Long food chains are in general chaotic

The question whether chaos exists in nature is much debated. In this paper we prove that chaotic parameter regions exist generically in food chains of length greater than three. While nonchaotic dynamics is also possible, the presence of chaotic parameter regions indicates that chaotic dynamics is likely. We show that the chaotic regions survive even at high exponents of closure. Our results have been obtained using a general food chain model that describes a large class of different food chains. The existence of chaos in models of such generality can be deduced from the presence of certain bifurcations of higher codimension.     

Chaotic Red Queen coevolution in three-species food chains

Coevolution between two antagonistic species follows the so-called ‘Red Queen dynamics’ when reciprocal selection results in an endless series of adaptation by one species and counteradaptation by the other. Red Queen dynamics are ‘genetically driven’ when selective sweeps involving new beneficial mutations result in perpetual oscillations of the coevolving traits on the slow evolutionary time scale. Mathematical models have shown that a prey and a predator can coevolve along a genetically driven Red Queen cycle. We found that embedding the prey–predator interaction into a three-species food chain that includes a coevolving superpredator often turns the genetically driven Red Queen cycle into chaos. A key condition is that the prey evolves fast enough. Red Queen chaos implies that the direction and strength of selection are intrinsically unpredictable beyond a short evolutionary time, with greatest evolutionary unpredictability in the superpredator. We hypothesize that genetically driven Red Queen chaos could explain why many natural populations are poised at the edge of ecological chaos. Over space, genetically driven chaos is expected to cause the evolutionary divergence of local populations, even under homogenizing environmental fluctuations, and thus to promote genetic diversity among ecological communities over long evolutionary time.     

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.     

Omnivory and stability of food webs

The role and prevalence of omnivory, defined as feeding on more than one trophic level, are critical to understand food web structure and dynamics. Whether omnivory stabilizes or destabilizes food webs depends on the assumptions of theoretical models. Recently, Tanabe and Namba [Tanabe, K., Namba, T., 2005. Omivory creates chaos in simple food web models. Ecology 86, 3411–3414] found that omnivory can create chaos in a simple food web model with linear functional responses and 12 model parameters. In this paper, first we numerically examined bifurcation diagrams with all the parameters as bifurcation parameters, including self-limitation of the intermediate consumer and predator. Chaos spontaneously appears when the intraguild predator’s consumption rates are low for nutrient-rich intraguild prey and high for nutrient-poor basal resource and the intraguild prey reproduces efficiently feeding on the basal resource. Second, we investigated effects of the addition of a species into the basic model food web which exhibits chaos. The additional species is assumed to consume only one of the basal resource, intermediate consumer, or omnivorous predator. Consequences of the addition greatly depend on the trophic level on which the additional species feeds. While the increased diversity of predators feeding on the intermediate consumer stabilizes the web, the increased diversity of prey feeding on the basal resource induces collapse of the food web through exploitative competition for the basal resource. The food chain with the top predator feeding on the omnivorous predator is highly unstable unless the mortality of the top predator is extremely low. We discuss the possibility of real-world chaos and the reason why stability of food webs strongly depends on the topological structure of the webs. Finally, we consider the implications of our results for food web theory and resource management.     

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.     

The role of fish life histories in allometrically scaled food‐web dynamics

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Pattern, process and prediction in aquatic ecology. A limnological view of some general ecological problems

1. New scientific concepts such as models of chaos, complex dynamics and non-linear interactions have the potential to contribute to an improved understanding of ecological patterns and processes. This paper discusses some of the known dynamics of phytoplankton, pelagic food chains and nutrient cycles in the light of some of these new concepts. The paper brings these new conceptual models together with data from a wide range of sources in an attempt to produce a synthesis of system behaviour which allows us to understand why some things are inherently more predictable than others. In particular it looks at the limnological management tools of empirical biomass models and biomanipulation and at the need for prediction of species composition. 2. The structures observed in ecosystems (nutrient pools, sizes, species, temporal/spatial patterns) show properties at a spectrum of scales, as do the processes (fluxes, grazing, competition). Both respond to a spectrum of external perturbations that may be climatologically or anthropogenically induced. Empirical biomass models work because of the annual averaging of pattern and process and because of some inherent properties of the functioning of pelagic ecosystems. Many aspects of ecosystem pattern and process vary in a regular way with trophic state. Examination of empirical data sets can lead to an improved understanding of system behaviour if questions are asked about why things happen the way they do. 3. Feedbacks between pattern, process and periodicity are seen to be an inherent property of the system. Understanding the fundamental dynamics of non-linear interactions in ecosystems may make it possible to exploit the external spectrum of environmental perturbations and to control system function. For example, by imposing external physical perturbations on pelagic systems it may be possible to manipulate the species composition of the phytoplankton community. Because of the complexity of possible interactions both ‘horizontally’ between species and ‘vertically’ within the food chain, any prediction of species composition will necessarily be probabilistic.     

