Multihost, multiparasite systems: an application of bifurcation theory.

The local analysis of multihost multiparasite models has been hampered by algebraic intractability. There have been two responses to this difficulty: extensive numerical investigation, and simplification to a level where analytical techniques work. In this paper we describe another approach, based on bifurcation theory, in which the qualitative properties of the model equilibrium structure are realized on an array of maps drawn in parameter space. This approach is described in the context of two models: the basic two-host shared microparasite S-I model and the single-host two-microparasite S-I (susceptible-infective) model. The procedure involved does not require model simplification through a reduction in dimensionality. It can handle intraspecific as well as parasite-mediated competition and, in the second model, single-host parasite coexistence. The map arrays provide a concise catalogue of the possible modes of behaviour of a system and an explanation for changes in that behaviour. In particular, the reasons why the conjectures made about the behaviour of the first of these models do not hold throughout parameter space are immediately clear from the map structure, as are the conditions for collusive and competitive behaviour between the two types of parasite in the second model.