A Formal Derivation of the “Beddington” Functional Response

Abstract

In ecological modeling the interaction between a predator and its prey, is usually implemented as a linear or saturated function of the prey density. The main advantages of such a “functional response” are its simplicity, general applicability, and well understood mechanistic basis. In systems where predators compete directly for the available prey however, the functional response should depend not only on the prey density but also on the predator density. We aim here to devise a simple and generic “predator-dependent” functional response. We derive such a functional response by making quasi-steady-state assumptions for models, in which we allow predators and prey to form interaction complexes. We end up with the—previously proposed—“Beddington” functional response. Because of our formal derivation this simple predator-dependent functional response is now based on clear mechanistic reasoning. The direct predator interference of this functional response emerges from the interaction between a predator and a prey, and not from direct predator-predator interactions. We conclude however, that, although the Beddington functional response is generic for a two-dimensional system of one prey interacting with one predator, it is difficult to generalize it to higher dimensional systems consisting of several prey and predator species.