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When Leo Luckinbill (1973) grew Paramecium aurelia together with its predator Didinium nasutum in 6 mL of standard cerophyl medium, the Didinium consumed all the prey in a few hours. When the medium was thickened with methyl cellulose, the populations when through two or three diverging oscillations lasting several days before becoming extinct. When he used a half—strength cerophyl medium thickened with methyl cellulose, the populations maintained sustained oscillations for 33 d before the experiment was terminated. The data from this experiment provide a rare opportunity to test current predator—prey models. A standard differential equation predator—prey model with a carrying capacity for the prey and a saturating (Type 2) functional response predicts the outcome of Luckinbill's experiment qualitatively, but does not give a good quantitative fit to the data. Several modifications of this model are tested against the data for the populations grown in the medium thickened with methyl cellulose, using the Marquardt—Levenberg method to obtain the least squares best fit. Neither Leslie type models nor models with a ratio—dependent functional response do well, but adding either predator mutual interference or a sigmoid (Type 3) functional response improves the fit dramatically. Modeling the predator growth rate to depend on energy or nutrient storage instead of directly on the rate of consumption of prey, thus creating a delayed numerical response, along with predator mutual interference or a sigmoid functional response, produced the best models and gave excellent fits to the data. These models are further validated by the fact that changing only one or two parameter values to reflect the unthickened medium or the half—strength medium also gives reasonably good fits to the other data sets. The last model requires a more sigmoid functional response to fit the data in the thickened than in the unthickened medium, suggesting that an increase in the cost—benefit ratio of energy spent searching to energy gained capturing prey inhibits the predator searching at low prey densities.

The empirical distribution of length of stay of patients in departments of geriatric medicine is fit extremely well by a sum of two exponentials. Most of the patients in a geriatric department are rehabilitated and discharged or they die within a few weeks of admission, but the few who become long-stay patients remain for months or even years. A model is presented for the flow of patients through a geriatric department, which has analogies to models of drug flow in pharmacokinetics. The theoretical model explains why the empirical pattern of length of stay in the occupied beds fits a sum of two exponentials; conversely, the empirical distribution, obtained from the midnight bed state report, can be used to study the effect of various policy decisions on both immediate and future admission rates for the department, and shows the benefits of policies which reduce long-stay patient numbers by improving long-stay rehabilitation.

A stochastic version of the Harrison-Millard multistage model of the flow of patients through a hospital division is developed in order to model correctly not only the average but also the variability in occupancy levels, since it is the variability that makes planning difficult and high percent occupancy levels increase the risk of frequent overflows. The model is fit to one year of data from the medical division of an acute care hospital in Adelaide, Australia. Admissions can be modeled as a Poisson process with rates varying by day of the week and by season. Methods are developed to use the entire annual occupancy profile to estimate transition rate parameters when admission rates are not constant and to estimate rate parameters that vary by day of the week and by season, which are necessary for the model variability to be as large as in the data. The final model matches well the mean, standard deviation and autocorrelation function of the occupancy data and also six months of data not used to estimate the parameters. Repeated simulations are used to construct percentiles of the daily occupancy distributions and thus identify ranges of normal fluctuations and those that are substantive deviations from the past, and also to investigate the trade-offs between frequency of overflows and the percent occupancy for both fixed and flexible bed allocations. Larger divisions can achieve more efficient occupancy levels than smaller ones with the same frequency of overflows. Seasonal variations are more significant than day-of-the-week variations and variable discharge rates are more significant than variable admission rates in contributing to overflows.

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