Extinction of top-predator in a three-level food-chain model

 In this paper we extend the Lyapunov functions, constructed by A. Ardito and P. Ricciardi for predator–prey system [1], to the three level food chain models. We first consider a general three-level food-chain model. A criterion for the extinction of top predator will be given. Then we restrict our attentions to the case in which the prey is of logistic growth and predators have Holling’s type II functional responses. Received: 10 October 1997     

Meal sharing among the Ye’kwana

In this study meal sharing is used as a way of quantifying food transfers between households. Traditional food-sharing studies measure the flow of resources between households. Meal sharing, in contrast, measures food consumption acts according to whether one is a host or a guest in the household as well as the movement of people between households in the context of food consumption. Our goal is to test a number of evolutionary models of food transfers, but first we argue that before one tests models of who should receive food one must understand the adaptiveness of food transfers. For the Ye’kwana, economies of scale in food processing and preparation appear to set the stage for the utility of meal sharing. Evolutionary models of meal sharing, such as kin selection and reciprocal altruism, are evaluated along with non-evolutionary models, such as egalitarian exchange and residential propinquity. In addition, a modified measure of exchange balance—proportional balance—is developed. Reciprocal altruism is shown to be the strongest predictor of exchange intensity and balance.     

Peak-to-peak dynamics in food chain models

We show in this paper that the chaotic regimes of many food chain models often enjoy a very peculiar property, known as peak-to-peak dynamics. This means that the maximum (peak) density of the populations of any trophic level can be easily forecasted provided the last two peaks of the same population are known. Moreover, extensive simulation shows that only the last peak is needed if the forecast concerns the population at the top of the food chain and that peaks variability often increases from bottom to top. All these findings bring naturally to the conclusion that top populations should be sampled in order to have higher chances to detect peak-to-peak dynamics. The analysis is carried out by studying ditrophic food chain models with seasonally varying parameters, tritrophic food chain models with constant parameters, and more complex food chain and food web models.     

Influence of consumer mutual interference on the stabilization of a three-level trophic system

In this paper, we present a three-level (food–prey–predator) trophic food chain which includes consumer mutual interference (MIF). In contrast with other analyses, we consider the effect of both prey and predator MIF on the dynamics of a three-level trophic system. MIF is generally considered to exert a stabilizing effect on population dynamics based on the predator–prey model. However, results from analytical and numerical simulations utilizing a simple three-species food chain model suggest that while the addition of prey MIF to the model provides a stabilizing influence, as the chaotic dynamics collapse to a stable steady state, adding only predator MIF to the model can only stabilize the system at intermediate MIF values. The three-species trophic food chain is also stabilized when combination of both prey and predator MIF is added to the model. Our work serves to provide insight into the effects of MIF in the real world.     

Cloned transgenic heart-healthy pork?

Here I comment on the production and uses of swine that express a humanized fat-1 gene. The gene product is a fatty acid desaturase that converts ω-6 fatty acids to ω-3 fatty acids. Omega-3 fatty acids have been implicated as being important for reproductive success, maintaining a healthy cardiovascular system, sustaining a functional immune system, and even preventing depression and cancer. The descendants of these hfat-1 transgenic swine will be very useful as models of the human condition, and if they are permitted to enter the food chain, they may improve human health.     

Trophic cascades and trophic trickles in pelagic food webs

Prey-dependent models, with the predation rate (per predator) a function of prey numbers alone, predict the existence of a trophic cascade. In a trophic cascade, the addition of a top predator to a two-level food chain to make a three-level food chain will lead to increases in the population size of the primary producers, and the addition of nutrients to three-level chains will lead to increases in the population numbers at only the first and third trophic levels. In contrast, ratio-dependent models, with the predation rate (per predator) dependent on the ratio of predator numbers to prey, predict that additions of top predators will not increase the population sizes of the primary producers, and that the addition of nutrients to a three-level food chain will lead to increases in population numbers at all trophic levels. Surprisingly, recent meta-analyses show that freshwater pelagic food web patterns match neither prey-dependent models (in pelagic webs, ''prey'' are phytoplankton, and ''predators'' are zooplankton), nor ratio-dependent models. In this paper we use a modification of the prey-dependent model, incorporating strong interference within the zooplankton trophic level, that does yield patterns matching those found in nature. This zooplankton interference model corresponds to a more reticulate food web than in the linear, prey-dependent model, which lacks zooplankton interference. We thus reconcile data with a new model, and make the testable prediction that the strength of trophic cascades will depend on the degree of heterogeneity in the zooplankton level of the food chain.     

