## Framing the Problem

How should a forager "track" environmental changes? The simplest model imagines an environment in which one resource fluctuates while another is stable (Arnold 1978; Bobisud and Potratz 1976; Steph ens 1987). The varying resource, called V, is sometimes in a good state, which yields g units ofbenefit per unit time, and sometimes in a bad state, which yields b units of benefit per unit time. The mediocre stable resource, called S, always provides s units of benefit per unit time. The states of the varying resource occur in runs, specified by a persistence parameter q, the probability that the state now (in time i) will persist in the next time interval (time i + 1). So ifq = 1/2, the state in the next time interval is just as likely to have changed as to have remained the same, while if q is close to 1, the current state is a good predictor of the state in the next time interval.

We assume thatg > s > b, so a forager should exploit the varying resource when it's in the good state, but switch to the stable resource as soon as the varying resource "goes bad." A forager might be able to follow this omniscient strategy if some externally visible cue signaled the state of the varying resource, but we will assume that the forager can detect the state of Vonly via direct experience. In other words, the forager must sample. To keep the problem simple, we assume that experience allows perfect discrimination, so a single sample tells the forager whether the varying resource is in its good or bad state.

Figure 2.5 shows the situation. The varying resource follows the pattern of a square wave that varies between g and b, while the stable resource is a flat line (at s) somewhere between g and b. Now consider what happens when V changes from good (g) to bad (b). The forager detects this immediately and switches to the stable resource, but how long should it stay there? Periodically, the forager needs to check V to see if a transition back to the good state (g) has occurred. An animal that checks too frequently will make many "sampling errors," obtaining b when it could have had s (this error costs s — b). On the other hand, an animal that doesn't check frequently enough will make overrun errors, missing the switch back tog and obtaining s wheng is available (this error costsg — s). We can summarize this logic in a single parameter that we'll call the error ratio, = (s — b)/(g — s) the cost of sampling errors divided by the cost of overrun errors. So, for example, a large error ratio means that sampling errors are relatively expensive, and we expect infrequent sampling. If, instead, the error ratio is small, we would expect frequent sampling to minimize overrun errors. The astute reader may have noticed some familiar elements of signal detection theory in our construction of the error ratio: the consequencesg, b, and s neatly fill out a "truth table," as in our development of signal detection (with s filling two cells), and the error ratio itself parallels the ratio of consequences in equation (2.1).

The environmental persistence ofa resource, q, also has an important effect on the economics of sampling frequency. One can understand this effect intuitively by considering two special cases. If q = 1/2, resource V changes from good to bad at random, and there is, quite literally, nothing to track. So we expect no sampling when q = 1/2; the forager should choose either to always exploit S or to always exploit V, whichever provides the higher average gain. On the other hand, if q = 1, the current state is a perfect predictor of future states, so we know that if the varying resource V provides g now, it will always provideg. The interesting thing about this "perfect predictor" case is that it makes a single sample extremely valuable—in theory, a single sample can point the forager to a lifetime of correct behavior.

The persistence parameter and error ratio combine to determine the sampling rate (i.e., the time before returning to V to sample its state) that maximizes the long-term rate of resource gain (the optimal sampling rate, *; Figure 2.6). The model predicts sampling in a trumpet-shaped region narrowest where q = 1/2 and widening as q approaches 1. A forager should not sample in the region above the trumpet; instead, it should exploit only the stable resource S. Another "don't sample" region lies below the trumpet, in Figure 2.5. Tracking a changing environment. (A) An environmentwith a varying resource alternating between states g and b in a square wave pattern and a mediocre stable resource in state s. (B, C) The economics of high and low sampling rates. (B) Sampling frequently leads to many sampling errors (s) but few overrun errors (o). (C) Less frequent sampling reduces the number of sampling errors but causes more overrun errors.

