Dynamics of a general model of host-symbiont interaction

We consider a model of host-symbiont interactions, in which symbionts can only live in association with their host and are transmitted both vertically from associated hosts to their offspring and horizontally from associated hosts to nearby unassociated hosts. The effect of the symbiont is modelled by a change in the birth rate of associated hosts. We analyze the two-dimensional dynamics in the resulting four-dimensional parameter space, and determine the qualitative behaviour for all parameter values. We find that for all but one choice of parameter values, solutions in the feasible region, apart from a 0- or 1-dimensional set of initial conditions, tend either to a unique equilibrium, or to one of two distinct equilibria. Moreover, the bistable case occurs only when the symbiont is a mutualist whose horizontal spread rate through the host population exceeds the positive change in the birth rate of associated hosts.