Propagation Dynamics for a Spatial Discrete Waterborne Pathogen Model With Temporal Periodicity

ABSTRACT This paper is concerned with the propagation dynamics for a spatial discrete waterborne pathogen model with temporal periodicity. We firstly establish the spreading speed of the model. Then, applying the asymptotic fixed‐point theorem via a pair of time‐periodic upper and lower solutions, we show that the model admits time‐periodic traveling waves when the basic reproduction number is larger than one and the wave speed is greater than a threshold speed. In addition, when the basic reproduction number is either smaller than one; or larger than one but the wave speeds are less than the threshold speed, we further prove the non‐existence of time‐periodic traveling waves by using the methods of comparison principle. From our result, one can see the minimum wave speed for the time‐periodic traveling waves is exactly the same with the spreading speed of the model.