Global stability, local stability and permanence in model food webs

The dynamical theory of food webs has been based typically on local stability analysis. The relevance of local stability to food web properties has been questioned because local stability holds only in the immediate vicinity of the equilibrium and provides no information about the size of the basin of attraction. Local stability does not guarantee persistence of food webs in stochastic environments. Moreover, local stability excludes more complex dynamics such as periodic and chaotic behaviors, which may allow persistence. Global stability and permanence could be better criteria of community persistence. Our simulation analysis suggests that these three stability measures are qualitatively consistent in that all three predict decreasing stability with increasing complexity. Some new predictions on how stability depends on food web configurations are generated here: a consumer-victim link has a smaller effect on the probabilities of stability, as measured by all three stability criteria, than a pair of recipient-controlled and donor-controlled links; a recipient-controlled link has a larger effect on the probabilities of local stability and permanence than a donor-controlled link, while they have the same effect on the probability of global stability; food webs with equal proportions of donor-controlled and recipient-controlled links are less stable than those with different proportions.     

Neutral stability in many-species food webs

The predator-prey systems of Lotka and Volterra are modified to include efficiencies e_ij of converting prey biomass to predator biomass. For three or more species, it is shown that neutral stability results if e_ik = e_ije_jk for all transfers. When the number of species is odd, consistency of the differential equation system for changes in logarithms of biomass is also required.     

Biodiversity maintenance in food webs with regulatory environmental feedbacks

Although the food web is one of the most fundamental and oldest concepts in ecology, elucidating the strategies and structures by which natural communities of species persist remains a challenge to empirical and theoretical ecologists. We show that simple regulatory feedbacks between autotrophs and their environment when embedded within complex and realistic food-web models enhance biodiversity. The food webs are generated through the niche-model algorithm and coupled with predator-prey dynamics, with and without environmental feedbacks at the autotroph level. With high probability and especially at lower, more realistic connectance levels, regulatory environmental feedbacks result in fewer species extinctions, that is, in increased species persistence. These same feedback couplings, however, also sensitize food webs to environmental stresses leading to abrupt collapses in biodiversity with increased forcing. Feedback interactions between species and their material environments anchor food-web persistence, adding another dimension to biodiversity conservation. We suggest that the regulatory features of two natural systems, deep-sea tubeworms with their microbial consortia and a soil ecosystem manifesting adaptive homeostatic changes, can be embedded within niche-model food-web dynamics.     

Omnivory and the stability of food webs

The ecological concept of omnivory, feeding at more than a single trophic level, is formulated as an intermediate stage between any two of three classical three-dimensional species interaction systems-tritrophic chain, competition, and polyphagy. It is shown that omnivory may be either stabilizing or destabilizing, depending, in part, on the conditions of the parent systems from which it derives. It is further conjectured that the tritrophic to competition gradient cannot be entirely stable, that there must be an instability at some level of intermediate omnivory.     

Biodiversity and the productivity and stability of ecosystems

Attempts to unveil the relationships between the taxonomic diversity, productivity and stability of ecosystems continue to generate inconclusive, contradictory and controversial conclusions. New insights from recent studies support the hypothesis that species diversity enhances productivity and stability in some ecosystems, but not in others. Appreciation is growing for the ways that particular ecosystem features, such as environmental variability and nutrient stress, can influence biotic interactions. Alternatives to the diversity-stability hypothesis have been proposed, and experimental approaches are starting to evolve to test these hypotheses and to elucidate the mechanisms underlying the functional role of species diversity.     

Omnivory and the stability of simple food webs

Traditional ecological theory predicts that the stability of simple food webs will decline with an increasing number of trophic levels and increasing amounts of omnivory. These ideas have been tested using protozoans in laboratory microcosms. However, the results are equivocal, and contrary to expectation, omnivory is common in natural food webs. Two recent developments lead us to re-evaluate these predictions using food webs assembled from protists and bacteria. First, recent modelling work suggests that omnivory is actually stabilizing, providing that interactions are not too strong. Second, it is difficult to evaluate the degree of omnivory of some protozoan species without explicit experimental tests. This study used seven species of ciliated protozoa and a mixed bacterial flora to assemble four food webs with two trophic levels, and four webs with three trophic levels. Protist species were assigned a rank for their degree of omnivory using information in the literature and the results of experiments that tested whether the starvation rate of predators was influenced by the amount of bacteria on which they may have fed and whether cannibalism (a form of omnivory) occurred. Consistent with recent modelling work, both bacterivorous and predatory species with higher degrees of omnivory showed more stable dynamics, measured using time until extinction and the temporal variability of population density. Systems with two protist species were less persistent than systems with one protist species, supporting the prediction that longer food chains will be less stable dynamically. Received: 28 December 1997 / Accepted: 22 June 1998     

