Complex pattern formation arising from wave instabilities in a three-agent chemical system with superdiffusion

This work investigates analytically and numerically the wave instabilities in a three-agent chemical system undergoing anomalous diffusion. Anomalous diffusion is modeled using higher-dimensional space-fractional operators. The model considers one activator and two inhibitors, and it is a superdiffusive extension of some reaction-diffusion systems in the literature (Phys D Nonlinear Phenom 199(1–2):264–277, 2004). The model under investigation is presented in generic form as a three-species reaction-diffusion system, so that it has broad applicability to a wide range of chemical and biological systems. A weakly nonlinear analysis is performed, dispersive curves of eigenvalues are plotted and their behavior is analyzed. This analysis reveals that the critical value of the wave number for wave instability increases when the superdiffusive exponent decreases. The numerical scheme is performed and some numerical simulations are conducted as evidence of the analytical predictions. We show how the evolution of spatio-temporal patterns is related to meaningful parameters. In particular, we demonstrate that the system exhibits the coexistence of regular and irregular structures, forming complex patterns that are not familiar in standard reaction-diffusion systems.