The effects of a pool of dispersers on host-parasitoid systems

When individuals migrate in a multi-patch environment, a considerable proportion of their lifetime might be spent in transit between patches. We investigate the effects such a pool of dispersers can have on local stability and dynamics for a variety of multi-patch host-parasitoid models. When an arbitrary number of patches with internal Lotka-Volterra dynamics is coupled via a global pool of dispersers, the equilibrium is globally stable. The global pool is stabilising if dispersal is by hosts only, by parasitoids only, or by both hosts and parasitoids. If dispersal is local such that individuals first enter a pool close to the patch where they originate and then disperse to adjacent pools, the equilibrium is locally stable. We also analyse the situation where the functional response of parasitoids within a patch is Holling type II which is known to destabilise host-parasitoid systems. Coupling this single patch to a pool of dispersers can produce a locally stable interaction, provided the handling time of hosts is not too long. However, the pool provides a biologically realistic example of an interaction that is locally stable but not permanent. The longer the handling time, the smaller the region of population densities within which populations converge to the equilibrium state. In a multi-patch environment with a global disperser pool, the dynamics of the system are not qualitatively different from the single patch case (i.e. the equilibrium can be locally stable but the system is not permanent). In a multi-patch environment with local disperser pools, true spatial interactions between patches can develop. In contrast to the global pool, local pools can destabilise the stable equilibrium of the single patch case. Limit cycles develop around this unstable equilibrium that lead to extremely complicated dynamics. In contrast to the global pool, a system of local pools can exhibit bounded fluctuations so that populations do not go extinct.