Traveling waves in a diffusive tick population dynamics model incorporating nonlocal acquired tick resistance

Abstract

We consider tick population dynamics subject to the spatial movement of relevant hosts at different stages of the life cycle, where the tick survival rate during a period of the life cycle is reduced due to acquired tick resistance in hosts. This gives rise to a diffusive model with two delays and nonlocal host-tick interactions for the first time. We investigate the issue of tick range expansion in the form of travelling wave fronts. By constructing an appropriate pair of upper and lower solutions and applying the monotone iteration argument, we obtain a sharp criterion to ensure the existence of traveling wave fronts. Furthermore, we rigorously characterize the exponential decay of these traveling wave fronts at negative infinity and prove the non-existence of travelling wave solutions under the threshold parameter condition. Finally, numerical simulations confirm excellent agreement with the theoretical results.