Methodological problems with estimating patch depression during resource depletion

Abstract

The Marginal Value Theorem (Charnov 1976) is a widely used paradigm for determining optimal exploitation under conditions of diminishing returns. One requirement for applying this model is estimation of the gain function, which is the cumulative gain as exploitation proceeds. The function is commonly derived from experiments in which one or a number of animals is allowed to deplete food patches either for set durations, set numbers of bites, or until they choose to stop. The cumulative food intake and time taken are then collated from a number of such trials, and some mathematical function is fitted to describe patch depression, which is the term used for an inverse relationship between instantaneous intake rate and elapsed time in the patch. For example, such a procedure was followed by A strom et al. (1990), who analysed intake in relation to exploitation time by moose foraging in an arena of trees varying in size. They derived a function that implicitly assumes patch depletion and depression while foraging, and reported the occurrence of patch depression. Similar or more controlled approaches have been taken with cattle (Laca et al. 1994, Ginnet et al. 1999). Two problems with such an approach are apparent: the unintentional effect of pauses by the animal and averaging across animal-patch combinations. Animals occasionally pause momentarily during foraging, perhaps because they are disturbed or because they habitually scan for predators whilst foraging. If pauses are randomly distributed during feeding, then the number of pauses will increase with time and will be Poisson-distributed. Depending on how pause length is distributed, the accumulated increase in patch residence time due to pauses will have some form of skewed distribution. Gathering together data on food intake and residence time from a number of trials would therefore inevitably produce a decelerating cumulative gain function, due to accumulation of pauses rather than resource depression. These points are illustrated by results from simulation, which were implemented using Genstat 5.4.1 (Genstat 5 Committee, 1993). To clarify the effect of pauses, a linear gain function was simulated, with each prey item having a constant search plus handling time (1 s). Pauses after consuming a prey item were assumed to occur at random with probability p=0.05, and to last for a mean time of 5 s, normally distributed with sd=1, and truncated at zero. The less realistic assumption of pauses lasting a constant 5 s made no difference to the results. Three rules for patch departure were compared, according to the number of food items consumed. These were simulated as either (a) a uniform random deviate between 0 and 100; (b) a normal random deviate with mean=50 items, sd=10; (c) after exactly 50 items had been taken (Fig. 1a, b and c, respectively). These rules were intended to simulate cases where animals terminate feeding (a) for a variety of unknown reasons; (b) after removing a roughly constant amount of food from each patch; (c) after removing a constant amount. Note that each case has the same expected value. Simulations were repeated for 100 trials, and the accumulated intake and elapsed time (i.e. foraging plus pauses) from each were analysed by non-linear fitting of function 2 of A strom et al. (1990). This gives food eaten in time T as: E(T)=s{1−1/[1+ (kT/s)]}, where s is asymptotic food intake and k is a parameter. Curvilinearity is slight with 0 k/s 0.005 and marked when k/s 0.01. With random patch departure, data generated were rather similar to those presented by A strom et al. (1990), and parameters of the gain function were k/s= 0.002 in the example shown (Fig. 1a). Departure from