Skewed attacks, stability, and host suppression

It is well known that the model equilibrium in the Nicholson-Bailey framework for parasitoid-host interactions can be stabilized if there is variation in the risk of parasitism among individual hosts. We show that the shape of the distribution of risk among hosts is crucial in determining stability. Stability for all values of the reproduction rate R of the host requires the distribution to be skewed, with a modal risk of zero. As a consequence, a distribution of risk with nonzero mode but otherwise sufficiently high variability will not stabilize the host-parasitoid equilibrium for certain values of R. We also develop a new, simple, and general criterion for stability, according to which the host-parasitoid equilibrium is stable if and only if the adult host equilibrium density increases as host reproductive rate R increases. This reinforces previous results that skewed risk of the type needed for stability reduces parasitoid efficiency and hence the parasitoid's ability to suppress the host. Finally, we identify appropriate measures that can be made in the field to relate skew in risk of parasitism to stability.