Food chains in the chemostat: Relationships between mean yield and complex dynamics

Atritrophic food-chain chemostat model composed of a prey with Monod-type nutrient uptake, a Holling Type II predator and a Holling Type II exploited superpredator is considered in this paper. The bifurcations of the model show that dynamic complexity first increases and then decreases with the nutrient supplied to the bottom of the food chain. Extensive simulations prove that the same holds for food yield, i.e., there exists an optimum nutrient supply which maximizes mean food yield. Finally, a comparative analysis of the results points out that the optimum nutrient supply practically coincides with the nutrient supply separating chaotic dynamics from high-frequency cyclic dynamics. This reinforces the idea, already known for simpler models, that food yield maximization requires that the system behaves on the edge of chaos.     

Top-predator abundance and chaos in tritrophic food chains

Nonlinear models and laboratory experiments suggest that populations can be chaotic, whereas field data show that a fair proportion of observed populations are not too far from being chaotic. Thus, a natural question arises: do ecosystems enjoy special properties at the edge of chaos? By limiting the analysis to three classes of tritrophic food chains and to the role played by the nutrient available to the bottom of the ecosystem, we show that top-predator mean abundance is maximum at the edge of chaos. The geographical variability of species abundance along nutrient gradients and the dynamic complexity of observed natural populations are consistent with our findings. Effective guidelines for the sustainability of exploited ecosystems are also derived.     

Carnivore identity mediates the effects of light and nutrients on aquatic food‐chain efficiency

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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.     

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.     

Morphological and physiological traits for higher biomass production in perennial rhizomatous grasses grown on marginal land

A widespread criticism of growing energy crops is that they displace much needed food crops and cause upward pressure on food prices. One solution is the use of marginal land that is unfavourable for food production and is currently underutilized. However, the yield of crops growing on marginal land is reduced because they are subjected to a range of abiotic stresses such as extremes of temperature and rainfall and edaphic factors such as increased soil salinity. Therefore, to achieve acceptable economic returns on the use of this land it will be necessary to, on the one hand improve management practices and on the other to select for plant genotypes which are able to tolerate and potentially overcome the stressful conditions they are exposed to. Here, we review the morphological and physiological traits of perennial rhizomatous grasses that could be modified to overcome these stresses and to maximize biomass production on marginal land. The traits include aspects of crop phenology, canopy and leaf photosynthesis, biomass partitioning, nutrient and water use efficiency and heat, cold and salt tolerance. It is proposed that newly developed biotechnological methods combined with high‐throughput plant phenotyping offer opportunities to rapidly select new genotypes that could achieve economic yields on large areas of marginal land.     

The responses of unstable food chains to enrichment

Summary This article investigates the mean abundances of trophic levels in simple models of two- and three-level food chains as a function of the rate of input of nutrients. The analysis concentrates on cases in which the equilibrium point with all species present is unstable. In most of the models, the instability arises because the consumer species become satiated when food density is high. In unstable two-level systems, bottom level abundance generally increases with increased nutrient input. The abundance of the second level may decrease with increased input. Changes in the intrinsic rate of increase and carrying capacity of the bottom level can have qualitatively opposite effects on trophic level abundances. Refuges for or immigration of the bottom level usually cause both levels to increase in mean abundance with an increased carrying capacity. A variety of different predator—prey models are discussed briefly and the results suggest that increased nutrient input will often increase the abundance of both levels; however, several circumstances can cause the top level to decrease. In three-level systems, an increased carrying capacity can cause extinction of the top level. Extinction may or may not be conditional on the initial densities of the three levels. These results may help explain the observed lack of correlation between productivity and the number of trophic levels in natural food webs, as well as the lack of very long food chains. The results suggest that patterns of abundances across productivity gradients cannot be used to assess the importance of top-down vs bottom-up effects.     

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     

Resilience and local stability in a nutrient-limited resource-consumer system

The “paradox of enrichment” predicts that increasing the growth rate of the resource in a resource-consumer dynamic system, by nutrient enrichment, for example, can lead to local instability of the system—that is, to a Hopf bifurcation. The approach to the Hopf bifurcation is accompanied by a decrease in resilience (rate of return to equilibrium). On the other hand, studies of nutrient cycling in food webs indicate that an increase in the nutrient input rate usually results in increased resilience. Here these two apparently conflicting theoretical results are reconciled with a model of a nutrient-limited resource-consumer system in which the tightly recycled limiting nutrient is explicitly modelled. It is shown that increasing nutrient input may at first lead to increased resilience and that resilience decreases sharply only immediately before the Hopf bifurcation is reached.     

