Local dynamical equivalence of certain food webs

Abstract

An important challenge in theoretical ecology is to find good coarse-grained representations of complex food webs. Here, we use the approach of generalized modeling to show that it may be possible to formulate a coarse-graining algorithm that conserves the local dynamics of the model exactly. We show examples of food webs with a different number of species that have exactly identical local bifurcation diagrams. Based on these observations, we formulate a conjecture governing which populations of complex food webs can be grouped together into a single variable without changing the local dynamics. As an illustration, we use this conjecture to show that chaotic regions generically exist in the parameter space of a class of food webs with more than three trophic levels. While our conjecture is at present only applicable to relatively special cases, we believe that its applicability could be greatly extended if a more sophisticated mapping of parameters were used in the model reduction.

Mesoscale symmetries explain dynamical equivalence of food webs

“…Extension of the results in [33] to directed and weighted networks allows application of them to study the food web dynamics. This reveals the topological mechanism of the reduction rule identified heuristically in [30] and provides more general rules for food web aggregation. We note that the previously observed identity is of limited use for the analysis of food web data.…”

“…A GM for ecological food webs [31] was used in [30] to study critical points in parameter space in which the dynamics around steady states changes in bifurcations. The authors noted…”

“…In the symbolic representation, nodes indicate populations and arrows indicate biomass flows due to predator-prey interactions. Using this notation the dynamical equivalence conjectured in [30] is shown symbolically in the lower row.…”

“…This goal is generally hard to achieve as it seems to imply knowledge of the dynamics of the original system. However, in a recent publication [30] it was shown in a class of generalized models (GMs) [31,32] that under certain conditions an aggregation is possible that conserves the local dynamics around steady states and at least certain features of the global dynamics exactly. Although the previous paper demonstrated this aggregation in several examples, it failed to provide a detailed analysis of the underlying symmetries that make the aggregation possible.…”

Boolean versus continuous dynamics on simple two-gene modules

“…This method allows to investigate the occurrence of different dynamical regions without the need to specify the precise form of interaction terms and equilibrium concentrations. This method has already been successfully applied to ecological food webs [11,20,21], socio-economic models [11] and metabolic networks [22,23].…”