Effect of migrations on synchrony in host-parasitoid system

Insect outbreaks can cause large scale defoliation of forest trees or destruction of crops, leading to ecosystem degradation and economic losses. Some outbreaks occur simultaneously across large geographic scales and some outbreaks occur periodically every few years across space. Parasitoids are a natural enemy of these defoliators and could help mitigate these pest outbreaks. A holistic understanding of the host-parasitoid interactions in a spatial context would thus enhance our ability to understand, predict and prevent these outbreaks. We use a discrete time deterministic model of the host parasitoid system with populations migrating between 2 patches to elucidate features of spatial host outbreaks. We show that whenever populations persist indefinitely, host outbreaks in both patches can occur alternatively (out of phase) at low migration between patches whereas host outbreaks always occur simultaneously (in phase) in both patches at high migration between patches. We show that our results are robust across a large range of parameters across different modelling approaches used typically to model intraspecific competition among hosts and parasitism, in the host-parasitoid literature. We give an analytical expression for the period of oscillations when the migration is low i.e., when host outbreaks in both patches are out of phase, show it is in agreement with numerical results. We end our paper by showing that we get the same results whether we include the biologically rooted formulations from May et al. (1981) or a general cellular automata model with qualitative rules.