Stability of Grazing Systems: An Application of Predator-Prey Graphs

In a now classical paper, Rosenzweig & MacArthur (1963) have shown how the general stability properties of simple predator-prey systems can be studied by graphical techniques, supplemented by mathematical analysis of the behaviour near equilibrium points. A similar, graphical analysis of predation functions was suggested by Holling (1965). The stability analysis of predator and prey isoclines in the predator-prey 'phase diagram' has been discussed further by MacArthur & Connell (1966) and Rosenzweig (1969, 1971). The mathematical and graphical analysis of stability in predator-prey systems, and even three-trophic-levels systems, have been extended by Canale (1970), Rosenzweig (1973), Hubbell (1973), May (1971, 1972), Vandermeer (1973), Strebel & Goel (1973) and Maynard Smith & Slatkin (1973). Thus a considerable body of theory has been developed to deal with systems of populations at two or more trophic levels, or 'exploitation' systems (Rosenzweig 1973). So far only few attempts (Salt 1967; Maly 1969; McAllister, Le Brasseur & Parsons 1972) have been made to apply this theory to real ecological systems, whether by directing experiments or observations to test it, or even by comparing its predictions with existing data. Partly this slowness in application may have been due to the feeling that these theoretical models are still too simplified to be expected to apply directly to the manyspecies, variable-environment and spatially heterogeneous predator-prey systems found in nature. However, as pointed out by Rosenzweig (1973), scientific theory often starts by testing rather simple models, even of complex systems. Grazing systems used and controlled by man, from intensive pastures to extensive range, may be considered as a special case of'predator-prey' systems. Much of the theory was developed with explicit or implicit reference to two animal populations (e.g. Rosenzweig & MacArthur 1963; Holling 1965). But herbivore ('predator')-plant ('prey') interaction is sufficiently similar in its general features to make the same approach useful. Some important modifications are necessary, on the basis of biological considerations, but some results are directly transferable between predator-prey and herbivore-plant systems. Grazing systems have some advantages as relatively simple test cases for general ecological theories: the number of species is limited (usually only one herbivore, in some cases one or a few plant species), environmental heterogeneity within a system is often low and movement of animals is controlled. There is a large and increasing number of observations and experiments on pasture and range systems in many parts of the world. On the other hand, grazing systems are one of the types of ecosystems which are of greatest importance to man; if theoretical ecology could contribute to their understanding and to the solution of their practical management problems, this would be a very useful contribution indeed.