Top predator persistence in differential equation models of food chains: The effects of omnivory and external forcing of lower trophic levels

Persistence criteria are given for the highest trophic level predator in ordinary differential equation models of food chains exhibiting arbitrary omnivory and external supplementation of food source or an intermediate predator. The results are expressed in terms of inequalities involving the bounds on the intrinsic growth and interaction rates. Whether omnivory or external forcing enhances persistence is discussed, particularly for the examples of three-, four-, and five-link Lotka-Volterra food chains.     

Chaos in three species food chains

We study the dynamics of a three species food chain using bifurcation theory to demonstrate the existence of chaotic dynamics in the neighborhood of the equilibrium where the top species in the food chain is absent. The goal of our study is to demonstrate the presence of chaos in a class of ecological models, rather than just in a specific model. This work extends earlier numerical studies of a particular system by Hastings and Powell (1991) by showing that chaos occurs in a class of ecological models. The mathematical techniques we use are based on work by Guckenheimer and Holmes (1983) on co-dimension two bifurcations. However, restrictions on the equations we study imposed by ecological assumptions require a new and somewhat different analysis.     

Ecological consequences of global bifurcations in some food chain models

Food chain models of ordinary differential equations (ode’s) are often used in ecology to gain insight in the dynamics of populations of species, and the interactions of these species with each other and their environment. One powerful analysis technique is bifurcation analysis, focusing on the changes in long-term (asymptotic) behaviour under parameter variation. For the detection of local bifurcations there exists standardised software, but until quite recently most software did not include any capabilities for the detection and continuation of global bifurcations. We focus here on the occurrence of global bifurcations in four food chain models, and discuss the implications of their occurrence. In two stoichiometric models (one piecewise continuous, one smooth) there exists a homoclinic bifurcation, that results in the disappearance of a limit cycle attractor. Instead, a stable positive equilibrium becomes the global attractor. The models are also capable of bistability. In two three-dimensional models a Shil’nikov homoclinic bifurcation functions as the organising centre of chaos, while tangencies of homoclinic cycle-to-cycle connections ‘cut’ the chaotic attractors, which is associated with boundary crises. In one model this leads to extinction of the top predator, while in the other model hysteresis occurs. The types of ecological events occurring because of a global bifurcation will be categorized. Global bifurcations are always catastrophic, leading to the disappearance or merging of attractors. However, there is no 1-on-1 coupling between global bifurcation type and the possible ecological consequences. This only emphasizes the importance of including global bifurcations in the analysis of food chain models.     

Trophic exploitation in grassland food chains: simple models and a field experiment

This study provides insight into the importance of top carnivores (top-down control) and nutrient inputs (bottom-up control) in structuring food chains in a terrestrial grassland system. Qualitative predictions about food chain structure are generated using 4 simple models, each differing in assumptions about some key component in the population dynamics of the herbivore trophic level. The four model systems can be classified broadly into two groups (1) those that assume plant resource intake by herbivores is limited by search rate and handling time as described by classic Lotka-Volterra models; and (2) those that assume plant resource intake by herbivores is limited externally by the supply rate of resources as described by alternatives to Lotka-Volterra formulations. The first class of models tends to ascribe greater importance to top-down control of food chain structure whereas the second class places greater weight on bottom-up control. I evaluated the model predictions using experimentally assembled grassland food chains in which I manipulated nutrient inputs and carnivore (wolf spider) abundance to determine the degree of top-down and bottom-up control of grassland plants and herbivores (grasshoppers). The experimental results were most consistent with predictions of the second class of models implying a predominance of bottom-up control of food chain structure.     

Joint effects of nutrients and contaminants on the dynamics of a food chain in marine ecosystems

We analyze the joint effect of contaminants and nutrient loading on population dynamics of marine food chains by means of bifurcation analysis. Contaminant toxicity is assumed to alter mortality of some species with a sigmoidal dose-response relationship. A generic effect of pollutants is to delay transitions to complex dynamical states towards higher nutrient load values, but more counterintuitive consequences arising from indirect effects are described. In particular, the top predator seems to be the species more affected by pollutants, even when contaminant is toxic only to lower trophic levels.     

