An adjustment of the extended contingency model of Farnsworth & Illius (1998)

Farnsworth & Illius (1998) modified the classical contingency model (Stephens & Krebs 1986) so that it could take into account the overlap between searching and handling time observed in large herbivores (Spalinger & Hobbs 1992; Laca, Ungar & Demment 1994). Their model predicts an optimal diet based on estimates of prey encounter rate (λ, prey/min), digestible energy (e, in kJ/prey), handling time (h, in min/prey), and the proportion of h exclusive from searching activity (η). Given that 1 > η > 0, which will be assumed throughout this paper, animals can search for the next prey while chewing the last bite during the portion (1 − η)h of handling time, whereas they cannot search during the portion ηh of handling time devoted to cropping a food item. Farnsworth & Illius (1998) also distinguished between foraging that is limited by encounter rate and by handling time. Interestingly, they showed that if more than one prey is required to make a diet handling-limited, the last prey should often be consumed at a rate lower than the encounter rate. This contrasts with the 0–1 rule characteristic of the classical foraging models (Stephens & Krebs 1986). The model of Farnsworth & Illius (1998) illuminates our understanding of foraging decisions in large herbivores, but algebraic errors in some of its equations lead to inaccurate predictions during handling-limited foraging. My objective is to provide revised equations that indicate (1) when a diet is handling-limited, (2) the rate at which the last type of a multi-prey type diet should be accepted to make the diet just handling-limited, and (3) the energy intake rate provided by a handling-limited diet based on the partial consumption of a given prey type. Farnsworth & Illius (1998) stated on p. 76 that a prey type (i) is subject to handling-limited foraging when ‘hI ≥ 1/λi’, which could also be written as (1 − ηi)hi + ηihi ≥ 1/λi. Because herbivores can search for new bites while they are masticating a previous bite, the proper conflict is between expected time to the next bite to be encountered 1/λi and the expected time to chew the last bite (1 − ηi)hi. When (1 − ηi)hi ≥ 1/λi, the forager is limited by handling, because bite rate = 1/hi. In contrast, when (1 − ηi)hi < 1/λi, then the bite rate = λi/(1 + λiηihi). Extending this argument to multispecies foraging, and based on equation 9 of Farnsworth & Illius (1996) and equation 2 of Farnsworth & Illius (1998), it can be derived that handling-limited foraging should rather occur when: where m is the total number of prey types included in the diet. The optimal diet is obtained by ranking prey by increasing profitability (e/h), and then expanding the diet until foraging becomes handling-limited. If consumption of the most profitable prey type is limited by handling time, the animal should specialize on this prey type. Hence, the intake rate would simply correspond to profitability of the highest ranked prey (e1/h1). When more than one prey type are required to reach handling limitation, the last type should be partially accepted at a rate that relates to the encounter rate for the more highly ranked items already in the diet. The acceptance rate of prey type m can be expressed as: . By definition, partial acceptance of prey type m entails that these plants are accepted at a rate lower than their encounter rate, and thus that . Because nm represents the minimal amount of prey m that would make the diet handling-limited, the following equality is implied: With the rearrangement of equation 2 we can find nm from: which corresponds to the time that would be spent searching after chewing (i.e. during ‘pure’ search) if only m − 1 prey types were accepted in the diet divided by the time that can be spent searching while handling prey m. Because ηi > 0, equation 3 would provide higher nm than equation 17 of Farnsworth & Illius (1998). However, their equation reflected encounter-limited foraging rather than handling-limited foraging, as it should have. Given the acceptance of all the m − 1 prey encountered and the partial acceptance rate of prey m, the energy intake rate of the handling-limited diet becomes: The diet would be optimum only if the handling-limited foraging provides a greater intake rate than the rate for encounter-limited foraging without inclusion of the last prey type: If inequality (5) does not hold, only m − 1 prey types should be included in the diet. I believe that the study of Farnsworth & Illius (1998) constitutes an important contribution to foraging theory, and my revision of their equations now allows the prediction of optimal diet also during the handling-limited foraging of large herbivores. The funding for this study was provided by Parks Canada, University of Guelph, and scholarships from FCAR and OGS. I thank John Fryxell for his advice and constructive comments on this paper. I am grateful to Keith Farnsworth and Andrew Illius for encouraging me to pursue this work.

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