Minimal wave speed and spread speed in a system modelling the geographic spread of black-legged tick Ixodes scapularis

Abstract

In a recent work [1], a mathematical model is derived to explore the role that the white-tail deer plays in the geographic spread of the black-legged tick Ixodes scapularis in the northeast of the United States of America. The threshold dynamics is rigorously investigated in terms of the basic reproduction number, for the cases of the 1-D whole space Ω=R and general bounded spatial domain Ω with homogeneous Neumann and Direchlet boundary conditions. However, the minimal wave speed and spread speed of the model, which are the motivation of this model and thus most important, are only explored numerically. In the present paper, we offer a rigorous theoretical confirmation of what are numerically observed in [1], concluding that if the basic reproduction number is larger than one, the model allows a spread speed which is also the minimal speed of traveling wave fronts, and this speed is linearly deterministic.