Population Cycles

Ecological Monographs 62:119-142 (1992)

H. Resit Akçakaya
Department of Ecology and Evolution, Stony Brook University, Stony Brook, NY 11794

Present address: Applied Biomathematics, 100 North Country Road, Setauket, NY 11733, USA

Abstract

        Populations of certain mammal species and their predators show cyclic fluctuations in northern latitudes, and the amplitude of cycles in some cases increases towards the north. The evidence reviewed suggests that (i) abiotic factors and intrinsic mechanisms are unable to explain cycles; (ii) quantity and quality of food resources of the herbivore population have important effects on population dynamics, although plant-herbivore interaction cannot explain the cycles by itself; (iii) predation is another important factor for these populations, and probably essential for cyclicity of herbivore populations. A number of mathematical models have shown that prey-predator models can produce limit cycles, but they have not demonstrated that the cyclic fluctuations observed in natural populations can be explained by the mechanisms they incorporate, and the parameters they define.

        In this study, a mathematical model is developed to predict specific patterns of prey-predator cycles observed in nature with independently estimated parameters. This prey-predator model is based on the concept of ratio-dependence: the trophic functions (functional and numerical responses) are modeled as functions of prey:predator ratio rather than as functions of prey density only, as in traditional prey-predator models. This approach incorporates the concept of interference in a simple way by describing trophic interactions as functions of per capita resources. The parameters of the model are estimated from studies on the biology of cyclic lynx and hare populations, rather than by fitting time-series data to the model. Parameters of the model give rise to limit cycles when they are changed in the way they are expected to change from south to north, which is consistent with the observations on the latitudinal patterns in cyclicity. The major quantitative prediction of the model is the cycle period. The period is predicted to be around 10 years, which is the observed period of hare-lynx fluctuations.