Supporting Online Material for Generalized Models Reveal Stabilizing Factors in Food Webs

Generalized modeling. GM is based on the insight that, in general, the computation of steady states is much more difficult and computationally expensive than the investigation of the local dynamics around them. Once a steady state is given, its stability is determined by the corresponding Jacobian matrix, which can be analyzed at low computational cost. In the GM of food webs, computation of steady states can be avoided as follows: for every arbitrary steady state, we formally map the species densities Xi and the functions describing production, predation, and mortality to 1 by a suitable normalization. The Jacobian of the normalized system contains a number of unknown terms, which can be identified as free parameters with clear biological interpretations. These GM parameters can be treated just like parameters in conventional modeling. GM parameters fall into two classes: (a) scale parameters, which determine the topology and magnitude of biomass fluxes, and (b) exponent parameters, which measure the local nonlinearity of the considered functions. For mononomial functions, the corresponding exponent parameter simply is the monomial’s exponent. For instance, a linear function corresponds to a parameter value of 1, a quadratic function to a value of 2, and a square-root function to a value of 0.5. However, we do not restrict the functional forms in our model to monomials. For general functions, the exponent parameter measures the sensitivity of a process, say predation, to a variable, e.g., prey density (for details, see Ref. S1). Exponent parameters are called elasticities in the context of metabolic control theory. It is always possible to step back and forth between a conventional model and the corresponding generalized model. For a given steady state in the conventional model, the corresponding GM parameters are unique and can be computed straightforwardly. Conversely, for a given generalized model, one can always construct a class of conventional models that generate the given GM parameters. For instance, an exponent parameter γ = 1, indicating a locally linear functional response, corresponds to a Lotka-Volterra functional response for all prey densities, to a Holling type-II functional response for low prey densities, and to a Holling type-III functional response for intermediate prey densities. Analyzing a single set of GM parameters therefore reveals information on a large class of different conventional models. Example: Single species. To illustrate the GM approach, we consider a single population with density X. We assume that this population grows due to reproduction at rate S( X ), while also suffering from predation at rate G( X ), and from natural mortality at rate M( X ). In this simple example, we do not model the population of predators explicitly. The population dynamics is therefore given by a single differential equation,