Host-parasitoid coexistence and egg-limited encounter rates

Most models of host-parasitoid interactions consider parasitoid attack rates or, more accurately, encounter rates to be limited by the ability of the parasitoids to find suitable hosts. Some models extend this limitation to include the length of time it takes a parasitoid to handle each host. Here we consider host-parasitoid dynamics in the context of parasitoid encounter rates being limited by the number of eggs that each parasitoid has to lay when the host is at high densities and by the ability of individual parasitoids to find hosts when the host is at low densities. Although the encounter rate function we obtain is mathematically equivalent to previously obtained encounter rate functions that include handling time, the stability properties of the resulting host-parasitoid system have heretofore not been fully explored. Our analysis indicates in the absence of host density self-regulating mechanism that the well-known condition in which host-parasitoid interactions cannot be stable unless the proportion of hosts escaping attack has a sufficiently clumped distribution (i.e., k less than or equal to 1 in the negative binomial model, where k is the negative binomial parameter) still applies and that the intrinsic growth rate of the parasitoid population must exceed the intrinsic growth rate of the host population by a factor that both is greater than one and increases as the degree of dumping associated with the proportion of hosts that escape attack increases (i.e., as k --> 0 in the negative binomial model).