Lars Rudolf

Generalized Models Reveal Stabilizing Factors in Food Webs

Untangling Food Webs The factors affecting the stability of food webs are important in conservation and ecological restoration. Gross et al. (p. 747) used a generalized modeling approach to evaluate billions of replicates of food webs in order to reveal the properties that stabilize (or destabilize) food webs. Variability in the strength of trophic links between predator and prey strength affected stability in different ways depending on the size of the web—stabilizing only in relatively small food webs and destabilizing in larger ones. Universal topological rules were extracted for the patterns of network links that enhance food-web stability. Analysis of several billion replicates of food webs reveals universal topological rules affecting their stability. Insights into what stabilizes natural food webs have always been limited by a fundamental dilemma: Studies either need to make unwarranted simplifying assumptions, which undermines their relevance, or only examine few replicates of small food webs, which hampers the robustness of findings. We used generalized modeling to study several billion replicates of food webs with nonlinear interactions and up to 50 species. In this way, first we show that higher variability in link strengths stabilizes food webs only when webs are relatively small, whereas larger webs are instead destabilized. Second, we reveal a new power law describing how food-web stability scales with the number of species and their connectance. Third, we report two universal rules: Food-web stability is enhanced when (i) species at a high trophic level feed on multiple prey species and (ii) species at an intermediate trophic level are fed upon by multiple predator species.

Vertical distribution and composition of phytoplankton under the influence of an upper mixed layer

The vertical distribution of phytoplankton is of fundamental importance for the dynamics and structure of aquatic communities. Here, using an advection-reaction-diffusion model, we investigate the distribution and competition of phytoplankton species in a water column, in which inverse resource gradients of light and a nutrient can limit growth of the biomass. This problem poses a challenge for ecologists, as the location of a production layer is not fixed, but rather depends on many internal parameters and environmental factors. In particular, we study the influence of an upper mixed layer (UML) in this system and show that it leads to a variety of dynamic effects: (i) Our model predicts alternative density profiles with a maximum of biomass either within or below the UML, thereby the system may be bistable or the relaxation from an unstable state may require a long-lasting transition. (ii) Reduced mixing in the deep layer can induce oscillations of the biomass; we show that a UML can sustain these oscillations even if the diffusivity is less than the critical mixing for a sinking phytoplankton population. (iii) A UML can strongly modify the outcome of competition between different phytoplankton species, yielding bistability both in the spatial distribution and in the species composition. (iv) A light limited species can obtain a competitive advantage if the diffusivity in the deep layers is reduced below a critical value. This yields a subtle competitive exclusion effect, where the oscillatory states in the deep layers are displaced by steady solutions in the UML. Finally, we present a novel graphical approach for deducing the competition outcome and for the analysis of the role of a UML in aquatic systems.

Collapse of an ecological network in Ancient Egypt.

Significance The composition of animal communities directly impacts the stability of ecosystems. Here, we use historical information of species extinctions in Egypt over 6,000 years to reconstruct predator–prey interactions and determine to what extent observed changes in species composition influence predictions of community stability. Our study reveals that the roles of species and the stability of the community have fundamentally changed throughout the Holocene, and provides compelling evidence that local dynamic stability is informative of species persistence over time. The dynamics of ecosystem collapse are fundamental to determining how and why biological communities change through time, as well as the potential effects of extinctions on ecosystems. Here, we integrate depictions of mammals from Egyptian antiquity with direct lines of paleontological and archeological evidence to infer local extinctions and community dynamics over a 6,000-y span. The unprecedented temporal resolution of this dataset enables examination of how the tandem effects of human population growth and climate change can disrupt mammalian communities. We show that the extinctions of mammals in Egypt were nonrandom and that destabilizing changes in community composition coincided with abrupt aridification events and the attendant collapses of some complex societies. We also show that the roles of species in a community can change over time and that persistence is predicted by measures of species sensitivity, a function of local dynamic stability. To our knowledge, our study is the first high-resolution analysis of the ecological impacts of environmental change on predator–prey networks over millennial timescales and sheds light on the historical events that have shaped modern animal communities.

How to predict community responses to perturbations in the face of imperfect knowledge and network complexity

Recent attempts to predict the response of large food webs to perturbations have revealed that in larger systems increasingly precise information on the elements of the system is required. Thus, the effort needed for good predictions grows quickly with the systems complexity. Here, we show that not all elements need to be measured equally well, suggesting that a more efficient allocation of effort is possible. We develop an iterative technique for determining an efficient measurement strategy. In model food webs, we find that it is most important to precisely measure the mortality and predation rates of long-lived, generalist, top predators. Prioritizing the study of such species will make it easier to understand the response of complex food webs to perturbations.

