Host exclusion and coexistence in apparent and direct competition: An application of bifurcation theory

Recent empirical studies have focused attention on the interplay in multi-host systems of parasite-mediated apparent competition and direct competition between hosts. However, theoretical investigation of such systems has been hindered by the onset of algebraic intractability with the increase in system dimensionality. In this paper we circumvent this problem by using a geometric approach in which arrays of bifurcation maps are constructed, each map being structured by the set of (bifurcation) points in parameter space at which qualitative changes in system behaviour take place. From these maps can be compiled a concise catalogue of the possible modes of system behaviour, enabling an investigation of the interaction of apparent and direct competitive forces to be carried out. Of importance is the identification of those situations where increasing one or both of these competitive forces leads to a change in the stability state. The maps provide an efficient way of determining whether, and, if so, under what conditions, specific modes of behaviour are allowed by the model. Two field phenomena of particular interest, discussed in the paper, are host invasion and dominance reversal resulting from the introduction of the pathogen into a directly competitive system.