Complex dynamic behaviour of autonomous microbial food chains

 The dynamic behaviour of food chains under chemostat conditions is studied. The microbial food chain consists of substrate (non-growing resources), bacteria (prey), ciliates (predator) and carnivore (top predator). The governing equations are formulated at the population level. Yet these equations are derived from a dynamic energy budget model formulated at the individual level. The resulting model is an autonomous system of four first-order ordinary differential equations. These food chains resemble those occuring in ecosystems. Then the prey is generally assumed to grow logistically. Therefore the model of these systems is formed by three first-order ordinary differential equations. As with these ecosystems, there is chaotic behaviour of the autonomous microbial food chain under chemostat conditions with biologically relevant parameter values. It appears that the trajectories on the attractors consists of two superimposed oscillatory behaviours, a slow one for predator–top predator and a fast one for the prey–predator on one branch at which the top predator increases slowly. In some regions of the parameter space there are multiple attractors. Received 8 November 1995; received in revised form 7 January 1997     

Rapid prey evolution and the dynamics of two-predator food webs

Traits affecting ecological interactions can evolve on the same time scale as population and community dynamics, creating the potential for feedbacks between evolutionary and ecological dynamics. Theory and experiments have shown in particular that rapid evolution of traits conferring defense against predation can radically change the qualitative dynamics of a predator–prey food chain. Here, we ask whether such dramatic effects are likely to be seen in more complex food webs having two predators rather than one, or whether the greater complexity of the ecological interactions will mask any potential impacts of rapid evolution. If one prey genotype can be well-defended against both predators, the dynamics are like those of a predator–prey food chain. But if defense traits are predator-specific and incompatible, so that each genotype is vulnerable to attack by at least one predator, then rapid evolution produces distinctive behaviors at the population level: population typically oscillate in ways very different from either the food chain or a two-predator food web without rapid prey evolution. When many prey genotypes coexist, chaotic dynamics become likely. The effects of rapid evolution can still be detected by analyzing relationships between prey abundance and predator population growth rates using methods from functional data analysis.     

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.     

Cascading effects of larval Crucian carp introduction on phytoplankton and microbial communities in a paddy field: top-down and bottom-up controls

Effects of fish predation propagate through aquatic food webs, where the classical grazing food chain and microbial loop are interwoven by trophic interactions. The overall impact on aquatic food webs is further complicated because fish may also exert bottom-up controls through nutrient regeneration. Yet, we still have limited information about cascading effects among fish, zooplankton, phytoplankton, and microbes. In this study, we performed a mesocosm experiment to evaluate effects of fish introduction on plankton communities. Six plots were set in factorial combination with fish introduction and rice straw plowing in a paddy field, and the experiment was continued for 4 weeks. Introduction of fish significantly increased chlorophyll a concentrations in smaller size fractions (<15 μm) and abundances of filamentous bacteria (>5 μm in length) and heterotrophic nanoflagellates in 3–15 μm fraction. Microbes in 0.8–3 μm fraction showed increasing but not significant trends in response to fish introduction. These results indicate cascading effects of fish predation operating via two pathways, one through grazing food chain and the other through microbial food web. Phytoplankton community compositions shifted in similar fashion in all plots until 1 week after fish introduction, and then diverged between plots with and without fish thereafter. Bottom-up effects of fish introduction were suggested by increases of total chlorophyll a and inedible phytoplankton species in response to fish introduction. This study provides an example of how fish predation regulates biomass and structure of phytoplankton and microbial communities.     

Determination of the daily selenium intake in slovakia

Three models were used to determine the daily dietary Selenium intake in Slovakia. The Selenium content of food produced and consumed in the Slovak Republic was used to estimate and calculate the daily Selenium intake based on food consumption data per capita and seven days, (24 h) eating protocol models. In a duplicate portion model, Selenium was analyzed in a whole day hospital diet during an eight-day period. According to these models the daily dietary Selenium intake was 38.2 μg; 43.3 ±6.5 μg for men and 32.6 ±6.6 μg for women; 27.1 ±7.8 μg for normal and 32.3 ±4.8 μg for nourishing hospital diets. The main contributors of Selenium to daily intake were the following: eggs, pork, and poultry. The obtained results indicate that the daily dietary intake of Selenium of the Slovak people is below the recommended values.     

