Complex predator invasion waves in a Holling–Tanner model with nonlocal prey interaction

We consider predator invasions for the nonlocal Holling–Tanner model. Predators are introduced in a small region adjacent to an extensive predator-free region. In its simplest form an invasion front propagates into the predator-free region with a predator–prey coexistence state displacing the predator-free state. However, patterns may form in the wake of the invasion front due to instability of the coexistence state. The coexistence state can be subject to either oscillatory or cellular instability, depending on parameters. Furthermore, the oscillatory instability can be either at zero wave number or finite wave number. In addition, the (unstable) predator-free state can be subject to additional cellular instabilities when the extent of the nonlocality is sufficiently large. We perform numerical simulations that demonstrate that the invasion wave may have a complex structure in which different spatial regions exhibit qualitatively different behaviors. These regions are separated by relatively narrow transition regions that we refer to as fronts. We also derive analytic approximations for the speeds of the fronts and find qualitative and quantitative agreement with the results of computations.