An Invulnerable Age Class and Stability in Delay-Differential Parasitoid-Host Models

Abstract

Reviewing some examples of biological control of insect pests, Murdoch et al. (1985) concluded that success was commonly characterized by unstable populations of pest and parasitoid or predator and, perhaps, by a tendency toward local extinction. The only exception appeared to be control by the parasitoid Aphytis melinus of red scale, a pest insect on citrus trees in coastal southern California. A classic example of biological control, this system appears to persist around a low, stable equilibrium density of pest and parasitoid. It is, of course, difficult to determine whether or not a population in the field is stable. The difference between the conditions of biological control in different scale insects, however, lends support to the idea that some pests may be controlled in a stable manner and others not. The olive scale in groves in northern California seems unstable in that all scales in twig samples, in trees sampled, and in whole groves can be parasitized, with the populations showing a tendency toward local extinction; in one grove, density fluctuated strongly and not apparently around an equilibrium (Murdoch et al. 1984). In southern California citrus groves, however, A. melinus appears unable to parasitize more than about 30% of the red scale in any small sample, and living scales are found in most samples. In addition, long-term data on scale populations (DeBach 1974) show only small fluctuations around the mean density. Reeve and Murdoch (1985) confirmed that both scale density and the parasitism rate are remarkably constant, and they later showed (Reeve and Murdoch 1986) that the per capita rate of change of the scale population is density-dependent. These observations do not establish rigorously that this system is stable, but at least the contrast with olive scale and other examples discussed by Murdoch et al. (1985) suggests that the maintenance of relative constancy and the avoidance of local extinction in red scale require an explanation. Two other differences between these two scale-insect systems may help explain the contrast in their dynamic behavior: the presence or absence of an invulnerable