Effects of invasive macrophyte on trophic diversity and position of secondary consumers

Invasive species are one of the widespread stressors of aquatic ecosystems. Several studies document food web effects of invasive fish, but little information is available on the effects of invasive macrophytes. We studied differences in food chain length as well as trophic position and trophic diversity of fish and odonates in lakes dominated by native plants or invasive Eurasian watermilfoil. Trophic position and food chain length were determined using baseline-adjusted δ¹⁵N isotope signatures. Trophic diversity, or isotope niche width, was estimated from convex hull area analysis. Results show that trophic position of secondary consumers was not affected by the invasive macrophyte, whereas trophic diversity was greater in watermilfoil-dominated lakes. The direction of isotopic niche expansion was different in fish and odonates, suggesting potential decoupling in predator–prey interactions. This study shows that dominant non-native macrophytes may cause significant changes in food web structure of invaded ecosystems. Trophic diversity may be a more sensitive indicator of environmental stress than trophic position and has the potential to be used for assessment of invasive species impacts and restoration success.     

Field biology, food web models, and management: challenges of context and scale

Managers are increasingly aware of the need for science to inform the stewardship of natural lands and resources. If ecologists are to address this need, we must increase the scope of our inferences, while maintaining sufficient resolution and realism to predict trajectories of specific populations or ecosystem variables. Food chain and simple food web models, used either as core or component hypotheses, can help us to meet this challenge. The simple mass balance logic of dynamic food chain or food web models can organize our thinking about a range of applied problems, such as evaluating controls over populations of concern, or of biotic assemblages that affect important ecosystem properties. In other applications, a food chain or web may be incorporated as one element in models of regional mass balances affecting resources or environments. Specific predictions of food web models will often fail because of inadequate resolution (e.g., of functionally significant differences among taxa within "trophic levels") or insufficient scope (e.g., of spatio-temporal variation over scales relevant to management). Increasing use of tracers to delimit spatial scales of food web interactions will reduce, but not eliminate, this limitation. If used with skepticism and vigilance to local natural history, however, food chain or simple food web models can promote the iterative feedback between prediction, falsification by observation, and new prediction central to hypothetico-deductive science and adaptive management. Experience argues that this stepwise path is the fastest towards better understanding and control of our impacts on nature.     

Cladoceran filtration rate-body length relations: model improvements developed for a Microcystis-dominated hypertrophic reservoir

Filtering rates were measured for zooplankton species in Situon single-celled Chlorella and on four Microcystis colony sizefractions (5–20, 20–40, 40–60 and 60–100µm) in a hypertrophic reservoir. Natural-log-transformedfiltration rates of five cladoceran species, one copepod andone rotifer were included in an all-food-particle, all-speciesmultiple regression model which explained 43% of the variancein filtration rate as a function of animal body length. An additional14% and 7.6% of the variance was attributable to food type andzooplankton species respectively, with temperature accountingfor <4% of the variance. Restricting the filtration ratemodel to cladocerans alone explained 51% of the variance asa function of animal length, 16% as a function of food type,7.5% as a function of species and only 0.2% as a function oftemperature. In linear filtration rate models for each foodtype, cladoceran body length explained 70% of the variance whenfeeding on Chlorella and between 57 and 67% of the varianceon the four Microcystis colony fractions. Models describingcladoceran filtration rates on Chlorella and the 5–20µm Microcystis colony fraction were significantly differentfrom the three models on larger colonies due to cladoceran responsesto increasing food particle size. Accordingly, a combined modelfor Microcystis colonies >20 µm was developed. Inclusionof food quality factors such as cyanophyte colony size seemsjustified in models aimed at estimating clearance rates, resourceutilization and phytoplankton grazing losses in plankton orecosystem studies when applied to eutrophic or hypertrophiclakes where large cyanophyte particles are abundant.     

Transition of entheropathogenic and saprotrophic bacteria in the niche cycle: Animals-excrement-soil-plants-animals

The possibility of transition of saprotrophic and enteropathohenic bacterial populations following the chain of naturally related habitats—fodder-animal gastrointestinal tract (GIT)-animals excrement-soil-plants and again animals with a cyclic formation—has been investigated quantitatively. All bacteria used in the experiments have been shown to successfully overcome all the mechanical, physical-chemical, and biological barriers in the food chain and to come out into the environment with a quite high number. It has been demonstrated that the same bacterial population can pass the whole cycle without additional introduction of similar populations from the outside.     

