Generalized modeling of complex dynamical systems: an application to the stability of ecological networks

Abstract

Understanding the stability of food webs is crucial for resource sustainability and conservation of ecosystems, especially in the context of climate change. Specific models describe the biomass flow in food webs by a set of ordinary differential equations that require the explicit reconstruction of a mathematical expression for each of the interactions or processes, like predator-prey interactions, primary production, and mortality. Although specific models are rich with time-evolution information, limited access for empirical observation of these typically immense systems induce uncertainty in the data and approximations in the corresponding models that can threaten robustness or relevance of results. Generalized models can produce the stability of all the equilibria of specific models that have the same vague structure and bypasses the requirement to specify every function by evoking a normalizing transformation. The analysis is subsequently computationally efficient and can be used to study large food webs with a great number of replicates. Often generalized ecological network studies confine the scope to a small subset of the variable dynamical scenarios, but this limits the interpretations that can be inferred. With this in mind, we develop a deterministic food-web generator that can be used to compare large food webs that differ by only a single link and maintain an expansive dynamical scope. We found behavior that indicates the existence of critical links and a grander theory on topological equivalence. We explicitly show how we can create hypothetical paths the system may traverse upon enrichment of lower trophic levels using the expanded dynamical scope. Generalized modeling is unable to produce evolution solutions among other things, but it has an unlimiting access to the stability of equilibria while specific models provide only a subset of stability data. Generalized modeling is a relatively new method and its relation to specific model outcomes/results is not clearly understood. Specific models can inform generalized modeling studies on properties like coexistence of fixed points or actually occurring relative weighting of flows between ecosystem members. We combined the methods and demonstrated the validity of the abstract technique of generalized modeling in emphasis to its usefulness/power for the analysis of network stability. The specific model provided a unifying explanation to a conglomerate of related microcosm experiments that showed conflicting results on enrichment and implied stabilization upon the hampering of predatory efficiency. We identified the conditions by which enrichment is stabilizing to a steady state when basal species are in a resource-deprived environment but destabilizing if resources become more abundant. A prevalent issue in ecology involves discrepancies between simulation and empirical observation about food-web stability such as how intuition says enrichment or complexity in some way are favorable to stability but mathematical models find it predominately the opposite. A common rationalization for these discrepancies includes discourse on reductionistic versus holistic rhetoric. The idea being that as models become better representations of ecosystems that capture more intricacies and detail, they will help to resolve the issue. We constructed over a million food webs that reveal positive effects on fixed-point stability from the incorporation of more realistic ecosystem features that include species specialization, habitat modularity, and predator's prey preferences. Arctic warming is a portent to changes in species composition and ecological theory predicts the existence of key ecosystem members that have extraordinary influence on overall ecosystem function or the state of the system. Motivated by sea-ice loss and northward expansion of species distributions, ice-obligate species are removed from the food webs and southern competitors are introduced. Although it is common understanding that apex predators can enhance biodiversity, we find the presence of "super killers" significantly destabilizes food webs. Ecosystems have immense complexity with thousands of species, but ecosystem models condense and consider only a few species that are of the most interest or abundance, neglecting the many weak interactions comprising the larger ecosystem. Considering this, we suppose a food web is subsumed by a larger phantom ecological network that represents hypothetically rare species or predator-prey relationships. Each link from the phantom network contributes a variably weak perturbation, but collectively, induce a net positive effect on the average stability of the food webs, considerably so near the optimal perturbation strength.