Delayed feedback and multiple attractors in a host-parasitoid system

Continuous-time, age structured, host-parasitoid models exhibit three types of cyclic dynamics: Lotka-Volterra-like consumer-resource cycles, discrete generation cycles, and "delayed feedback cycles" that occur if the gain to the parasitoid population (defined by the number of new female parasitoid offspring produced per host attacked) increases with the age of the host at tacked. The delayed feedback comes about in the following way: an increase in the instantaneous density of searching female parasitoids increases the mortality rate on younger hosts, which reduces the density of future older and more productive hosts, and hence reduces the future per head recruitmen t rate of searching female parasitoids. Delayed feedback cycles have previo usly been found in studies that assume a step-function for the gain functio n. Here, we formulate a general host-parasitoid model with an arbitrary gai n function, and show that stable, delayed feedback cycles are a general phe nomenon, occurring with a wide range of gain functions, and strongest when the gain is an accelerating function of host age. We show by examples that locally stable, delayed feedback cycles commonly occur with parameter value s that also yield a single, locally stable equilibrium, and hence their occ urrence depends on initial conditions. A simplified model reveals that the mechanism responsible for the delayed feedback cycles in our host-parasitoi d models is similar to that producing cycles and initial-condition-dependen t dynamics in a single species model with age-dependent cannibalism.