Global redistribution and local migration in semi-discrete host-parasitoid population dynamic models.

Host-parasitoid population dynamics is often probed using a semi-discrete/hybrid modeling framework. Here, the update functions in the discrete-time model connecting year-to-year changes in the population densities are obtained by solving ordinary differential equations that mechanistically describe interactions when hosts become vulnerable to parasitoid attacks. We use this semi-discrete formalism to study two key spatial effects: local movement (migration) of parasitoids between patches during the vulnerable period; and yearly redistribution of populations across patches outside the vulnerable period. Our results show that in the absence of any redistribution, constant density-independent migration and parasitoid attack rates are unable to stabilize an otherwise unstable host-parasitoid population dynamics. Interestingly, inclusion of host redistribution (but not parasitoid redistribution) before the start of the vulnerable period can lead to stable coexistence of both species. Next, we consider a Type-III functional response (parasitoid attack rate increases with host density), where the absence of any spatial effects leads to a neutrally stable host-parasitoid equilibrium. As before, density-independent parasitoid migration by itself is again insufficient to stabilize the population dynamics and host redistribution provides a stabilizing influence. Finally, we show that a Type-III functional response combined with density-dependent parasitoid migration leads to stable coexistence, even in the absence of population redistributions. In summary, we have systematically characterized parameter regimes leading to stable/unstable population dynamics with different forms of spatial heterogeneity coupled to the parasitoid's functional response using mechanistically formulated semi-discrete models.