Revisiting the spatiotemporal dynamics of a diffusive predator-prey system: An analytical approach

Abstract

This study investigates some analytic solutions and phase portraits to the diffusive predator-prey system in studying the spatiotemporal dynamics of a predator-prey community in ecology through an analytical approach and a qualitative theory of planar dynamical systems, respectively. To accomplish such aims, a simple wave transformation is applied to the diffusive predator-prey system for converting it into a system of ordinary differential equations with a planar dynamical system for analyzing the behavior of bifurcation properties and analytic solutions. Then, the analytical unified method is employed to the attained system. The applied wave transformation is put back to the obtained solutions of the system of ordinary differential equations. Finally, the analytic solutions namely, kink, anti-kink, singular, periodic-singular, and sinusoidal wave solutions are attained to the considered system. All the constructed wave solutions are found to be new from the viewpoint of the application of the unified method. In order to verify the biological wave phenomena of predator and prey populations, some graphs are presented for illustrating the analytic solutions, which have interesting implications in ecology. The effects of free parameters and wave celerity on the attained solutions are demonstrated graphically along with their physical descriptions. The graphical outputs reveal that the predator and prey population densities are changed with the change in the free parameters. Wave solutions to the fractional diffusive predator-prey system are also reported with Atangana conformable derivative sense. As the value of the fractional parameter increases, the smoothness of the anti-kink wave profile is found to increase gradually, but the steepness decreases. For the sinusoidal wave profile, the periodicity and smoothness increase as the value of the fractional parameter increases. Thus, the present study may enrich the interpretation of the spatiotemporal dynamics of predator-prey interaction in a real environment. It is also found that the applied method and the relevant transformation are effective and easy to use for acquiring new analytic solutions to the diffusive predator-prey system over the other analytic methods. The method of interest is novel and efficient because it overcomes the weaknesses and deficiencies of the other analytic methods. Therefore, the method can be applied to further studies to explain various physical phenomena arising in ecology.