Normal form of Turing–Turing bifurcation for the diffusive Bazykin system with prey-taxis

In this paper, we introduce prey-taxis to the diffusive Bazykin system and study the codimension-two Turing–Turing bifurcation of this modified system. For the local system, i.e. without diffusion terms and the prey-taxis term, we investigate the stability of the unique positive equilibrium. Then, for the diffusive Bazykin system with prey-taxis, we examine the stability of the unique positive constant steady state, Turing instability, and Turing bifurcation. Moreover, by choosing the random diffusion coefficients as bifurcation parameters, we investigate the Turing–Turing bifurcation of the diffusive Bazykin system with prey-taxis. Furthermore, to demonstrate and classify the spatiotemporal dynamics near the Turing–Turing bifurcation point, we derive an algorithm for calculating the third-order truncated normal form of this bifurcation using the center manifold theorem and normal form theory. Finally, to verify the theoretical analysis results and demonstrate the effectiveness of the derived algorithm, we perform several numerical simulations using matlab software. We also investigate the influence of the prey-taxis coefficient on the Turing instability of the diffusive Bazykin system with prey-taxis.