A mechanistic derivation of the DeAngelis–Beddington functional response Stefan Geritz, Mats Gyllenberg n Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland HIGHLIGHTS c We derive the DeAngelis–Beddington functional response in terms of behavior at the individual level. c The derivation is based on a predator-induced change in individual state of prey from exposed to unexposed. c The result can be reduced to the Holling-II but not to the ratio-dependent functional response. article info Article history: Received 31 May 2012 Received in revised form 20 August 2012 Accepted 23 August 2012 Available online 2 September 2012 Keywords: Predator–prey Functional response Refuge Mechanistic modelling abstract We give a derivation of the DeAngelis–Beddington functional response in terms of mechanisms at the individual level, and for the first time involving prey refuges instead of the usual interference between predators. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction The well-known Holling (1959) type II functional response F 1 ðxÞ¼ ax 1 þ ahx , ð1:1Þ which gives the average number of prey consumed by one predator per unit of time when the prey density is x, is usually derived using a time-budgeting argument. Indeed, (1.1) is based on the assumption that predators divide their time into searching (hunting) for prey and handling (i.e., killing, eating and digesting) the prey caught. The parameters a and h have clear-cut and important biological interpretations: a is the rate at which searching predators attack prey and h is the expected time a predator handles its prey. An easy computation, see Holling (1959), shows that, on average, a predator spends a fraction 1=ð1 þ ahxÞ of its time searching. This immediately leads to the functional response (1.1). An alternative derivation, based on separating the short time scale of searching and handling prey from the long time scale of giving birth and dying, can be found in Metz and Diekmann (1986, p. 6–7). DeAngelis et al. (1975) and independently Beddington (1975) proposed a generalisation of the Holling functional response (1.1), namely F 2 ðx, yÞ¼ mx ax þ by þ g : ð1:2Þ Since the publication of the papers (Beddington, 1975; DeAngelis et al., 1975), and especially in the past 20 years, predator–prey models with the functional response (1.2) have been studied in detail (Cantrell and Cosner, 2001; Li and Takeuchi, 2011; Zeng and Fan, 2008; Zhang and Chen, 2006). Purely mathematical results include determination of steady states, investigation of their stability, permanence and bifurcation theory. Proponents of the DeAngelis–Beddington functional response (1.2) emphasise that it generalises both the Holling type II functional response and the ratio-dependent functional response advocated by Arditi and Ginzburg (1989). Indeed, taking b ¼ 0 one arrives at (1.1) and taking g ¼ 0 one arrives at the ratio-dependent functional response. However, the ratio-dependent functional response has been criticized, e.g., by Abrams (1994, 1997) on sound theoretical and empirical grounds. Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/yjtbi Journal of Theoretical Biology 0022-5193/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jtbi.2012.08.030 n Corresponding author. Tel.: þ358 9 191 51480. E-mail address: mats.gyllenberg@helsinki.fi (M. Gyllenberg). Journal of Theoretical Biology 314 (2012) 106–108