Analysis of a West Nile virus model with nonlocal diffusion and free boundaries

Abstract

<p>We consider a West Nile virus model with nonlocal diffusion and free boundaries, in the form of a cooperative evolution system that can be viewed as a nonlocal version of the free boundary model of Lin and Zhu (2017 <i>J. Math. Biol.</i> <b>75</b> 1381-1409). The model is a representative of a class of 'vector-host' models. We prove that this nonlocal model is well-posed, and its long-time dynamical behaviour is characterised by a spreading-vanishing dichotomy. We also find the criteria that completely determine when spreading and vanishing can happen, revealing some significant differences from the model in Lin and Zhu (2017 <i>J. Math. Biol.</i><b> 75</b> 1381-1409). It is expected that the nonlocal model here may exhibit accelerated spreading (see remark 1.4 part (c)), a feature contrasting sharply to the corresponding local diffusion model, which has been shown by Wang <i>et al</i> (2019 <i>J. Math. Biol. </em <b>79</b> 433-466) to have finite spreading speed whenever spreading happens. Many techniques developed here are applicable to more general cooperative systems with nonlocal diffusion.</p>