Turing–Turing Bifurcation and Normal Form in a Predator–Prey Model with Predator-Taxis and Prey Refuge

This paper investigates a predator–prey reaction–diffusion model incorporating predator-taxis and a prey refuge mechanism, subject to homogeneous Neumann boundary conditions. Our primary focus is the analysis of codimension-2 Turing–Turing bifurcation and the calculation of its associated normal form for this model. First, employing the maximum principle and Amann’s theorem, we rigorously prove the local existence and uniqueness of classical solutions. Second, utilizing linear stability theory and bifurcation theory, we conduct a thorough analysis of the existence and stability properties of the positive constant steady-state. Furthermore, we derive precise conditions under which the model undergoes a Turing–Turing bifurcation. Third, by applying center manifold reduction and normal form theory, we derive the method for calculating the third-truncated normal form characterizing the dynamics near the Turing–Turing bifurcation point. Finally, we present numerical simulations to validate the theoretical findings, confirming the correctness of the analytical results concerning the bifurcation conditions and the derived normal form.