Local dynamical equivalence of certain food webs
Thilo Gross and Ulrike Feudel
Ocean Dynamics 59, 417-427, 2008
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.
Figure 1: Top row: Equivalence of complex food webs. If parameters are chosen appropriately (see text) the bifurcation diagrams of certain complex food webs are exactly identical. The respective topologies are indicated in the top right corner.
Both diagrams are identical to the corresponding diagram of a four-trophic food chain. Bottom row: Example of two food webs that cannot be mapped to food chains in the proposed way. In all four diagrams the red, yellow, and green surfaces correspond to Hopf bifurcations and the blue surface to bifurcations of saddle-node type.