Preytaxis is the attraction (or repulsion) of predators along prey density gradients and a potentially important mechanism for predator movement. However, the impact preytaxis has on the spatial spread of a predator invasion or of an epidemic within the prey has not been investigated. We investigate the effects preytaxis has on the wavespeed of several different invasion scenarios in an eco-epidemiological system. In general, preytaxis cannot slow down predator or disease invasions and there are scenarios where preytaxis speeds up predator or disease invasions. For example, in the absence of disease, attractive preytaxis results in an increased wavespeed of predators invading prey, whereas repulsive preytaxis has no effect on the wavespeed, but the wavefront is shallower. On top of this, repulsive preytaxis can induce spatiotemporal oscillations and/or chaos behind the invasion front, phenomena normally only seen when the (non-spatial) coexistence steady state is unstable. In the presence of disease, the predator wave can have a different response to attractive susceptible and attractive infected prey. In particular, we found a case where attractive infected prey increases the predators' wavespeed by a disproportionately large amount compared to attractive susceptible prey since a predator invasion has a larger impact on the infected population. When we consider a disease invading a predator-prey steady state, we found some counter-intuitive results. For example, the epidemic has an increased wavespeed when infected prey attract predators. Likewise, repulsive susceptible prey can also increase the infection wave's wavespeed. These results suggest that preytaxis can have a major effect on the interactions of predators, prey and diseases.
Keywords: Biological invasions; Pattern formation; Spatiotemporal oscillations; Wavespeed.