The implicit assumptions of classic functional responses and their multi-species extensions

Background/Question/Methods

Functional responses are central to describe consumer-resource interactions. Defined as the per capita average feeding rate of consumers, since Holling's seminal papers, they have been widely used in ecology. Holling's central observation was that functional responses often saturate as resource density increases. In addition, if the interference between consumers is strong, they also decrease with consumer density. Although these empirical observations hold qualitatively in most systems, different functional forms can quantitatively describe these density-dependencies. The assumptions on which such functional forms rely are not always explicitly stated or are expressed in a phenomenological way. Here we clarify the common assumptions involved in the derivation of functional responses. We do so by revisiting a classic approach based on the formulation of feeding interactions in terms of individual based reaction schemes, using the formalism of chemical kinetics. This approach has at least two advantages. First, the underlying hypotheses and particular assumptions leading to the different functional forms become evident. Second, through the use of stochastic processes in continuous time, not only expected average feeding rates but also their whole distribution across replicates can be calculated.

Results/Conclusions

We show that framing classic functional responses and their multi-species extensions in terms of chemical-like reactions schemes relies on three fundamental assumptions: (1) mass action (2) stationarity, and (3) what we call “chemostatic conditions”. The two last assumptions involve a strong time scale separation between feeding and population dynamics, where the behavioral time scale is so fast that the feeding process reaches stationarity before population numbers of both consumers and resources have significantly changed. By using this unifying framework, surprisingly, we discover that the classic Beddington-DeAngelis functional response does not result from the underlying individual feeding processes in the same consistent way as Holling type II functional response does, and, as a consequence, it should be interpreted rather phenomenologically, or, at most, as an approximation. We provide instead a more general expression that can be regarded as the natural extension of Holling type II functional response when consumers interfere. Our work has clear implications, on the one hand, for both model selection and parameter inference from feeding experiments, and, on the other, for the use of multi-species extensions of these functional responses in population-level food web dynamic models.