Abstract
Abstract A delayed reaction-diffusion system with Neumann boundary conditions modelling the spatial spread of bacterial and viral diseases is considered. Sufficient conditions independent of diffusion and delay are obtained for the asymptotical stability of the spatially homogeneous positive steady state. We also perform a detailed Hopf bifurcation analysis by analyzing the corresponding characteristic equation and derive some formulae determining the direction of bifurcation and the stability of the bifurcating periodic solution by calculating the normal form on the center manifold. The delay driven instability of the positive steady state and the diffusion-driven instability of the spatially homogeneous periodic solution are investigated. Our results complement the main results in Tan et al. (2018) [10]. Some examples and numerical simulations are presented to illustrate our theoretical results.
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