Host-Parasitoid Systems in Patchy Environments: A Phenomenological Model

Abstract

SUMMARY (1) A host-parasitoid model is presented which is intermediate in complexity between the Nicholson-Bailey model (in which the parasitoids search independently randomly in a homogenous environment) and complicated models for incorporating environmental patchiness (in which the overall distribution of parasitoid attacks is derived from detailed assumptions about their searching behaviour and about the spatial distribution of the hosts). The model assumes the overall distribution of parasitoid attacks per host to be of negative binomal form. There are consequently three biological parameters: two are the usual parasitoid 'area of discovery', a, and the host 'rate of increase', F; the third is the negative binomial clumping parameter, k. Such intermediate-level models have proved useful in sorting out ideas in the related disciplines of epidemiology and parasitology. (2) Empirical and theoretical arguments for using the negative binomial to give a phenomenological description of the essential consequences of spatial patchiness in models are surveyed. (3) A biological interpretation of the parameter k in host parasitoid models is offered. If the parasitoids be distributed among patches according to some arbitrary distribution which has a coefficient of variation CVp, and if the parasitoid attack distribution within a patch be Poisson, then the ensuing compound distribution can be approximated by a negative binomial which will have the same variance as the exact distribution provided k is identified as k = (I I/CVp)2. (4) Expressions are obtained for the equilibrium values of host and parasitoid popula- tions. These equilibria are stable if, and only if, k < 1; that is, provided there is sufficient clumping. (5) The dynamical effects of parasitoid aggregation in some respects mimic those introduced by mutual interference among parasitoids; the appropriate coefficient of 'psuedo-interference' is calculated.