The role of host refuge and strong Allee effects in a host-parasitoid system

We propose and study a discrete host-parasitoid model of difference equations with a spatial host refuge where hosts in the refuge patch are free from parasitism but undergo a demographic strong Allee effect. If the growth rate of hosts in the non-refuge patch is less than one, a host Allee threshold is derived below which both populations become extinct. It is proven that both populations can persist indefinitely if the host growth rate in the non-refuge patch exceeds one and the maximum reproductive number of parasitoids is greater than one. Numerical simulations reveal that the host refuge can either stabilize or destabilize the host-parasitoid interactions, depending on other model parameters. A large number of parasitoid turnover from a parasitized host may be detrimental to the parasitoids due to Allee effects in the hosts within the refuge patch. Moreover, it is demonstrated numerically that if the host growth rate is not small, the population level of parasitoids may suddenly drop to a small value as some parameters are varied.