Time series analysis of oscillations in a model population: The effects of plague, pestilence and famine

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Penrith (Cumbria) during 1557–1812 has been treated as a model ecological population and the parish registers have been studied by time series analysis which reveals three different oscillations. Mean births and deaths per annum remained remarkably constant over the first 200 years, in spite of the severe mortality of two plagues; thereafter the population rose exponentially. Matrix modelling of the system, with the inclusion of feedback, explains well the different causes of the three oscillations and shows that this semi-isolated population was maintained at a maximum sustainable level that was permitted by the limited food resources available; high mortality associated with famine acted as the density-dependent feedback. Births constitute a simple loop formed from preadult mortality and fecundity with a delay associated with maturation. The death series shows 7-year cycles of increased mortality throughout the 250 years which generate 7-year oscillations in the birth series. These oscillations occur both under steady-state conditions and during exponential growth, and the model confirms that this simple response of the loop is independent of feedback. A plague with a major mortality at Penrith produced a decaying oscillation in the birth series with a wavelength of 32 years which was revealed by linear filtering. It is predicted from the model that this oscillation occurs independently of feedback and is generated by the delay in the loop (i.e. the time from birth to the average age at which a woman had her median child). Feedback drives the population back after a plague to its earlier steady-state level; under these specific conditions, a plague also generates persistent, longer wavelength cycles in both births and deaths and, as predicted by matrix modelling, the cross correlation function shows zero lag and the two oscillations are of the same amplitude. Modelling also shows that the gain of the feedback and the underlying mortality function determine both the wavelength and whether the oscillation decays.

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