beverton--Holt model, and the Nicholson--Bailey model.A complete stability analysis of the equilibria is given with the conclusion that the stability properties of these models are closely related to Gause-type predator-prey systems. The numerical part of our analysis brings out remarkable qualitative differences concerning the initial value sensitivity of the oscillations of the two basic models. Initial value sensitivity was measured by Lyapunov exponents and the observed differences cannot be brought to light by a study of limiting cases or by equilibrium analysis. The actual reasons behind these results remain a delicate mathematical question.Properties associated directly with the time series, like...

On the Stabilizing Role of Stage Structure in Piscene Consumer–Resource Interactions

A stage-structured consumer–resource model is investigated using plausible biological parameter estimates. The model, developed from patterns in energy allocation across fish life-history strategies, explicitly considers the effects of delays in maturation on juvenile growth and mortality. It is found that stage structure in the consumer fish population tends to locally stabilize consumer–resource dynamics for realistic parameters. Additionally, it is shown that stage structure bounds nonequilibrium behavior relative to the case without stage structure. Finally, it is shown that the increased stability and bounding of solutions has the seemingly paradoxical consequence of promoting nonequilibrium dynamics when even small amounts of noise are added to the system.

Ontogenetic Scaling of Foraging Rates and the Dynamics of a Size-Structured Consumer-Resource Model

The ontogenetic scaling of foraging capacity strongly influences the competitive ability of differently sized individuals within a species. We develop a physiologically structured model to investigate the effect of different ontogenetic size scalings of the attack rate on the population dynamics of a consumer-resource system. The resource is assumed to reproduce continuously whereas the consumer only reproduces at discrete time instants. Depending on the ontogenetic size scaling, the model exhibited recruit-driven cycles, stable fixed point dynamics, non-recruit juvenile-driven cycles, quasiperiodic orbits, or chaotic dynamics. The kind of dynamics observed was related to the maintenance resource levels required of differently sized individuals. Stable fixed point dynamics was, besides at the persistence boundary, only observed when the minimum resource levels were similar for newborns and mature individuals. The tendency for large population fluctuations over a wide range of the parameter space was due to the consumer's pulsed reproduction. Background mortality and length of season were major determinants of cycle length. Model dynamics strongly resembled empirically observed dynamics from fish and Daphnia populations with respect to both patterns and mechanisms. The non-recruit juvenile-driven dynamics is suggested to occur in populations with size-dependent interference or preemptive competition like cicada populations.