Multi-objective optimization of an ecological assembly model

The aim of the present work is to use multi-objective evolutionary algorithms (MOEA) to parameterise an ecological assembly model based on Lotka–Volterra dynamics. In community assembly models, species are introduced from a pool of species according to a sequence of invasion. By manipulating the assembly sequences, we look at the structure of the final communities obtained by a multi-objective process where the goal is to optimize the productivity of the final communities. The MOEA must also meet the constraint that the communities constructed in this fashion have a specified connectance. The Non-dominated Sorting Algorithm (NSGA-II) and the Strength Pareto Evolutionary Algorithm (SPEA2) were employed to optimize sequences according to the multi-objective optimization problem. The results show that the assembly process using optimized sequences generated different community structure than those generated via random sequences. First, the assembled communities are much more productive than those obtained from random sequences. We show that this increase of productivity is due to the degree distribution of the community food web, which was reshaped by the optimization process. In addition, using identical regional species pools the MOEAs were able to generate communities of different expected connectances. These results demonstrate the effectiveness of NSGA-II and SPEA2 for optimizing parameters in ecological models.     

Dynamic complexities in host-parasitoid interaction

In the 1970s ecological research detected chaos and other forms of complex dynamics in simple population dynamics models, initiating a new research tradition in ecology. However, the investigations of complex population dynamics have mainly concentrated on single populations and not on higher dimensional ecological systems. Here we report a detailed study of the complicated dynamics occurring in a basic discrete-time model of host-parasitoid interaction. The complexities include (a) non-unique dynamics, meaning that several attractors coexist, (b) basins of attraction (defined as the set of the initial conditions leading to a certain type of an attractor) with fractal properties (pattern of self-similarity and fractal basin boundaries), (c) intermittency, (d) supertransients, (e) chaotic attractors, and (f) "transient chaos". Because of these complexities minor changes in parameter or initial values may strikingly change the dynamic behavior of the system. All the phenomena presented in this paper should be kept in mind when examining and interpreting the dynamics of ecological systems. Copyright 1999 Academic Press.     

Forecasting species ranges by statistical estimation of ecological niches and spatial population dynamics

Aim The study and prediction of species–environment relationships is currently mainly based on species distribution models. These purely correlative models neglect spatial population dynamics and assume that species distributions are in equilibrium with their environment. This causes biased estimates of species niches and handicaps forecasts of range dynamics under environmental change. Here we aim to develop an approach that statistically estimates process‐based models of range dynamics from data on species distributions and permits a more comprehensive quantification of forecast uncertainties. Innovation We present an approach for the statistical estimation of process‐based dynamic range models (DRMs) that integrate Hutchinson's niche concept with spatial population dynamics. In a hierarchical Bayesian framework the environmental response of demographic rates, local population dynamics and dispersal are estimated conditional upon each other while accounting for various sources of uncertainty. The method thus: (1) jointly infers species niches and spatiotemporal population dynamics from occurrence and abundance data, and (2) provides fully probabilistic forecasts of future range dynamics under environmental change. In a simulation study, we investigate the performance of DRMs for a variety of scenarios that differ in both ecological dynamics and the data used for model estimation. Main conclusions Our results demonstrate the importance of considering dynamic aspects in the collection and analysis of biodiversity data. In combination with informative data, the presented framework has the potential to markedly improve the quantification of ecological niches, the process‐based understanding of range dynamics and the forecasting of species responses to environmental change. It thereby strengthens links between biogeography, population biology and theoretical and applied ecology.     

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.     

Does predator interference cause alternative stable states in multispecies communities?