食物链中抗生素耐药性基因的转移

抗生素的使用,一方面起到预防和治疗疾病的作用,另一方面,饲料、食品和环境中抗生素残留增加,使抗生素耐药性菌产生并进化,从而导致将来治疗某些疾病时无有效抗生素可用,比如,结核病已经卷土重来,而且现在就有许多患者就不能用抗生素治愈。随着人们生活水平的提高,安全、健康问题受到人们广泛的关注和重视。本文初步对耐药性基因在食物链中怎样产生和转移进行综述。     

Spatial models of virus-immune dynamics

To date, the majority of theoretical models describing the dynamics of infectious diseases in vivo are based on the assumption of well-mixed virus and cell populations. Because many infections take place in solid tissues, spatially structured models represent an important step forward in understanding what happens when the assumption of well-mixed populations is relaxed. Here, we explore models of virus and virus-immune dynamics where dispersal of virus and immune effector cells was constrained to occur locally. The stability properties of our spatial virus-immune dynamics models remained robust under almost all biologically plausible dispersal schemes, regardless of their complexity. The various spatial dynamics were compared to the basic non-spatial dynamics and important differences were identified: When space was assumed to be homogeneous, the dynamics generated by non-spatial and spatially structured models differed substantially at the peak of the infection. Thus, non-spatial models may lead to systematic errors in the estimates of parameters underlying acute infection dynamics. When space was assumed to be heterogeneous, spatial coupling not only changed the equilibrium properties of the uncoupled populations but also equalized the dynamics and thereby reduced the likelihood of dynamic elimination of the infection. In line with experimental and clinical observations, long-lasting oscillation periods were virtually absent. When source-sink dynamics were considered, the long-term outcome of the infection depended critically on the degree of spatial coupling. The infection collapsed when emigration from source sites became too large. Finally, we discuss the implications of spatially structured models on medical treatment of infectious diseases, and note that a huge gap exists in data accurately describing infection dynamics in solid tissues.     

一个海洋食物链能流的初步研究

选择我国北方海岸带水域中属于第一、二、三、四营养级的４个经济种,构成一个简单的人工食物链,即金藻→卤虫→玉筋鱼→黑鱼君．测定了各营养级间物质（湿重、干重）、能量的转换效率,求出了食物链各环节的生产量比值．结果表明,在这个食物链运转过程中,生产１ｋｇ湿重的黑鱼君（第四营养级）需要消耗相当于初级生产力２３５．２ｋｇ湿重的金藻（第一营养级）;生产１ｋｇ干重的黑鱼君需要消耗相当于１４８．３ｋｇ干重的金藻;黑鱼君富集１ｋＪ能量需要消耗含有１１０．７ｋＪ的金藻．     

Hydrocarbon chain dynamics in lipid bilayers

A model is proposed for hydrocarbon chain dynamics in lipid bilayers. In the upper and middle parts of the chain all motion occurs by concerted rotations around at least two carbon carbon bonds at a time, preserving a structural with kinks (that is gauche^±trans gauche^? conformations) as the only deviations from the all-trans chain. At the end, independent rotations around carboncarbon bonds play a larger and larger part. This gives a reasonable interpretation of deuterium NMR data.     

Remarks on antipredator behavior and food chain dynamics

When consumers feeding on a resource spend time in avoiding high risks of predation, the predator functional response declines with predator density. While this is well established, less attention has been paid to the dependence of the consumer functional response on predator density. Here we show how the separation of behavioral and ecological timescales allows one to determine both responses starting from an explicit behavioral model. Within the general set-up considered in this paper, the two functional responses can tend toward Holling type II responses when consumers react only weakly to predation. Thus, the main characteristics of the standard Rosenzweig-MacArthur tritrophic food chain (logistic resource and Holling type II consumer and predator) remain valid also when consumers have weak antipredator behavior. Moreover, through numerical analysis, we show that in a particular but interesting case pronounced antipredator behaviors stabilize the system.     

