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.7 1997 Academic Press Limited

Interspecific interactions and range limits: contrasts among interaction types

“…There are formal derivations of the Beddington-DeAngelis formulation from mechanistic assumptions for both mutualistic (Fishman and Hadany 2010) and predator/prey interactions (Beddington 1975;Huisman and De Boer 1997 ) (4b)…”

“…This could also formally describe parasitism, such as brood parasitism. The quantity ci is the maximum benefit species i can obtain from species j per unit time, 265 while qi is the maximum harm species i can receive from species j per unit time (Huisman and De Boer 1997). The parameters b2, h2, e1 and a2 expresses how the impact of species 2 on species 1 saturates when either species 1 or species 2 are at high densities see Table 2 for a full list of terms.…”

Revealing the role of predator interference in a predator–prey system with disease in prey population

“…Predator interference is also important at very low and high prey and predator densities (Kratina et al, 2009;Skalski and Gilliam, 2001). Moreover, previous studies have shown that interference among predators is a dominant driver of food-web stability (Chakraborty and Chattopadhyay, 2011;Rall et al, 2008;van Voorn et al, 2008;Huisman and De Boer, 1997) and also has the ability to generate patchiness in a homogeneous environment (Alonso et al, 2002). In spite of such huge importance, the effects of predator interference on the predator-prey-disease interactions have never been thoroughly investigated.…”

“…However, several previous researchers have suggested in favor of using Beddington-DeAngelis functional response which is similar to Holling type II functional response, but contains an extra term describing mutual interference among predators (Kratina et al, 2009;Skalski and Gilliam, 2001;Huisman and De Boer, 1997).…”

On the impulsive controllability and bifurcation of a predator–pest model of IPM

“…All these prey-predator (target pest-its natural enemy) models do, however, assume that predators do not interfere with one another and consequently their functional and numerical responses depend upon the size of the prey population only. That is, following the terminology given in (Arditi and Ginzburg, 1989;Huisman and DeBoer, 1997), papers on the mathematical modeling of IPM strategies have focused on prey-dependent models. However, a significant shortcoming of prey-dependent models is that they do not account for the effects of interference or cooperation between predators, which is why they have been challenged by the mathematical modeling community (Abrams and Ginzburg, 2000;Arditi and Ginzburg, 1989;Kuang and Beretta, 1998;Gutierrez, 1992).…”