Some Case Studies

One of the exciting phenomena in ecology is population cycles. This inspired people like A. J. Lotka of the United States and V. Vol terra of France more than half a century ago into a theoretical study, which later developed into a scientific discipline. Their well-known models (see §2.2) for cyclical predator-prey patterns provided an explanation that seemed satisfactory for a brief period. However, it has been pointed out in §2.2 that the simple Lotka-Volterra type equation has a phase diagram in the form of a centre, which is in general a point away from the origin. The trajectory is, by the nature of the problem, confined to the first quadrant, and it turns out to be such that in order to produce very large fluctuations in the numbers of the two interacting species described by the equation, it is sufficient to start with only a few members of each species. This is often not the case with ecological observations. This type of model is now superseded by more realistic ones. A comprehensive review of the subject is given by Oster and Ipaktchi (1978), in which a fundamental conservation law of population biology has been postulated.