## Scalar Timing and Patch Departure Sudden Patch Exhaustion

Brunner and Kacelnik (Brunner et al., 1992, 1996; Kacelnik et al., 1990; Kacelnik and Brunner, 2002) set out to analyze the timing problem suggested by Davies' spotted flycatchers using European starlings foraging on an operant analog of the patch departure scenario in the lab. Their design simulated an environment in which food was distributed in patches. Each patch contained a random number of prey items (N = 0 to 4) that were delivered according to a fixed-interval schedule until the patch ended with sudden depletion. The time elapsed since the last prey item was the only cue the bird had to detect patch depletion. Once the patch had depleted, the bird could leave the patch and travel to a new patch by flying between two perches. As described above, the optimal patch departure rule, given perfect timing, is to abandon the patch as soon as exhaustion is detected, namely, when a prey is not encountered after waiting for the programmed fixed interval.

Brunner and Kacelnik tested how the patch departure of starlings was affected by the value of the fixed-interval schedule in the patch by examining the behavior of the birds tested with six different values of the fixed interval, ranging between 0.8 and 25.6 sec. As predicted by the unconstrained optimality model, the giving-in time at which a bird stopped attempting to obtain food from a patch in each trial (defined as the last peck in the patch) increased linearly as the fixed interval increased. However, in contrast to the predictions of the unconstrained model, the line relating the value of the fixed interval to the giving-in time had a slope of 1.49 rather than 1.00. Thus the starlings waited for a fixed proportion of the fixed interval before abandoning the patch. This result is particularly interesting in the light of data showing that the starlings knew accurately when food should have been expected. An analysis of the patterns of pecking in the final, unreinforced interval of each patch revealed that the birds showed a peak of pecks centered accurately on the value of the fixed interval: the line relating the value of the fixed interval to the peak time had a slope that was not significantly different from 1.00. Therefore, despite apparently knowing accurately when food should have been delivered, the birds still chose to wait 1.49 times the usual interfood interval before giving in; i.e., a multiplicative bias was introduced by the decision mechanism that controls the bird's decision to give up searching for more food in the patch.

How can we explain the multiplicative bias in the giving-in rule? It is possible to answer this question either by considering the functional implications of adopting different giving-in rules or by considering the timing processes responsible for the decision. We will start with the functional approach. If the birds knew accurately when food should have been delivered in the patch, why did they not give in and leave the patch as soon as it did not appear? The answer to this question is to be found in the behavioral variability of the birds. Although the peak times of the birds show perfect accuracy, the standard deviations of the pecking functions are linearly related to the fixed interval: in other words, the timing functions of the birds display the scalar property evident in all timing data. The giving-in times also display the scalar property because their interquartile range also increased linearly with the fixed interval. In order to understand why it is optimal to introduce a multiplicative bias in the giving-in rule, we need to consider the potential costs of giving in early (before the patch is exhausted) vs. the costs of giving in late, as was observed. It is easy to show that from the perspective of rate of energy intake, it is much more costly to give in too early than too late. This asymmetry occurs because an animal that gives in early fails to get the last food item available in the patch, and this has a much bigger impact on the rate of energy intake than the relatively small time cost imposed by giving in late. Thus we can predict that if imprecision in timing is a constraint,

FIGURE 5.4 Variance in timing behavior and optimal giving-in time strategies. In each panel the solid line has a slope of 1.00 and indicates the optimal giving-in time predicted if timing is both perfectly accurate and precise. The top two panels (a and b) show constant variability in giving-in time, and the lower two panels (c and d) proportional or scalar variability in giving-in time. When timing is imprecise, the optimal strategy is biased in order to minimize the probability of leaving the patch before it is exhausted (panels b and d). When variability is constant, the optimal strategy is: giving-in time = fixed interval + bias (panel b); however, when variability is scalar, the optimal strategy is: giving-in time = fixed interval*bias (panel d). Brunner et al.'s (1992) starlings displayed a multiplicative bias as in panel (d) consistent with their also displaying the scalar property in their estimates of when food should have occurred. (Redrawn from Brunner, D., Kacelnik, A., and Gibbon, J., Anim. Behav., 44, 597-613, 1992. Copyright © 1992 by the Association for the Study of Animal Behavior. With permission.)