Whereas it is well known that simple ecological mechanisms may promote stability in simple species models, their consequences for stability and resilience in multispecies communities are largely unexplored. Here, we studied the effect of predator interference on the occurrence of alternative attractors and complex dynamics in randomly constructed multispecies predator-prey communities. We studied three types of interference: random interference (“asymmetric”), random interference but symmetrical between pairs of predators (“symmetric”), and interference among only the same species (“conspecific”). In all cases predator interference increased the average number of alternative attractors, whereas at the same time it reduced the emergence of oscillatory or chaotic dynamics. Our findings demonstrate a contrasting effect of predator interference on the stability of a community: on the one hand it reduces cycles and chaos in the dynamics, on the other hand predator interference increases the likelihood that communities may undergo critical transitions between multiple stable states.     

Exploring the temporal variability of a food web using long‐term biomonitoring data

Ecological communities are constantly being reshaped in the face of environmental change and anthropogenic pressures. Yet, how food webs change over time remains poorly understood. Food web science is characterized by a trade‐off between complexity (in terms of the number of species and feeding links) and dynamics. Topological analysis can use complex, highly resolved empirical food web models to explore the architecture of feeding interactions but is limited to a static view, whereas ecosystem models can be dynamic but use highly aggregated food webs. Here, we explore the temporal dynamics of a highly resolved empirical food web over a time period of 18 years, using the German Bight fish and benthic epifauna community as our case study. We relied on long‐term monitoring ecosystem surveys (from 1998 to 2015) to build a metaweb, i.e. the meta food web containing all species recorded over the time span of our study. We then combined time series of species abundances with topological network analysis to construct annual food web snapshots. We developed a new approach, ‘node‐weighted’ food web metrics by including species abundances to represent the temporal dynamics of food web structure, focusing on generality and vulnerability. Our results suggest that structural food web properties change through time; however, binary food web structural properties may not be as temporally variable as the underlying changes in species composition. Further, the node‐weighted metrics enabled us to detect that food web structure was influenced by changes in species composition during the first half of the time series and more strongly by changes in species dominance during the second half. Our results demonstrate how ecosystem surveys can be used to monitor temporal changes in food web structure, which are important ecosystem indicators for building marine management and conservation plans.     

Trophic cascades alter eco-evolutionary dynamics and body size evolution

Trait evolution in predator–prey systems can feed back to the dynamics of interacting species as well as cascade to impact the dynamics of indirectly linked species (eco-evolutionary trophic cascades; EETCs). A key mediator of trophic cascades is body mass, as it both strongly influences and evolves in response to predator–prey interactions. Here, we use Gillespie eco-evolutionary models to explore EETCs resulting from top predator loss and mediated by body mass evolution. Our four-trophic-level food chain model uses allometric scaling to link body mass to different functions (ecological pleiotropy) and is realistically parameterized from the FORAGE database to mimic the parameter space of a typical freshwater system. To track real-time changes in selective pressures, we also calculated fitness gradients for each trophic level. As predicted, top predator loss generated alternating shifts in abundance across trophic levels, and, depending on the nature and strength in changes to fitness gradients, also altered trajectories of body mass evolution. Although more distantly linked, changes in the abundance of top predators still affected the eco-evolutionary dynamics of the basal producers, in part because of their relatively short generation times. Overall, our results suggest that impacts on top predators can set off transient EETCs with the potential for widespread indirect impacts on food webs.     

Behaviour of simple population models under ecological processes

The two most popular and extensively-used discrete models of population growth display the generic bifurcation structure of a hierarchy of period-doubling sequence to chaos with increasing growth rates. In this paper we show that these two models, though they belong to a general class of one-dimensional maps, show very different dynamics when important ecological processes such as immigration and emigration/depletion, are considered. It is important that ecologists recognize the differences between these models before using them to describe their data—or develop optimization strategies—based on these models.     

The effects of space and diversity of interaction types on the stability of complex ecological networks

The relationship between structure and stability in ecological networks and the effect of spatial dynamics on natural communities have both been major foci of ecological research for decades. Network research has traditionally focused on a single interaction type at a time (e.g. food webs, mutualistic networks). Networks comprising different types of interactions have recently started to be empirically characterized. Patterns observed in these networks and their implications for stability demand for further theoretical investigations. Here, we employed a spatially explicit model to disentangle the effects of mutualism/antagonism ratios in food web dynamics and stability. We found that increasing levels of plant-animal mutualistic interactions generally resulted in more stable communities. More importantly, increasing the proportion of mutualistic vs. antagonistic interactions at the base of the food web affects different aspects of ecological stability in different directions, although never negatively. Stability is either not influenced by increasing mutualism—for the cases of population stability and species’ spatial distributions—or is positively influenced by it—for spatial aggregation of species. Additionally, we observe that the relative increase of mutualistic relationships decreases the strength of biotic interactions in general within the ecological network. Our work highlights the importance of considering several dimensions of stability simultaneously to understand the dynamics of communities comprising multiple interaction types.     