Homoclinic and heteroclinic orbits to a cycle in a tri-trophic food chain

 The asymptotic behavior of a tri-trophic food chain model is studied. The analysis is carried out numerically, by finding both local and global bifurcations of equilibria and limit cycles. The existence of transversal homoclinic orbits to a limit cycle is shown. The appearance of homoclinic orbits, by moving through a homoclinic bifurcation point, is associated with the sudden disappearance of a chaotic attractor. A homoclinic bifurcation curve, which bounds a region of extinction, is continued through a two-dimensional parameter space. Heteroclinic orbits from an equilibrium to a limit cycle are computed. The existence of these heteroclinic orbits has important consequences on the domains of attraction. Continuation of non-transversal heteroclinic orbits through parameter space shows the existence of two codimension-two bifurcations points, where the saddle cycle is non-hyperbolic. The results are summarized by dividing the parameter space in subregions with different asymptotic behavior. Received: 25 February 1998 / Revised version: 19 August 1998     

Inducible defenses and rotifer food chain dynamics

Theoretical studies have predicted that inducible defenses affect food chain dynamics and persistence. Here we review and evaluate laboratory experiments that tested hypotheses developed from these theoretical studies. This review specifically focuses on the effects of inducible defenses in phytoplankton-rotifer food chain dynamics. First, we describe the occurrence of colony formation within different strains of green algae (Scenedesmaceae) in response to infochemicals released during grazing by the herbivorous rotifer Brachionus calyciflorus. Then we examined the effects of inducible defenses on the population dynamics of this planktonic system in which algal strains that differed in their defense strategies were used. Simple food chains were composed of green algae (Scenedesmaceae), herbivorous rotifers (Brachionus calyciflorus) and carnivorous rotifers (Asplanchna brightwellii). In this system B. calyciflorus exhibits an inducible defense against predation by developing long postero-lateral spines. Experimental studies showed that inducible defenses, as opposed to their absence, could prevent high-amplitude population fluctuations. We discuss the dual effects of induced defenses on extinction probabilities and consider the fit of a theoretical model to experimental data to understand the mechanisms that underlie the observed dynamics. Guest editors: S. S. S. Sarma, R. D. Gulati, R. L. Wallace, S. Nandini, H. J. Dumont & R. Rico-Martínez Advances in Rotifer Research     

土壤—小麦食物链硒转移模式及其应用

利用~(75)Se为示踪剂,研究了土壤-小麦系统中Se的迁移和分布规律。土壤Se植物可利用系数SA=0.3—1.9,土壤不同,SA值也不同。土壤固液两相中Se的分配系数Kd=141—395,说明土壤对Se有强烈的吸附作用,Kd大小顺序为:暗棕色森林土>黑土>草甸棕壤。小麦对土壤Se的浓集系数CF=0.09—0.14,小麦不是浓集Se的作物;Se在小麦各器官间CF值>1,表明其在小麦体内运转比较容易。作者提出了一个食物链Se转移模式,评价了土壤有效Se水平,指出草甸棕壤有效Se充足,黑土和暗棕色森林土则不足。估算了病区每人每天摄Se量为26.6μg,远低于我国成人安全摄Se量建议值40—240μg/人·日的下限值。     

Population dynamics and the color of environmental noise: A study on a three-species food chain system