Biodiversity, ecosystem functioning and food webs in fresh waters: assembling the jigsaw puzzle

1. Dramatic advances have been made recently in the study of biodiversity–ecosystem functioning (B-EF) relations and food web ecology. These fields are now starting to converge, and this fusion has the potential to improve our understanding of how environmental stressors modulate ecosystem processes and the supply of 'goods and services'.
2. Food web structure and dynamics can exert particularly strong influences on B-EF relations in fresh waters, as consumer–resource interactions (e.g. trophic cascades) are often more important than horizontal interactions within trophic levels. For instance, many freshwater food webs are size structured, with large organisms tending to occupy the higher trophic levels and often exerting powerful effects on ecosystem processes. However, because they are also vulnerable to perturbations, non-random losses of these large taxa can alter both food web structure and ecosystem functioning profoundly.
3. Recently, the focus of food web research has shifted away from exploring patterns, towards developing an understanding of processes (e.g. quantifying fluxes of individuals, biomass, energy, nutrients) and how the two interact. Many of the best-characterized food webs are from fresh waters, and these ecosystems are now being used to address some of the shortcomings of earlier B-EF studies. I have identified several key gaps in our current knowledge and highlighted potentially fruitful avenues of future B-EF and food web research.
4. A major challenge for this newly emerging research is to place it within a unified theoretical framework. The application of metabolic theory and ecological stoichiometry may help to achieve this goal by considering biological systems within the constraints imposed upon them by physical and chemical laws.     

Biomass diversity and stability of food webs in aquatic ecosystems

The diversity–stability hypothesis in ecology asserts that biodiversity begets stability of ecological systems. This hypothesis has been supported by field studies on primary producers in grasslands, in which the interaction between species is mostly competition. As to ecosystems with multitrophic predatory interaction, however, no definite consensus has been arrived at for the relation between trophic diversity and ecosystem stability. The stability index suitable to ecosystems with predatory interaction is given by MacArthurs idea of stability and its formulation by Rutledge et al. More suitable indices of stability (relative conditional entropy) are proposed in this study for the comparison of different ecosystems, and the validity of the diversity–stability hypothesis for food webs (networks of predation) with many trophic compartments in natural aquatic ecosystems is examined. Results reveal that an increase in the biomass diversity of trophic compartments causes an increase in the whole systemic stability of food webs in aquatic ecosystems. Hence, evidence of the whole systemic validity of the diversity–stability hypothesis for natural aquatic ecosystems with ubiquitous multitrophic predatory interaction was obtained for the first time.     

Linking saturation,stability and sustainability in food webs with observed equilibrium structure

Stability of a dynamic equilibrium in a predator-prey system depends both on the type of functional response and on the point of equilibrium on the response curve. Saturation effects from Holling type II responses are known to destabilise prey populations, while a type III (sigmoid) response curve has been shown to provide stability at lower levels of saturation. These effects have also been shown in multi-trophic model systems. However, stability analyses of observed equilibria in real complex ecosystems have as yet not assumed non-linear functional responses. Here, we evaluate the implications of saturation in observed balanced material-flow structures, for system stability and sustainability. We first make the effects of the non-linear functional responses on the interaction strengths in a food web transparent by expressing the elements of Jacobian ‘community’ matrices for type II and III systems as simple functions of their linear (type I) counterparts. We then determine the stability of the systems and distinguish two critical saturation levels: (1) a level where the system is just as stable as a type I system and (2) a level above which the system cannot be stable unless it is subsidised, separating a stable materially sustainable regime from an unsustainable one. We explain the stabilising and destabilising effects in terms of the feedbacks in the systems. The results shed light on the robustness of observed patterns of interaction strengths in complex food webs and suggest the implausibility of saturation playing a significant role in the equilibrium dynamics of sustainable ecosystems.     

