  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.