Effects of temporal heterogeneity of water supply and nutrient levels on plant biomass growth depend on the plant’s relative size within its population

Temporal heterogeneity of water supply can alter the biomass growth of plants, even when the same total amount of water is provided. Most studies of heterogeneous watering have focused on responses of whole populations rather than individuals in a population. The effects of water supply heterogeneity may also depend on nutrient levels. Thus, we investigated the integrated effects of water supply heterogeneity and nutrient levels on plants within a population. Six plants of Perilla frutescens per pot were grown under different combinations of frequency of water supply and nutrient level. The effects on yield per pot, individual biomass, and allocation to roots were analyzed after a 44-day watering regime. A homogeneous water supply resulted in a greater yield per pot and greater biomass of individual plants than a heterogeneous supply. However, the interaction between water supply heterogeneity and nutrient level was significant only in larger individuals, not in smaller plants or at the p. Water supply heterogeneity affected the growth of all plants, but the effects differed among individuals depending on their relative size within their population. It is therefore important to focus not only on whole-population characteristics such as yield but also on individuals in a population in order to reveal the detailed effects of water supply heterogeneity.     

Resistance of a food chain to invasion by a top predator

We study the invasion of a top predator into a food chain in a chemostat. For each trophic level, a bioenergetic model is used in which maintenance and energy reserves are taken into account. Bifurcation analysis is performed on the set of nonlinear ordinary differential equations which describe the dynamic behaviour of the food chain. In this paper, we analyse how the ability of a top predator to invade the food chain depends on the values of two control parameters: the dilution rate and the concentration of the substrate in the input. We investigate invasion by studying the long-term behaviour after introduction of a small amount of top predator. To that end we look at the stability of the boundary attractors; equilibria, limit cycles as well as chaotic attractors using bifurcation analysis. It will be shown that the invasibility criterion is the positiveness of the Lyapunov exponent associated with the change of the biomass of the top predator. It appears that the region in the control parameter space where a predator can invade increases with its growth rate. The resulting system becomes more resistant to further invasion when the top predator grows faster. This implies that short food chains with moderate growth rate of the top predator are liable to be invaded by fast growing invaders which consume the top predator. There may be, however, biological constraints on the top predator's growth rate. Predators are generally larger than prey while larger organisms commonly grow slower. As a result, the growth rate generally decreases with the trophic level. This may enable short food chains to be resistant to invaders. We will relate these results to ecological community assembly and the debate on the length of food chains in nature.     

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.     

Variable littoral‐pelagic coupling as a food‐web response to seasonal changes in pelagic primary production

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Biodiversity and ecosystem functioning in food webs: the vertical diversity hypothesis

One challenge in merging community and ecosystem ecology is to integrate the complexity of natural multitrophic communities into concepts of ecosystem functioning. Here, we combine food‐web and allometry theories to demonstrate that primary production, as measured by the total nutrient uptake of the multitrophic community, is determined by vertical diversity (i.e. food web's maximum trophic level) and structure (i.e. distributions of species and their abundances and metabolic rates across trophic levels). In natural ecosystems, the community size distribution determines all these vertical patterns and thus the total nutrient uptake. Our model suggests a vertical diversity hypothesis (VDH) for ecosystem functioning in complex food webs. It predicts that, under a given nutrient supply, the total nutrient uptake increases exponentially with the maximum trophic level in the food web and it increases with its maximum body size according to a power law. The VDH highlights the effect of top–down regulation on plant nutrient uptake, which complements traditional paradigms that emphasised the bottom–up effect of nutrient supply on vertical diversity. We conclude that the VDH contributes to a synthetic framework for understanding the relationship between vertical diversity and ecosystem functioning in food webs and predicting the impacts of global changes on multitrophic ecosystems.     

Bottom‐up and top‐down effects of browning and warming on shallow lake food webs

Productivity and trophic structure of aquatic ecosystems result from a complex interplay of bottom‐up and top‐down forces that operate across benthic and pelagic food web compartments. Projected global changes urge the question how this interplay will be affected by browning (increasing input of terrestrial dissolved organic matter), nutrient enrichment and warming. We explored this with a process‐based model of a shallow lake food web consisting of benthic and pelagic components (abiotic resources, primary producers, grazers, carnivores), and compared model expectations with the results of a browning and warming experiment in nutrient‐poor ponds harboring a boreal lake community. Under low nutrient conditions, the model makes three major predictions. (a) Browning reduces light and increases nutrient supply; this decreases benthic and increases pelagic production, gradually shifting productivity from the benthic to the pelagic habitat. (b) Because of active habitat choice, fish exert top‐down control on grazers and benefit primary producers primarily in the more productive of the two habitats. (c) Warming relaxes top‐down control of grazers by fish and decreases primary producer biomass, but effects of warming are generally small compared to effects of browning and nutrient supply. Experimental results were consistent with most model predictions for browning: light penetration, benthic algal production, and zoobenthos biomass decreased, and pelagic nutrients and pelagic algal production increased with browning. Also consistent with expectations, warming had negative effects on benthic and pelagic algal biomass and weak effects on algal production and zoobenthos and zooplankton biomass. Inconsistent with expectations, browning had no effect on zooplankton and warming effects on fish depended on browning. The model is applicable also to nutrient‐rich systems, and we propose that it is a useful tool for the exploration of the consequences of different climate change scenarios for productivity and food web dynamics in shallow lakes, the worldwide most common lake type.     

