Killeen and Hall (2001) showed that a common factor called strength underlies the key dependent variables of response probability, latency, and rate, and that overall response rate is a good predictor of strength. In a search for the mechanisms that underlie those correlations, this article shows that (a) the probability of responding on a trial is a two-state Markov process; (b) latency and rate of responding can be described in terms of the probability and period of stochastic machines called clocked Bernoulli modules, and (c) one such machine, the refractory Poisson process, provides a functional relation between the probability of observing a response during any epoch and the rate of responding. This relation is one of proportionality at low rates and curvilinearity at higher rates.

Rats' leverpressing was reinforced on variable-ratio (VR) schedules. As ratio values increased, response rates initially increased with them, then eventually decreased. In Experiment 1, rates were uniformly higher with one-pellet reinforcers than with two-pellet reinforcers-the paradoxical incentive effect. Killeen's (1994) mathematical principles of reinforcement (MPR) described the data quantitatively but failed to predict the advantage for the one-pellet condition. In Experiment 2, rats receivedone-, two-, and three-pellet reinforcers with counterbalanced preloads of pellets; the continued superiority of the smaller reinforcers ruled out a satiation explanation. Experiment 3 introduced a 20-sec intertrial interval (ITI), and Experiment 4 filled the ITI with an alternate response to test a memorial/overshadowing explanation. In Experiment 5, the rats received one or two standard grain pellets or one sucrose pellet as reinforcers over an extended range of ratios. Once again, rates were higher for one than for two pellets at short to moderate VR values; thereafter, two pellets supported higher response rates. The diminution of the effect in Experiment 3 and its reversal in Experiment 4 and in Experiment 5 at large ratios provided evidence for overshadowing and reconciled the phenomenon with MPR.

Three experiments investigated foraging by rats and pigeons. In Experiment 1, each response on a manipulandum delivered food to a cup, with the distance between the manipulandum and the cup varying across conditions. The number of responses made before traveling to collect and eat the food increased with distance for rats, but not for pigeons. In Experiment 2, two manipulanda were placed at different distances from a fixed food source; both pigeons and rats preferentially used the manipulandum closest to the food source. Experiment 3 was a systematic replication of Experiment 1 with pigeons. In different conditions, each peck on the left key increased the upcoming hopper duration by 0.5, 1.5 or 2.5 s. Completing a ratio requirement on the right key of 1, 4, 8, 16 or 32 pecks, depending on the condition, then produced the food hopper for a duration that depended on the number of prior left pecks. As the ratio requirement increased on the right key, pigeons responded more on the left key and earned more food. Overall, the results replicate previous research, underlining similarities and differences between these species. The results are discussed in terms of optimal foraging, reinforcer sensitivity and delay discounting.