Hydra effect and paradox of enrichment in discrete-time predator-prey models

We develop three discrete-time predator-prey models from the Nicholson-Bailey host-parasitoid framework, assuming a type II functional response and logistic prey growth in form of the Beverton-Holt map. Our models show many similarities with the continuous-time Rosenzweig-MacArthur model, not only the same equilibria and sequence of bifurcations, but also phenomena such as the hydra effect and paradox of enrichment. Our three models differ in the order of events, in which the processes of density-dependent prey regulation and predation take place. When their order is reversed, but their relative order remains the same such that only census timing is changed, we observe quantitative differences in population size, but no differences in qualitative behaviour. When a modified order of events induces delayed density dependence, we observe increased stability of population dynamics, which is somewhat contrary to conventional expectation. Overall, our models exhibit behaviour analogous to the Rosenzweig-MacArthur model and highlight the importance of the order of events in discrete-time models.