On Assessing the Role of Spatially-Heterogeneous Density-Independent Host Mortality on the Stability of Host-Parasitoid Systems

Abstract

In an already classical work, Beddington et al. (1978) analysed several cases of successful biological control and suggested that their common feature was a displacement to a new equilibrium with a strong depression of the host population (low q values, q = N*/K, where K is the equilibrium density before the introduction of the parasitoid and N* the one afterwards). Since then, one of the aims of theoretical studies on hostparasitoid dynamics was considered to be the identification of processes that lead to the stability of these systems at low q values (for a review see Hassell 1978). Spatial heterogeneity in the distribution of hosts and parasitoids has been singled out as the most important process leading to the stability of host-parasitoid systems at low values of q, either when it results in spatial density-dependent (Hassell and May 1973, 1974, Murdoch and Oaten 1975, Murdoch 1977) or inverse density-dependent mortality of the host (Hassell 1984). Parasitoid aggregation, which leads to spatially-heterogeneous but density-independent mortality, has also been identified as an important stabilizing mechanism (May 1978, Murdoch et al. 1984). This view has been challenged by Murdoch et al. (1984, 1985, Reeve and Murdoch 1985) on the grounds that neither stability nor sufficiently strong spatial differences in host mortality are common features of successful biological control. Murdoch et al. (1984, and Reeve and Murdoch 1985) devised two methods (one graphical, the other statistical) to measure the strength of spatially-heterogeneous, density-independent host mortality based on May's (1978) model. In this model, host and parasitoid survival rates are considered as density-independent and f, the fraction of hosts evading parasitism (see Appendix), is given by the zero term of the negative binomial distribution (f = (1 + a P,/k)-k, where a is the search efficiency, k determines the degree of clumping in parasitoid attacks and P, is the mean predator density at generation t). This model is stable if and only if k < 1. A similar conclusion is attained for realistically low values of q with models in which density dependent survival is represented by a logistic function (May et al. 1981). The methods proposed by Murdoch et al. allow determination of whether k is lower than 1 in natural host-parasitoid systems. They used these methods in two studies of systems formed by scale insects and introduced parasitoids. In the system formed by the olive scale (Parlatoria oleae Culvee) and two introduced parasitoids (Murdoch et al. 1984), the host density was reduced from over 200 cales per twig to one or two per 50 twigs, (q = 0.0001). Whilst Murdoch et al. (1984) suggested that the system is unstable, Huffaker et al. (1986), on the basis of a long-term study, argued in the opposite sense. The second system (the California red scale, Aonidiella aurantii, and its parasitoid Aphytis melinus) is said to be stable (Reeve and Murdoch 1985), and the data provided by the authors suggest a q value between 0.1 and 0.01. No clear evidence of density-dependent or inverse density-dependent mortality was found in either system. In both host-parasitoid interactions k is greater than 1, and so Murdoch et al. (1984, Reeve and Murdoch 1985) concluded that spatial heterogeneity does not play a key role in the stability of either system. Their conclusions are discussed in the following paragraphs. May (1978) proposed the negative binomial distribution to describe parasitoid aggregation on phenomenological grounds (it fits many data sets), but pointed out that it could be derived by supposing that the parasitoids are gamma distributed among spatially discrete patches and the searching of parasitoids within patches is totally random. The same conclusion was reached by Murdoch et al. (1984; Chesson and Murdoch 1986): If ut is the number of parasitoids in the vicinity of each host and s(u) the probability density function of ut, then the average or expected proportion of unparasitized hosts (f) is given by