An invulnerable age class and stability in delay-differential parasitoid-host models

W.W. Murdoch, R.M. Nisbet, S.P. Blythe, William Gurney, J. Reeve

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The models were motivated by a study of red scale, an insect pest of citrus in southern California, and its successful parasitoid, Aphytis melinus. The system appears to be stable, but the usual stabilizing mechanisms suggested appear to be absent. We hypothesized that, in combination with overlapping generations, the scale's invulnerability to attack by Aphytis at several stages, particularly the long-lived reproductive adult stage, might explain the system's stability. In the model both species have a juvenile and an adult stage. Generations overlap and the host and the parasitoid are not synchronous. The model has the form of coupled, delayed differential equations representing the two populations. The key parameters determining stability are the duration of the juvenile parasitoid stage (the developmental lag) and of the invulnerable adult stage of the scale, relative to the duration of the juvenile scale stage. We conclude that an invulnerable adult stage in the scale is always stabilizing because in its presence some stable parameter space exists that does not occur in its absence. The longer the invulnerable adult stage is relative to the predator developmental lag, the more likely stability is. Stability is unlikely unless the adult scale stage is significantly longer than the juvenile stage. Stability is less likely as adult parasitoids live longer or as scale fecundity increases, but the system is less sensitive to these parameters than to the duration of the invulnerable class. An invulnerable juvenile stage can also add stability, but only for a narrow range of parameter values that are not likely in real systems. Given the parameter values in the red scale-Aphytis interaction in southern California, the invulnerable adult class is not likely to explain the observed stability, but probably contributes to it. This mechanism buys increased stability at the cost of higher pest equilibrium; we discuss the apparent tension between stability and successful pest control.
Original languageEnglish
Pages (from-to)263-282
Number of pages20
JournalAmerican Naturalist
Issue number2
Publication statusPublished - Feb 1987


  • biological pest control
  • red scale
  • delayed differential equations


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