Figure 2.5. Tracking a changing environment. (A) An environmentwith a varying resource alternating between states g and b in a square wave pattern and a mediocre stable resource in state s. (B, C) The economics of high and low sampling rates. (B) Sampling frequently leads to many sampling errors (s) but few overrun errors (o). (C) Less frequent sampling reduces the number of sampling errors but causes more overrun errors. Figure 2.6. The effects of error ratio (s - b)/(g - s) and environmental persistence (q) on the optimal sampling rate (a *). The parameter a * gives the optimal sampling rate; it is the probably of checking the varying resources during a run of bad luck. Each curve shows combinations of error ratio and environmental persistence that imply a particular optimal sampling rate as shown on the figure. A forager should always exploit the stable resource S in the region above the a * = 0.0 line and should always exploit the varying resource V in the region below the a * = 1.0 line. Sampling, therefore, is predicted only in the trumpet-shaped region bounded by the a * = 0.0 and a * = 1.0 lines.

Figure 2.6. The effects of error ratio (s - b)/(g - s) and environmental persistence (q) on the optimal sampling rate (a *). The parameter a * gives the optimal sampling rate; it is the probably of checking the varying resources during a run of bad luck. Each curve shows combinations of error ratio and environmental persistence that imply a particular optimal sampling rate as shown on the figure. A forager should always exploit the stable resource S in the region above the a * = 0.0 line and should always exploit the varying resource V in the region below the a * = 1.0 line. Sampling, therefore, is predicted only in the trumpet-shaped region bounded by the a * = 0.0 and a * = 1.0 lines.

which the forager should exploit only the varying resource V. As the predictability of the environment (q) increases toward 1, the region in which we predict sampling increases.

While most readers will recognize the logic of this result, it seems surprising if we step back from the particulars and consider the larger context. Animals need to sample because they live in varying environments, yet the conditions that favor sampling steadily broaden as the environment approaches fixity! It seems that sampling is as much about environment regularity as it is about environmental change (see Stephens 1991 for an application of these ideas to learning). The model makes three key predictions:

1. Sampling rates should decrease with s, the value of the stable but mediocre resource, because a decrease in s makes sampling errors more costly while reducing the cost of overrun errors.

2. Sampling rates should increase withg, the value of the varying resource's good state, because an increase in g makes overrun errors more costly.

3. Sampling rates should decrease with q, because q increases the duration of states.

Three separate studies have tested this basic tracking model (Inman 1990 using starlings; Shettleworth et al. 1988 using pigeons; Tamm 1987 using ru- flu Did Irl a je uJJ .1 » en tn 4M Hl m «-UÍ g<iw <-o,nj g-üwj o-an ffttw

Figure 2.7. Results ofthree experimental tests of the tracking model. The qualitative effects ofthe s and q variables are as predicted, but the effect of g seems to contradict the model.

en tn 4M Hl m «-UÍ g<iw <-o,nj g-üwj o-an ffttw

Figure 2.7. Results ofthree experimental tests of the tracking model. The qualitative effects ofthe s and q variables are as predicted, but the effect of g seems to contradict the model.

fous hummingbirds). The Shettleworth et al. and Inman studies asked whether the components of —especially g and s—affect sampling behavior as predicted, while Tamm studied the combined effects of and q. In all three studies, the bad state was "no food," giving b = 0 and = s/(g — s).

Figure 2.7 presents graphical summaries for these three studies. The figure shows a straightforward pattern: the effects of s and q agree with the theory. Observed sampling rates decrease with increases in both s and q. However, the effect of g does not agree with the model's predictions. Moreover, the effect ofg shows no clear pattern: in one case (Shettleworth et al., experiment 1), sampling rates decrease with increasing g in direct contradiction of the model; in another (Shettleworth et al., experiment 2), g has no effect; in a third (Inman),g shifts sampling rates in the predicted direction; in the fourth (Tamm), there is no consistent effect ofg. The data also suggest several other contradictions. For example, both the Inman and Shettleworth et al. studies had some treatments in which differentg and s values predicted the same error ratio ( ); one can do this by changing bothg and s by the same factor k (that is, s ks g — s kg — ks

In both studies, observed sampling rates were lower when k was greater, suggesting possible hunger effects (because when k is large, the subjects obtain more food on average and may be less motivated to feed).