Traveling waves reflecting various processes represented by reaction–diffusion equations

The aim of this paper is to discover analytically the interactional responses of populations in a dynamic region where the reaction–diffusion process with forcing effects takes place through traveling wave solutions. An expansion method is considered here to properly capture the responses for the first time. In order to profoundly analyze the physical and mathematical discussions, some illustrative behavioral results are exhibited for various values of physical parameters. Especially for the different values of diffusion coefficients in the model under consideration, their effects on the behavior of the solitary wave are discussed and observationally supported by considering various illustrations. It is also seen that the solutions representing the diffusion seen to be in the form of the behavior of hexagonal Turing patterns in different time periods. The application of this study in mathematical biology is to analyze the relationship between the population density of certain species in any local region and the specific population density with invasion characteristics. In addition, the formation of the extinction vortex of the invading population, depending on the characteristics of the solutions presented, is also descriptively discussed.