In this work, we investigate both the mathematical and numerical studies of the fractional reaction–diffusion system consisting of spatial interactions of three components’ species. Our main result is based on the analysis of the model for linear stability. Mathematical analysis of the main equation shows that the dynamical system is both locally and globally asymptotically stable. We further propose a theorem which guarantees the existence and permanence of the three species. We formulate a viable numerical methods in space and time. By adopting the Fourier spectral approach to discretize in space, the issue of stiffness associated with the fractional-order spatial derivatives in such system is removed. The resulting system of ordinary differential equations (ODEs) is advanced with the exponential time-differencing method of ADAMS-type. The complexity of the dynamics in the system which we discussed theoretically are numerically presented through some numerical simulations in 1D, 2D, and 3D to address the points and queries that may naturally arise.