Abstract In this paper, we study the existence of travelling wave solutions in a system of delayed reaction–diffusion equations (DRDEs) satisfying the semi-mixed quasi-monotonicity condition. As an application, we analyse a mathematical model of a prion disease, whose reaction terms fall within the framework of our approach. Under biologically motivated assumptions, we construct appropriate upper and lower solutions and apply an existence theorem tailored to such systems. We then use the contracting rectangle method to describe the asymptotic behaviour of the solutions as they converge to a non-trivial positive equilibrium. The paper concludes with numerical simulations.
Adimy et al. (Sun,) studied this question.