1. Despite the strong impact parasites can have, only few models of phytoplankton ecology or aquatic food webs have specifically included parasitism. 2. Here, we provide a susceptible-infected model for a diatom-chytrid host–parasite system that explicitly includes nutrients, infected and uninfected hosts, reproduction of the parasite on the hosts and free-living infective stages. 3. A distinguishing feature of the model is that parasite reproduction on host increases with nutrient availability to the infected host, as has been observed for many parasites and viruses. 4. It follows from this assumption that the parasite’s basic reproduction number, R0, increases with nutrient concentration, because at higher nutrient concentrations, infected hosts consume more nutrients that are used for the reproduction of the parasite. 5. Another important result is that there may be two alternative states to which population densities can converge: one with only the host and one with host and parasite co-existing. In the latter, the parasite can invade a host population only if it is introduced above a threshold density. 6. Furthermore, the model shows a strong tendency for host–parasite cycles, which may be chaotic. Nutrient enrichment leads to increasing amplitude of these cycles, which may cause host or parasite population extinction caused by stochastic fluctuations during periods of low population density, which is the Paradox of Enrichment. 7. Finally, if alternative states and cycles co-occur, increased population cycle amplitude may drive the parasite below its threshold density for successful invasion, causing parasite extinction in a ‘deterministic Paradox of Enrichment’. Published results confirm that host–parasite cycles and collapse of host–parasite systems may occur in real plankton communities. 8. Our results underline that ecological detail in host–parasite models may have consequences for disease dynamics that may be overlooked when ecological interactions between environment, host and parasite are not explicitly taken into account.