Biological Cycles and the Fivefold Way

One of the more curious natural phenomena is the regular periodic variation in the populations of certain interacting species, a variation which does not correlate with known periodic external forces such as the cycles of darkness and light, the seasons, or weather cycles. Most commonly, these species are involved in a predator-prey relationship. The phenomenon has been observed in the microcosm of a laboratory culture (paramecium aurelia (predator) and saccharomyces exiguus (prey)—see D’Ancona [11]) and in the macrocosm of the Canadian coniferous forests (Canadian lynx (predator) and snowshoe hare (prey)—see Keith [17]). Other examples include the budworm-larch tree cycle in the Swiss Alps [2] and a lemming-vegetation cycle in Scandinavia [19]. We shall discuss only the lynx-hare cycle, but the mathematical models we shall develop are in principle applicable to any predator-prey interaction. In the course of our analysis we shall study the equilibria of coupled differential systems and the possible limit behavior of bounded orbits of such systems. This will all then be applied to the question of the existence of cycles in a predator-prey system.