Exponential stability of delayed multi-group model with reaction–diffusion and multiple dispersal based on Razumikhin technique and graph theory

•The effect of three factors is considered in MGM.•A new approach is combining Razumikhin method with graph theory.•Three different kinds of stability criteria are obtained.•The relation between stability and topological structure of MGM is given.

In this paper, we investigate the delayed multi-group model with reaction–diffusion and multi-dispersal. In this multi-group model, each single group model and dispersal schema among multiple groups are affected by time delays and reaction–diffusion. The main method is the combining of Razumikhin technique and graph theory. Based on the main method, several stability criteria are presented, including the Razumikhin-type theorem, the Lyapunov-type theorem and a sufficient criterion in the form of coefficients. Finally, a numerical example is provided to illustrate the effectiveness of the results developed.