Body size and food web structure: testing the equiprobability assumption of the cascade model

The cascade model successfuly predicts many patterns in reported food webs. A key assumption of this model is the existence of a predetermined trophic hierarchy; prey are always lower in the hierarchy than their predators. At least three studies have suggested that, in animal food webs, this hierarchy can be explained to a large extent by body size relationships. A second assumption of the standard cascade model is that trophic links not prohibited by the hierarchy occur with equal probability. Using nonparametric contingency table analyses, we tested this ”equiprobability hypothesis” in 16 published animal food webs for which the adult body masses of the species had been estimated. We found that when the hierarchy was based on body size, the equiprobability hypothesis was rejected in favor of an alternative, ”predator-dominance” hypothesis wherein the probability of a trophic link varies with the identity of the predator. Another alternative to equiprobabilty is that the probability of a trophic link depends upon the ratio of the body sizes of the two species. Using nonparametric regression and liklihood ratio tests, we show that a size-ratio based model represents a significant improvement over the cascade model. These results suggest that models with heterogeneous predation probabilities will fit food web data better than the homogeneous cascade model. They also suggest a new way to bridge the gap between static and dynamic food web models. Received: 3 February 1999 / Accepted: 26 October 1999     

The Effects of Spatial Scale on Trophic Interactions

Food chain models have dominated empirical studies of trophic interactions in the past decades, and have lead to important insights into the factors that control ecological communities. Despite the importance of food chain models in instigating ecological investigations, many empirical studies still show a strong deviation from the dynamics that food chain models predict. We present a theoretical framework that explains some of the discrepancies by showing that trophic interactions are likely to be strongly influenced by the spatial configuration of consumers and their resources. Differences in the spatial scale at which consumers and their resources function lead to uncoupling of the population dynamics of the interacting species, and may explain overexploitation and depletion of resource populations. We discuss how changed land use, likely the most prominent future stress on natural systems, may affect food web dynamics by interfering with the scale of interaction between consumers and their resource.     

Stoichiometry in producer-grazer systems: Linking energy flow with element cycling

All organisms are composed of multiple chemical elements such as carbon, nitrogen and phosphorus. While energy flow and element cycling are two fundamental and unifying principles in ecosystem theory, population models usually ignore the latter. Such models implicitly assume chemical homogeneity of all trophic levels by concentrating on a single constituent, generally an equivalent of energy. In this paper, we examine ramifications of an explicit assumption that both producer and grazer are composed of two essential elements: carbon and phosphorous. Using stoichiometric principles, we construct a two-dimensional Lotka-Volterra type model that incorporates chemical heterogeneity of the first two trophic levels of a food chain. The analysis shows that indirect competition between two populations for phosphorus can shift predator—prey interactions from a (+, −) type to an unusual (−, −) class. This leads to complex dynamics with multiple positive equilibria, where bistability and deterministic extinction of the grazer are possible. We derive simple graphical tests for the local stability of all equilibria and show that system dynamics are confined to a bounded region. Numerical simulations supported by qualitative analysis reveal that Rosenzweig’s paradox of enrichment holds only in the part of the phase plane where the grazer is energy limited; a new phenomenon, the paradox of energy enrichment, arises in the other part, where the grazer is phosphorus limited. A bifurcation diagram shows that energy enrichment of producer—grazer systems differs radically from nutrient enrichment. Hence, expressing producer—grazer interactions in stoichiometrically realistic terms reveals qualitatively new dynamical behavior.     

On the use of the logistic equation in models of food chains

In food chain models the lowest trophic level is often assumed to grow logistically. Anomalous behaviour of the solution of the logistic equation and problems with the introduction of mortality have recently been reported. As predation on the lowest trophic level is a kind of mortality, one expects problems with these food chain models. In this paper we compare two formulations for the lowest trophic level: the logistic growth formulation and the mass balance formulation with resources modelled explicitly. We examine the effects of both models on the dynamic behaviour of a tri-trophic microbial food chain in a chemostat. For this purpose bifurcation diagrams, which give the existence and stability of the equilibria of the nonlinear dynamic system, are used. It turns out that the dynamic behaviours differ in a rather large region of the control parameter space spanned by the dilution rate and the concentration of the resources in the reservoir. We urge that mass balance equations should be used in modelling food chains in chemostats as well as in ecosystems.