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

Abstract 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 J . Math . Biol . 75 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 J . Math . Biol . 75 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 et al (2019 J . Math . Biol . 79 433–466) to have finite spreading speed whenever spreading happens. Many techniques developed here are applicable to more general cooperative systems with nonlocal diffusion.