In the present study, the effect of red, white and blue environmental noise on population dynamics in a simple three-species food chain system was analyzed. The colored noise was superimposed on the three-species food chain model first put forth by Rosenzweig in different ways and the resulting power spectra were investigated. We showed that the amplitude of environmental noise, the trophic level at which a population is positioned and whether a population is directly affected by environmental noise, are all important with respect to the way in which a population responds to noise with different colors. For the deterministic case, all population dynamics are red irrespective of the system dynamics. When all species are sensitive to environmental noise, the top predators dynamics always remain red regardless of the color of the noise and its amplitude, whereas the dynamics of the intermediate species turn blue under disturbances of any color with high amplitude, and those of the basal species may become blue only under blue noise. If only one species is sensitive to environmental noise, the dynamics of the insensitive species are always red, irrespective of the color of the noise and its amplitude. Unlike previous results obtained by studying single-species models, our results have almost nothing to do with deterministic system dynamics. In other words, changing the deterministic system dynamics from stable via periodic to chaotic does not qualitatively change the outcome. Our results are of importance in determining how we interpret the ubiquitous red power spectrum of natural ecological time series.     

Constraints on food chain length arising from regional metacommunity dynamics

Classical ecological theory has proposed several determinants of food chain length, but the role of metacommunity dynamics has not yet been fully considered. By modelling patchy predator-prey metacommunities with extinction-colonization dynamics, we identify two distinct constraints on food chain length. First, finite colonization rates limit predator occupancy to a subset of prey-occupied sites. Second, intrinsic extinction rates accumulate along trophic chains. We show how both processes concur to decrease maximal and average food chain length in metacommunities. This decrease is mitigated if predators track their prey during colonization (habitat selection) and can be reinforced by top-down control of prey vital rates (especially extinction). Moreover, top-down control of colonization and habitat selection can interact to produce a counterintuitive positive relationship between perturbation rate and food chain length. Our results show how novel limits to food chain length emerge in spatially structured communities. We discuss the connections between these constraints and the ones commonly discussed, and suggest ways to test for metacommunity effects in food webs.     

Phytoplankton food quality control of planktonic food web processes

We developed a mechanistic model of nutrient, phytoplankton, zooplankton and fish interactions to test the effects of phytoplankton food quality for herbivorous zooplankton on planktonic food web processes. When phytoplankton food quality is high strong trophic cascades suppress phytoplankton biomass, the zooplankton can withstand intense zooplanktivory, and energy is efficiently transferred through the food web sustaining higher trophic level production. Low food quality results in trophic decoupling at the plant-animal interface, with phytoplankton biomass determined primarily by nutrient availability, zooplankton easily eliminated by fish predation, and poor energy transfer through the food web. At a given nutrient availability, food quality and zooplanktivory interact to determine zooplankton biomass which in turn determines algal biomass. High food quality resulted in intense zooplankton grazing which favored fast-growing phytoplankton taxa, whereas fish predation favored slow-growing phytoplankton. These results suggest algal food quality for herbivorous zooplankton can strongly influence the nature of aquatic food web dynamics, and can have profound effects on water quality and fisheries production. Handling editor: D. Hamilton     

A chronology of plankton dynamics in silico: how computer models have been used to study marine ecosystems

Research on plankton ecology in the oceans has traditionally been conducted via two scientific approaches: in situ (in the field) and in vitro (in the laboratory). There is, however, a third approach: exploring plankton dynamics in silico, or using computer models as tools to study marine ecosystems. Models have been used for this purpose for over 60 years, and the innovations and implementations of historical studies provide a context for how future model applications can continue to advance our understanding. To that end, this paper presents a chronology of the in silico approach to plankton dynamics, beginning with modeling pioneers who worked in the days before computers. During the first 30 years of automated computation, plankton modeling focused on formulations for biological processes and investigations of community structure. The changing technological context and conceptual paradigms of the late-1970s and 1980s resulted in simulations becoming more widespread research tools for biological oceanographers. This period saw rising use of models as hypothesis-testing tools, and means of exploring the effects of circulation on spatial distributions of organisms. Continued computer advances and increased availability of data in the 1990s allowed old approaches to be applied to old and new problems, and led to developments of new approaches. Much of the modeling in the new millennium so far has incorporated these sophistications, and many cutting-edge applications have come from a new generation of plankton scientists who were trained by modeling gurus of previous eras. The future directions for modeling plankton dynamics are rooted in the historical studies.     