Global change alters the stability of food webs

Recent research has generally shown that a small change in the number of species in a food web can have consequences both for community structure and ecosystem processes. However ‘change’ is not limited to just the number of species in a community, but might include an alteration to such properties as precipitation, nutrient cycling and temperature. How such changes might affect species interactions is important, not just through the presence or absence of interactions, but also because the patterning of interaction strengths among species is intimately associated with community stability. Interaction strengths encompass such properties as feeding rates and assimilation efficiencies, and encapsulate functionally important information with regard to ecosystem processes. Interaction strengths represent the pathways and transfer of energy through an ecosystem. We review the best empirical data available detailing the frequency distribution of interaction strengths in communities. We present the underlying (but consistent) pattern of species interactions and discuss the implications of this patterning. We then examine how such a basic pattern might be affected given various scenarios of ‘change’ and discuss the consequences for community stability and ecosystem functioning.     

Relation between complexity and stability in food webs with adaptive behavior

We investigate the influence of functional responses (Lotka-Volterra or Holling type), initial topological web structure (randomly connected or niche model), adaptive behavior (adaptive foraging and predator avoidance) and the type of constraints on the adaptive behavior (linear or nonlinear) on the stability and structure of food webs. Two kinds of stability are considered: one is the network robustness (i.e., the proportion of species surviving after population dynamics) and the other is the species deletion stability. When evaluating the network structure, we consider link density as well as the trophic level structure. We show that the types of functional responses and initial web structure do not have a large effect on the stability of food webs, but foraging behavior has a large stabilizing effect. It leads to a positive complexity-stability relationship whenever higher "complexity" implies more potential prey per species. The other type of adaptive behavior, predator avoidance behavior, makes food webs only slightly more stable. The observed link density after population dynamics depends strongly on the presence or absence of adaptive foraging, and on the type of constraints used. We also show that the trophic level structure is preserved under population dynamics with adaptive foraging.     

Patterns in intraspecific interaction strengths and the stability of food webs

A common approach to analyse stability of biological communities is to calculate the interaction strength matrix. Problematic in this approach is defining intraspecific interaction strengths, represented by diagonal elements in the matrix, due to a lack of empirical data for these strengths. Theoretical studies have shown that an overall increase in these strengths enhances stability. However, the way in which the pattern in intraspecific interaction strengths, i.e. the variation in these strengths between species, influences stability has received little attention. We constructed interaction strength matrices for 11 real soil food webs in which four patterns for intraspecific interaction strengths were chosen, based on the ecological literature. These patterns included strengths that were (1) similar for all species, (2) trophic level dependent, (3) biomass dependent, or (4) death rate dependent. These four patterns were analysed for their influence on (1) ranking food webs by their stability and (2) the response in stability to variation of single interspecific interaction strengths. The first analysis showed that ranking the 11 food webs by their stability was not strongly influenced by the choice of diagonal pattern. In contrast, the second analysis showed that the response of food web stability to variation in single interspecific interaction strengths was sensitive to the choice of diagonal pattern. Notably, stability could increase using one pattern and decrease using another. This result asks for deliberate approaches to choose diagonal element values in order to make predictions on how particular species, interactions, or other food web parameters affect food web stability.     

The dynamics of top-down and bottom-up effects in food webs of varying prey diversity, composition, and productivity

Prey diversity is thought to mediate the strength of top-down and bottom-up effects, but few experiments directly test this hypothesis. I assembled food webs of bacteria and bacterivorous protist prey in laboratory microcosms with all combinations of five productivity levels, two top predator treatments (present or absent), and three prey compositions. Depauperate food chains contained one of two edible prey species, while more diverse food webs contained both edible prey species plus two additional less-edible/inedible prey. Equilibrium theory predicts that prey diversity should weaken the top-down and bottom-up effects on trophic level biomasses, due to density compensation among prey species. Top-down effects should increase with productivity in food chains, but decrease with productivity in food webs. Results revealed highly dynamic top-down effects, the strength of which varied more over time than among treatments. Further, top-down effects did not merely vary in absolute strength over time, but also in relative strength across different prey compositions and productivity levels. It might be expected that equilibrium models would qualitatively reproduce time-averaged results. However, time-averaged data largely failed to support equilibrium predictions. This failure may reflect strong temporal variability in treatment effects combined with nonlinear density dependence of species' per-capita growth rates. Strong temporal variability in the strength of top-down effects has not previously been demonstrated, but likely is common in nature as well.     