Patterns of cooperation: fairness and coordination in networks of interacting agents

We study the self-assembly of a complex network of collaborations among self-interested agents. The agents can maintain different levels of cooperation with different partners. Further, they continuously, selectively and independently adapt the amount of resources allocated to each of their collaborations in order to maximize the obtained payoff. We show analytically that the system approaches a state in which the agents make identical investments, and links produce identical benefits. Despite this high degree of social coordination, some agents manage to secure privileged topological positions in the network, enabling them to extract high payoffs. Our analytical investigations provide a rationale for the emergence of unidirectional non-reciprocal collaborations and different responses to the withdrawal of a partner from an interaction that have been reported in the psychological literature.

Mesoscale symmetries explain dynamical equivalence of food webs

A goal of complex system research is to identify the dynamical implications of network structure. While early results focused mainly on local or global structural properties, there is now growing interest in mesoscale structures that comprise more than one node but not the whole network. A central challenge is to discover under what conditions the occurrence of a specific mesoscale motif already allows conclusions on the dynamics of a network as a whole. In this paper, we investigate the dynamics of ecological food webs, complex heterogeneous networks of interacting populations. Generalizing the results of MacArthur and Sanchez-Garcia (2009 Phys. Rev. E 80 26117), we show that certain mesoscale symmetries imply the existence of localized dynamical modes. If these modes are unstable the occurrence of the corresponding mesoscale motif implies dynamical instability regardless of the structure of the embedding network. In contrast, if the mode is stable it means that the symmetry can be exploited to reduce the number of nodes in the model, without changing the dynamics of the system. This result explains a previously observed dynamical equivalence between food webs containing a different number of species.

Impact of dispersal on the stability of metapopulations

Dispersal is a key ecological process that enables local populations to form spatially extended systems called metapopulations. In the present study, we investigate how dispersal affects the linear stability of a general single-species metapopulation model. We discuss both the influence of local within-patch dynamics and the effects of various dispersal behaviours on stability. We find that positive density-dependent dispersal and positive density-dependent settlement are destabilizing dispersal behaviours while negative density-dependent dispersal and negative density-dependent settlement are stabilizing. It is also shown that dispersal has a stabilizing impact on heterogeneous metapopulations that correlates positively with the number of patches and the connectance of metapopulation networks.

The influence of dispersal on a predator-prey system with two habitats.

Dispersal between different habitats influences the dynamics and stability of populations considerably. Furthermore, these effects depend on the local interactions of a population with other species. Here, we perform a general and comprehensive study of the simplest possible system that includes dispersal and local interactions, namely a 2-patch 2-species system. We evaluate the impact of dispersal on stability and on the occurrence of bifurcations, including pattern forming bifurcations that lead to spatial heterogeneity, in 19 different classes of models with the help of the generalized modelling approach. We find that dispersal often destabilizes equilibria, but it can stabilize them if it increases population losses. If dispersal is nonrandom, i.e. if emigration or immigration rates depend on population densities, the correlation of stability with dispersal rates is positive in part of the models. We also find that many systems show all four types of bifurcations and that antisynchronous oscillations occur mostly with nonrandom dispersal.

Revealing instabilities in a generalized triadic supply network: A bifurcation analysis

Supply networks are exposed to instabilities and thus a high level of risk. To mitigate this risk, it is necessary to understand how instabilities are formed in supply networks. In this paper, we focus on instabilities in inventory dynamics that develop due to the topology of the supply network. To be able to capture these topology-induced instabilities, we use a method called generalized modeling, a minimally specified modeling approach adopted from ecology. This method maps the functional dependencies of production rates on the inventory levels of different parts and products, which are imposed by the network topology, to a set of elasticity parameters. We perform a bifurcation analysis to investigate how these elasticities affect the stability. First, we show that dyads and serial supply chains are immune to topology-induced instabilities. In contrast, in a simple triadic network, where a supplier acts as both a first and a second tier supplier, we can identify instabilities that emerge from saddle-node, Hopf, and global homoclinic bifurcations. These bifurcations lead to different types of dynamical behavior, including exponential convergence to and divergence from a steady state, temporary oscillations around a steady state, and co-existence of different types of dynamics, depending on initial conditions. Finally, we discuss managerial implications of the results.

Coordination, Differentiation and Fairness in a Population of Cooperating Agents

In a recent paper, we analyzed the self-assembly of a complex cooperation network. The network was shown to approach a state where every agent invests the same amount of resources. Nevertheless, highly-connected agents arise that extract extraordinarily high payoffs while contributing comparably little to any of their cooperations. Here, we investigate a variant of the model, in which highly-connected agents have access to additional resources. We study analytically and numerically whether these resources are invested in existing collaborations, leading to a fairer load distribution, or in establishing new collaborations, leading to an even less fair distribution of loads and payoffs.