Evidence for a Second Chemotactic System in the Cellular Slime Mold, Dictyostelium discoideum

An unknown substance found in bacteria (Escherichia coli) is especially effective in attracting the vegetative amoebae of the cellular slime mold, Dictyostelium discoideum. However, the aggregating amoebae are not attracted to it at all. On the other hand, the vegetative amoebae show very little chemotactic response to cyclic adenosine monophosphate (cyclic AMP), whereas the aggregating amoebae are exceptionally responsive to it. It is suggested that the new factor may be used in food seeking, whereas cyclic AMP, the chemotactic substance responsible for aggregation, is the acrasin of this species. The important point is that the amoebae are differentially stage-specific in their responses to these two chemotactic agents.     

Intraguild predation,invertebrate predators,and trophic cascades in lake food webs

The top-down and bottom-up properties of model food webs that include intraguild predation and self-limiting factors such as cannibalism are investigated. Intraguild predation can dampen or even reverse the top-down effects predicted by food chain theory. The degree of self-limitation among the intraguild prey is a key factor in determining the direction and strength of the top-down response. Intraguild predation and self-limiting factors can also substantially alter the bottom-up effects of enrichment. These results can help explain the disparate results of trophic cascade experiments in lakes, where cascades are usually seen when large Daphnia are the primary herbivores, but not when smaller-bodied herbivores are dominant. Top-down manipulations should cascade at least modestly to phytoplankton in those lakes whose food web can be reasonably approximated by a chain (typically, those where Daphnia is the dominant herbivore), as predicted by food chain theory. On the other hand, smaller-bodied zooplankton are often preyed upon heavily by invertebrate predators as well as by planktivorous fish, thereby introducing elements of intraguild predation into these food webs. In this case, conventional food chain theory is likely to give incorrect predictions. Very large cascade effects may be due primarily to regime shifts between intraguild predation-dominated food webs and those that more resemble food chains, rather than due to the simple food chain cascade usually considered.     

Chaotic motifs in gene regulatory networks

Chaos should occur often in gene regulatory networks (GRNs) which have been widely described by nonlinear coupled ordinary differential equations, if their dimensions are no less than 3. It is therefore puzzling that chaos has never been reported in GRNs in nature and is also extremely rare in models of GRNs. On the other hand, the topic of motifs has attracted great attention in studying biological networks, and network motifs are suggested to be elementary building blocks that carry out some key functions in the network. In this paper, chaotic motifs (subnetworks with chaos) in GRNs are systematically investigated. The conclusion is that: (i) chaos can only appear through competitions between different oscillatory modes with rivaling intensities. Conditions required for chaotic GRNs are found to be very strict, which make chaotic GRNs extremely rare. (ii) Chaotic motifs are explored as the simplest few-node structures capable of producing chaos, and serve as the intrinsic source of chaos of random few-node GRNs. Several optimal motifs causing chaos with atypically high probability are figured out. (iii) Moreover, we discovered that a number of special oscillators can never produce chaos. These structures bring some advantages on rhythmic functions and may help us understand the robustness of diverse biological rhythms. (iv) The methods of dominant phase-advanced driving (DPAD) and DPAD time fraction are proposed to quantitatively identify chaotic motifs and to explain the origin of chaotic behaviors in GRNs.     

Understanding patterns and processes in models of trophic cascades

Climate fluctuations and human exploitation are causing global changes in nutrient enrichment of terrestrial and aquatic ecosystems and declining abundances of apex predators. The resulting trophic cascades have had profound effects on food webs, leading to significant economic and societal consequences. However, the strength of cascades–that is the extent to which a disturbance is diminished as it propagates through a food web–varies widely between ecosystems, and there is no formal theory as to why this should be so. Some food chain models reproduce cascade effects seen in nature, but to what extent is this dependent on their formulation? We show that inclusion of processes represented mathematically as density‐dependent regulation of either consumer uptake or mortality rates is necessary for the generation of realistic ‘top‐down’ cascades in simple food chain models. Realistically modelled ‘bottom‐up’ cascades, caused by changing nutrient input, are also dependent on the inclusion of density dependence, but especially on mortality regulation as a caricature of, e.g. disease and parasite dynamics or intraguild predation. We show that our conclusions, based on simple food chains, transfer to a more complex marine food web model in which cascades are induced by varying river nutrient inputs or fish harvesting rates.     

Parametric Analysis of a Predator–prey System Stabilized by a Top Predator

We present a complete parametric analysis of a predator–prey system influenced by a top predator. We study ecosystems with abundant nutrient supply for the prey where the prey multiplication can be considered as proportional to its density. The main questions we examine are the following: (1) Can the top predator stabilize such a system at low densities of prey? (2) What possible dynamic behaviors can occur? (3) Under which conditions can the top predation result in the system stabilization? We use a system of two nonlinear ordinary differential equations with the density of the top predator as a parameter. The model is investigated with methods of qualitative theory of ODEs and the theory of bifurcations. The existence of 12 qualitatively different types of dynamics and complex structure of the parametric space are demonstrated. Our studies of phase portraits and parametric diagrams show that a top predator can be an important factor leading to stabilization of the predator-prey system with abundant nutrient supply. Although the model here is applied to the plankton communities with fish (or carnivorous zooplankton) as the top trophic level, the general form of the equations allows applications of our results to other ecological systems.