Pattern formations of an epidemic model with Allee effect and time delay

Allee effect widely exists for endangered plants and animals in ecosystem, which indicates that the minimum population density or size is necessary for population survival, namely, Allee threshold. In this paper, a delayed reaction-diffusion epidemic model with respect to Allee effect is investigated. The instability of the positive constant steady state is induced by two mechanisms, one is diffusion-induced instability, the other is delay-induced instability. The first case gives rise to Turing patterns. Moreover, Turing region becomes narrow as incubation delay being increased. We further observe that the range of Turing mode is enlarged with the increase of Allee threshold. The numerical simulations verify our theoretical results. The combined effects of Allee effect and disease on the spatial distributions of endangered species are studied, which provides new insights for human intervention in conservation management of these species.