A Spatial Model For The Spread Of Fire Blight During Bloom

Abstract

Through the coupling of two semi-linear reaction-diffusion equations and three ordinary differential equations we formulate the first spatial model to describe the spread of fire blight during bloom. First, it is shown that non-negative, unique and bounded solutions exist to our coupled system. We then discuss the stability of the disease-free equilibrium, the asymptotic behaviour of solutions and argue that patterns do not form in our model as time gets large. By using the method of upper and lower solutions together with Schauder’s fixed point theorem, we show that our model admits travelling wave solutions that connect a disease free state and a state in which all hosts are dead. Through simulation experiments we find that host density and the transfer of ooze to healthy flowers both have a significant impact on the early speed of spread.