Although superseded by more recent and biologically realistic studies, Crofton's (1971b) model of host–parasite associations remains of interest as the simplest model which captures the essentials. Even if its simplifying assumptions are all accepted, Crofton's model has two defects: the first is that its general conclusions are drawn from numerical simulations for a very restricted range of parameter values; the second is that the probability for a parasite transmission stage to succeed in establishing itself in a host is not constrained to be less than unity, as biologically it must be. The present paper remedies these two defects, by giving analytical results valid for all values of the parameters, and by demanding that the parasite transmission factor indeed saturates to unity. Some of Crofton's conclusions remain intact, others are significantly altered.
(1) Crofton (1971b) has presented a mathematical model which aims to exhibit some of the essential dynamical properties of host–parasite associations. The extreme biological simplicity of this model (e.g. hosts and parasites have the same generation time) makes it applicable to few real systems, and later models (Anderson & May, 1977; May & Anderson, 1977) have added many more general biological features in an effort to makecontact with empirical data. Nevertheless, Crofton's model retains pedagogical value as the basic model.
(2) Even within its own framework ofsimple assumptions, Crofton's model has two defects. The first is thatthe general conclusions about its dynamical behaviour are drawn from numerical stimulations for a re stricted, and not necessarily representative, range of parametervalues. The second is that the factor describingthe input of parasite transmission stages into the next generation of hostsdoes not saturate to unity, as its biological definition implies it must. Thepresent paper gives an analytical account of the dynamical behaviour of Crofton's model, valid for all values of the relevant biological parameters, and with a parasite trans mission factor that does saturate to unity. The ensuing conclusions are in several respects significantly differentfrom Crofton's
(3) The intrinsic growth rates of the host and parasite populations are defined as λ and A; the negative binomial parameter k measures the overdispersion of parasites among hosts (small k corresponds to high overdispersion); and L characterizes thelethal level of parasites per host.Then unless λ1+1/k λ A exp (– 1[L) no equilibrium state is possible, andthe host population undergoes Malthusiangrowth that the parasites cannot check. This inequality tends to be satisfied if k is not too small, λ not too large, and A significantly larger than λ: see Figs 1, 2, and 4.This aspect of the model derives from the saturation of the parasite transmission factor, and is omitted fro Crofton's discussion.
(4) When an equilibrium does exist, the following observations can be made. The equilibrium host population H* is given by eq. (15): it de creases with increasing A; increases with increasing λ; is roughly inde pendent of L; and increases with increasing parasite overdispersion for small k (k < In λ);, while being roughly independent of k for larger k. Theequilibrium number of parasites per host m* is given by eq. (9): it is independent of A; increases roughly linearly with L; increases with increasing overdispersion or λ for small k (k < in λ); and increases slowly with λ, and is roughly independent of k, for larger k. The totalpopulation of parasites at equilibrium is given by P* = H*m*.
(5) The stability of the equilibrium, i.e. its ability to recover from disturbance, depends mainly on λ and on k, as illustrated in Fig. 4. Except for values of λ and k perilously close to the boundary where no equili brium is possible, the disturbed host and parasite populations will return to their equilibrium values by undergoing damped oscillations. The damping will tend to be weak if k is large, or if λ is small.
(6) These conclusions accord with those derived from more detailed and realistic host–parasite models.
(7) The general process, whereby thehost–parasite association can be stabilized by overdispersion of parasites, is dynamically similar to that whereby prey–predator or host–parasitoid associationscan be stabilized by differential aggregation of predators or by explicit refuges for the prey.