The mesoscavenger release hypothesis and implications for ecosystem and human well‐being

Many apex scavenger species, including nearly all obligate scavengers, are in a state of rapid decline and there is growing evidence these declines can drastically alter ecological food webs. Our understanding of how apex scavengers regulate populations of mesoscavengers, those less‐efficient scavengers occupying mid‐trophic levels, is improving; yet, there has been no comprehensive evaluation of the evidence around the competitive release of these species by the loss of apex scavengers. Here we present current evidence that supports the mesoscavenger release hypothesis, the increase in mesoscavengers and increase in carrion in the face of declining apex scavengers. We provide two models of scavenger dynamics to demonstrate that the mesoscavenger release hypothesis is consistent with ecological theory. We further examine the ecological and human well‐being implications of apex scavenger decline, including carrion removal and disease regulation services.     

Bifurcations and chaos in a predator-prey system with the Allee effect

It is known from many theoretical studies that ecological chaos may have numerous significant impacts on the population and community dynamics. Therefore, identification of the factors potentially enhancing or suppressing chaos is a challenging problem. In this paper, we show that chaos can be enhanced by the Allee effect. More specifically, we show by means of computer simulations that in a time-continuous predator-prey system with the Allee effect the temporal population oscillations can become chaotic even when the spatial distribution of the species remains regular. By contrast, in a similar system without the Allee effect, regular species distribution corresponds to periodic/quasi-periodic oscillations. We investigate the routes to chaos and show that in the spatially regular predator-prey system with the Allee effect, chaos appears as a result of series of period-doubling bifurcations. We also show that this system exhibits period-locking behaviour: a small variation of parameters can lead to alternating regular and chaotic dynamics.     

Effects of network and dynamical model structure on species persistence in large model food webs

Four models of network structure are combined with models of bioenergetic dynamics to study the role of food web topology and nonlinear dynamics on species coexistence in complex ecological networks. Network models range from the highly structured niche model to loosely constrained energetically feasible random networks. Bioenergetic models differ in how they represent primary production, functional responses, and consumption by generalists. Network structure weakly influenced the ability of species to coexist. Species persistence is strongly affected by functional responses and generalists’ consumption rates but weakly affected by models and amounts of primary production. Despite these generalities, specific mechanisms that determine persistence under one dynamical regime, such as top-down control by consumers, may play an insignificant role under different dynamical conditions. Future research is needed to strengthen the weak empirical basis for various functional forms and parameter values that strongly influence whether species can coexist in complex food webs. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

When can noise induce chaos and why does it matter: a critique

Noise‐induced chaos illustrates how small amounts of exogenous noise can have disproportionate qualitative impacts on the long term dynamics of a nonlinear system. This property is particularly clear in chaotic systems but is also important for the majority of ecological systems which are nonchaotic, and has direct implications for analyzing ecological time series and testing models against field data. Dennis et al. point out that a definition of chaos which we advocated allows a noise‐dominated system to be classified as chaotic when its Lyapunov exponent λ is positive, which misses what is really going on. As a solution, they propose to eliminate the concept of noise‐induced chaos: chaos “should retain its strictly deterministic definition”, hence “ecological populations cannot be strictly chaotic”. Instead, they suggest that ecologists ask whether ecological systems are strongly influenced by “underlying skeletons with chaotic dynamics or whatever other dynamics”– the skeleton being the hypothetical system that would result if all external and internal noise sources were eliminated. We agree with Dennis et al. about the problem – noise‐dominated systems should not be called chaotic – but not the solution. Even when an estimated skeleton predicts a system's short term dynamics with extremely high accuracy, the skeleton's long term dynamics and attractor may be very different from those of the actual noisy system. Using theoretical models and empirical data on microtine rodent cycles and laboratory populations of Tribolium, we illustrate how data analyses focusing on attributes of the skeleton and its attractor – such as the “deterministic Lyapunov exponent”λ₀ that Dennis et al. have used as their primary indicator of chaos – will frequently give misleading results. In contrast, quantitative measures of the actual noisy system, such as λ, provide useful information for characterizing observed dynamics and for testing proposed mechanistic explanations.