The endogenous oscillations in baptisms and burials in the population at Penrith described in Chapter 3 were apparently triggered by the impact of the mortality of the plague. It can be predicted that any population that is also maintained in steady-state and which suffers a major mortality crisis would display similar oscillations (see section 3.8). These oscillations may not be readily detectable in other communities; they may decay rapidly and be obscured by the noise. The response of a discrete population in steadystate to a mortality crisis will depend on the gain of the feedback and can be classified as follows in order of increasing feedback gain.

1. (1) Overdamped (Fig. 5.1 A); when the gain is low, the birth rate recovers slowly to the pre-plague level without any oscillations.

2. (2) Just underdamped (Fig. 5.IB); with increased gain, the system recovers more rapidly and a low amplitude oscillation develops but is damped out.

3. (3) Very underdamped (Fig. 5.1C); with more gain and little damping the response decays slowly so that there are a number of periods of the oscillation. In general the period of the oscillation will be less than in type (2).

4. (4) Marginally stable (Fig. 5.ID); there is a further increase in gain of the feedback and the oscillation does not decay.

5. (5) Unstable (Fig. 5.IE); the amplitude of the oscillations increases progressively with the very high gain.

The long wavelength cycles in baptisms and burials at Penrith during the teady-state period, 1550–1750, are believed to be of type (4), with no decay in the oscillation.