Qualitative analysis and Hopf bifurcation of a generalized Lengyel–Epstein model

In this present paper, we deal with a generalized Lengyel–Epstein model with the zero-flux boundary conditions. Firstly, we give an attraction region and the boundedness estimates of the solutions to the parabolic equations. Hereafter, one performs the local and global stability of the unique positive equilibrium. The first Lyapunov exponent technique and the normal form theory are employed to investigate the directions of the Hopf bifurcation, respectively. It is found that the supercritical or the subcritical may occur in the generalized Lengyel–Epstein model. Finally, we explore the steady states of the elliptic equations. The boundedness, the nonexistence, and the existence of the steady states are performed. Numerical experiments well verify the theoretical analysis. Relevant theoretical results illustrate that the diffusion rates of the substance can affect the dynamical behaviors of such a generalized Lengyel–Epstein model.