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Individual cows (25 in each of four herds) were monitored 8-10 times weekly for 12 weeks (stable fly season) on a southern California dairy, with 100 observations per cow. The numbers of biting stable flies, Stomoxys calcitrans (L.) (Diptera: Muscidae) on the front legs and the frequencies of four fly-repelling behaviours per 2-min observation period [head throws, front leg stamps, skin twitches (panniculus reflex) and tail flicks] were recorded. Fly numbers varied, peaking at 3.0-3.5 flies per leg in week 9 (late May). Weekly herd mean frequencies of fly-repelling behaviours were highly dependent on fly numbers, with a linear regression r(2) > 0.8. Head throws and stamps were less frequent than skin twitches and tail flicks. Individual cows differed in numbers of stable flies and behaviours. Behaviours were correlated with flies for individual cows, but at a lower level than were herd means (r = 0.3-0.7). Cows that stamped more within a herd tended to have lower fly counts; other fly-repelling behaviours were less effective. Cows maintained ranks within a herd with regard to fly numbers (r = 0.47), head throws (0.48), leg stamps (0.64), skin twitches (0.69) and tail flicks (0.64). Older cows tended to harbour higher fly numbers and to stamp less relative to younger adult cows. Ratios of leg stamps and head throws to fly numbers dropped significantly through time, suggesting habituation to pain associated with fly biting. Tail flicks were not effective for repelling Stomoxys, but were easiest to quantify and may help in monitoring pest intensity. At this low-moderate fly pressure, no consistent impacts on milk yield were detected, but methods incorporating cow behaviour are recommended for future studies of economic impact.

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We propose a model for the analysis of non-stationary point processes with almost periodic rate of occurrence. The model deals with the arrivals of events which are unequally spaced and show a pattern of periodicity or almost periodicity, such as stock transactions and earthquakes. We model the rate of occurrence of a non-homogeneous Poisson process as the sum of sinusoidal functions plus a baseline. Consistent estimates of frequencies, phases and amplitudes which form the sinusoidal functions are constructed mainly by the Bartlett periodogram. The estimates are shown to be asymptotically normally distributed. Computational issues are discussed and it is shown that the frequency estimates must be resolved with order o.T 1 / to guarantee the asymptotic unbiasedness and consistency of the estimates of phases and amplitudes, where T is the length of the observation period. The prediction of the next occurrence is carried out and the mean-squared prediction error is calculated by Monte Carlo integration. Simulation and real data examples are used to illustrate the theoretical results and the utility of the model.