Microbial food chains and food webs

Mathematical models for simple microbial food chains and food webs in continuous culture are developed and analyzed. A model for competition of two microbial species for a single scarce resource is also presented as a degenerate case of the food web model. Two models for food chains are developed. The first is based on a model of microbial growth (Monod's) that is widely mentioned and used at the present time. The second is based on a generalization of that model that recent experimental results on microbial food chains seem to require. Experimental data for microbial food webs are almost entirely lacking but a tentative model having what are felt to be the right properties is developed and analyzed. The results obtained from these models seem to be consistent in most circumstances with current ecological thinking on community dynamics.     

Topological properties of food webs: from real data to community assembly models

We explore patterns of trophic connections between species in the largest and highest-quality empirical food webs to date, introducing a new topological property called the link distribution frequency (i.e. degree distribution), defined as the frequency of species S _L with L links. Non-trivial differences are shown in link distribution frequencies between species-rich and species-poor communities, which might have important consequences for the responses of ecosystems to disturbances. Coarse-grained topological properties observed, as species richness-connectance and number of links-species richness relationships, provide no support for the theory of links-species scaling law or constant connectance across empirical food webs investigated. We further explore these observations by means of simulated food webs resulting from multitrophic assembly models using different functional responses between species. Species richness-connectance and links-species richness relationships of empirical food webs are reproduced by our models, but degree distributions are not properly predicted, suggesting the need of new theoretical approximations to food web assembly. The best agreement between empirical and simulated webs occurs for low values of interaction strength between species, corroborating previous empirical and theoretical findings where weak interactions govern food web dynamics.     

Threshold extinction in food webs

Food web response to species loss has been investigated in several ways in the previous years. In binary food webs, species go secondarily extinct if no resource item remains to be exploited. In this work, we considered that species can go extinct before the complete loss of their resources and we introduced thresholds of minimum energy requirement for species survival. According to this approach, extinction of a node occurs whenever an initial extinction event eliminates its incoming links so it is left with an overall energy intake lower than the threshold value. We tested the robustness of 18 real food webs by removing species from most to least connected and considering different scenarios defined by increasing the extinction threshold. Increasing energy requirement threshold negatively affects food web robustness. We found that a very small increase of the energy requirement substantially increases system fragility. In addition, above a certain value of energy requirement threshold we found no relationship between the robustness and the connectance of the web. Further, food webs with more species showed higher fragility with increasing energy threshold. This suggests that the shape of the robustness–complexity relationship of a food web depends on the sensitivity of consumers to loss of prey.     

Keystone species and food webs

Different species are of different importance in maintaining ecosystem functions in natural communities. Quantitative approaches are needed to identify unusually important or influential, ‘keystone’ species particularly for conservation purposes. Since the importance of some species may largely be the consequence of their rich interaction structure, one possible quantitative approach to identify the most influential species is to study their position in the network of interspecific interactions. In this paper, I discuss the role of network analysis (and centrality indices in particular) in this process and present a new and simple approach to characterizing the interaction structures of each species in a complex network. Understanding the linkage between structure and dynamics is a condition to test the results of topological studies, I briefly overview our current knowledge on this issue. The study of key nodes in networks has become an increasingly general interest in several disciplines: I will discuss some parallels. Finally, I will argue that conservation biology needs to devote more attention to identify and conserve keystone species and relatively less attention to rarity.     

Making connections in food webs

Patterns in food web structure have provided an important, though contentious, testing ground for ideas about the population dynamics and energetics of multispecies systems. One of the most debated of these patterns is the apparent decrease in food web connectance as the number of species in a web Increases. Several contrasting mechanisms that might determine food web connectance have been suggested. These mechanisms, in combination with new, food web data, suggest that the conventional pattern, and explanations for it, may well be open to dispute. The true nature of the relationship between connectance and species number has implications for the explanation of other web patterns and for theories of food web structure, but a general explanation remains elusive.