Non-linear dynamics models characterizing long-term virological data from AIDS clinical trials

Human immunodeficiency virus (HIV) dynamics represent a complicated variant of the text-book case of non-linear dynamics: predator-prey interaction. The interaction can be described as naturally reproducing T-cells (prey) hunted and killed by virus (predator). Virus reproduce and increase in number as a consequence of successful predation; this is countered by the production of T-cells and the reaction of the immune system. Multi-drug anti-HIV therapy attempts to alter the natural dynamics of the predator-prey interaction by decreasing the reproductive capability of the virus and hence predation. These dynamics are further complicated by varying compliance to treatment and insurgence of resistance to treatment. When following the temporal progression of viral load in plasma during therapy one observes a short-term (1-12 weeks) decrease in viral load. In the long-term (more than 12 weeks from the beginning of therapy) the reduction in viral load is either sustained, or it is followed by a rebound, oscillations and a new (generally lower than at the beginning of therapy) viral load level. Biomathematicians have investigated these dynamics by means of simulations. However the estimation of the parameters associated with the dynamics from real data has been mostly limited to the case of simplified, in particular linearized, models. Linearized model can only describe the short-term changes of viral load during therapy and can only predict (apparent) suppression. In this paper we put forward relatively simple models to characterize long-term virus dynamics which can incorporate different factors associated with resurgence: (Fl) the intrinsic non-linear HIV-1 dynamics, (F2) drug exposure and in particular compliance to treatment, and (F3) insurgence of resistant HIV-1 strains. The main goal is to obtain models which are mathematically identifiable given only measurements of viral load, while retaining the most crucial features of HIV dynamics. For the purpose of illustration we demonstrate an application of the models using real AIDS clinical trial data involving patients treated with a combination of anti-retroviral agents using a model which incorporates compliance data.     

Biogeochemistry of selenium and its impact on food chain quality and human health

In areas where soils are low in bioavailable selenium (Se), potential Se deficiencies cause health risks for humans. Though higher plants have been considered not to require this element, the experience with low-Se soils in Finland has provided evidence that the supplementation of commercial fertilizers with sodium selenate affects positively not only the nutritive value of the whole food chain from soil to plants, animals and humans but also the quantity of plant yields. The level of Se addition has been optimal, and no abnormally high concentrations in plants or in foods of animal origin have been observed. Se levels in serum and human milk indicate that the average daily intake has been within limits considered to be safe and adequate. In fact, plants act as effective buffers, because their growth is reduced at high Se levels. They also tend to synthesize volatile compounds in order to reduce excess Se. On the other hand, when added at low concentrations, Se exerts a beneficial effect on plant growth via several mechanisms. As in humans and animals, Se strengthens the capacity of plants to counteract oxidative stress caused by oxygen radicals produced by internal metabolic or external factors. At proper levels it also delays some of the effects of senescence and may improve the utilization of short-wavelength light by plants. High additions are toxic and may trigger pro-oxidative reactions. Thus, the present supplementation of fertilizers with Se can be considered a very effective and readily controlled way to increase the average daily Se intake nationwide.     

Effects of noise on some dynamical models in ecology

We investigate effects of random perturbations on the dynamics of one-dimensional maps (single species difference equations) and of finite dimensional flows (differential equations for n species). In particular, we study the effects of noise on the invariant measure, on the correlation dimension of the attractor, and on the possibility of detecting the nonlinear deterministic component by applying reconstruction techniques to the time series of population abundances. We conclude that adding noise to maps with a stable fixed-point obscures the underlying determinism. This turns out not to be the case for systems exhibiting complex periodic or chaotic motion, whose essential properties are more robust. In some cases, adding noise reveals deterministic structure which otherwise could not be observed. Simulations suggest that similar results hold for flows whose attractor is almost two-dimensional.