Infected coexistence instability with and without density-dependent regulation

Numerical studies of two-host shared pathogen systems, using nonlinear dynamical models, have suggested that there are a set of simple rules governing the possible number of stable equilibria. In this paper we show by counterexample that these rules are not universally valid, whether or not the system is subject to density-dependent regulation. The analysis of diverse counterexamples reveals a variety of structures, including models with no relevant stable equilibria or with multiple stable infected coexistence equilibria. In the former case, asymptotic behaviour is characterised by single or multiple limit cycles, generating sustained oscillations. It has also been found possible to have limit cycles and stable equilibria coexisting in the same model. The parameter values defining the counterexamples reflect exceptional but, in most cases, plausible situations and provide insight into the mechanisms by which stability fails to be achieved. One biological implication of this analysis is that long-lived infective stages are not necessary to produce oscillations in the numbers of forest living invertebrates.