Fixed interval

FIGURE 5.4 Variance in timing behavior and optimal giving-in time strategies. In each panel the solid line has a slope of 1.00 and indicates the optimal giving-in time predicted if timing is both perfectly accurate and precise. The top two panels (a and b) show constant variability in giving-in time, and the lower two panels (c and d) proportional or scalar variability in giving-in time. When timing is imprecise, the optimal strategy is biased in order to minimize the probability of leaving the patch before it is exhausted (panels b and d). When variability is constant, the optimal strategy is: giving-in time = fixed interval + bias (panel b); however, when variability is scalar, the optimal strategy is: giving-in time = fixed interval*bias (panel d). Brunner et al.'s (1992) starlings displayed a multiplicative bias as in panel (d) consistent with their also displaying the scalar property in their estimates of when food should have occurred. (Redrawn from Brunner, D., Kacelnik, A., and Gibbon, J., Anim. Behav., 44, 597-613, 1992. Copyright © 1992 by the Association for the Study of Animal Behavior. With permission.)

then giving-in times should be biased to be longer than the fixed interval to guard against giving in too early. If, as the data revealed, the precision of timing is linearly related to the fixed interval (i.e., the scalar property applies), then the bias needs to be multiplicative, as opposed to a constant, in order to reduce the probability of giving in too early. These arguments are presented graphically in Figure 5.4.

It is also possible to analyze the behavior of the starlings from the mechanistic perspective of scalar timing theory. Scalar timing theory can be used to explain both the pecking patterns of the bird in the final unreinforced interval of the patch and the giving-in times. Both of these behavioral measures can be analyzed using the version of the scalar timing model proposed to deal with fixed-interval performance described above. The final unrewarded interval in the patch is exactly analogous to a probe trial from the peak-interval procedure. In order to use the scalar timing theory to model this performance, only a small modification to the decision mechanism is needed. Because birds start pecking at a high rate sometime prior to the FI and then stop pecking at this high rate sometime after the FI, the decision rule needs to produce both the start and stop of the high rate responding. This can be achieved by modifying the comparison made to mt - m*

The starling responds at the high rate when this absolute ratio is greater than or equal to b. When applied over several trials, this model will result in pecking functions that are centered on the value of the fixed interval with a standard deviation proportional to the fixed interval, exactly as observed in the starlings.

The giving-in times can be modeled with another small modification of the basic model. In this case, we assume that the birds are using the same reference memory of reinforcement times used above to generate the pecking patterns, but a different decision rule. The bird leaves the patch at time g such that mg - m*

where mg is the perception of the current time, g, and bg is a new threshold that is larger than b. Before Brunner et al.'s (1992) application of scalar timing theory to an optimal foraging problem, it had previously been assumed that thresholds were fixed (e.g., Gibbon 1977); however, one of the most important outcomes of Brunner et al.'s experiment is the suggestion that in fact a threshold such as bg may be optimized by natural selection to maximize the rate of energy intake of the birds given the constraint of their imprecise timing mechanisms. The predictions of such an optimality approach are that the threshold, bg, and consequently the giving-in times should be bigger when the cost associated with travel or the energy gain associated with the reinforcement are increased.

This study provides a beautiful example of the benefits of the ethological approach of integrating mechanistic and evolutionary analyses of the same problem. Without the scalar timing model, we would not have understood why the starlings' giving-in times are a fixed proportion of the interval between food items. Without the evolutionary approach, we would not have understood that the biases assumed in scalar timing are not arbitrarily chosen, but may actually be the outcome of an evolutionary optimization of the costs of leaving the patch too early and too late.

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