This paper analyzes a number of relatively general models of predator-prey adaptation and coadaptation. The motivation behind this work is, in part, to evaluate the "race analogy" that has been applied in analyzing predator-prey coevolution. The models are based on the assumption that increased investment in predation-related adaptations must be paid for by decreased adaptation to some other factor. Increased investment in predation-related adaptations by the prey lowers the predator's functional response, and increased investment by the predator increases the functional response. The models are used to determine how each species should respond to an increase in the predation-related investment of the other species. Several broad classes of population-dynamics models and several alternatives for the cost of predation-related adaptation are investigated. The results do not support the general applicability of the race analogy. In the type of model analyzed in greatest detail here, predator and prey adaptations combine multiplicatively in determining the predator's capture-rate constant. In such models, prey usually increase investment in predator avoidance or escape when predators increase their investment in capture. However, predators often do not change or decrease their investment in response to an increase in the prey's investment. The direction of the predator's response depends on the particular parameter that pays the cost of increased predation investment, the shape of the cost-benefit functions, and the assumptions about the population dynamics of the predator-prey system. Similar models are used to determine whether increased investment by one species should increase the rate of incorporation of mutations that improve the predation-related adaptations of the other species. The arms-race analogy also fails for this case. The results cast doubt on the usefulness of Dawkins and Krebs (1979) "life-dinner" principle.