Ecoepidemics with infected prey in herd defence: the harmless and toxic cases

We consider a predator-prey population model with prey gathering together for defence purposes. A transmissible unrecoverable disease affects the prey. We characterize the system behaviour, establishing that ultimately either only the susceptible prey survive, or the disease becomes endemic, but the predators are wiped out. Another alternative is that the disease is eradicated, with sound prey and predators thriving at an equilibrium or through persistent population oscillations. Finally, the populations can thrive together, with the endemic disease. The only impossible alternative in these circumstances is predators thriving just with infected prey. But this follows from the model assumptions, in that infected prey are too weak to sustain themselves. A mathematical peculiarity of the model is the singularity-free reformulation, which leads to three entirely new dependent variables to describe the system. The model is then extended to encompass the situation in which ingestion of diseased prey is fatal for the predators and to the cases where the predators find the infected prey less palatable.