Turing-Hopf bifurcation analysis and normal form in delayed diffusive predator–prey system with taxis and fear effect

In this paper, we investigate a general delayed diffusive predator–prey system with taxis and fear effect, which can have different functional response functions. By selecting the taxis coefficient and the discrete time delay as bifurcation parameters, we first derive an algorithm for calculating the third-order truncated normal form of Turing-Hopf bifurcation for this system. To demonstrate the effectiveness of the derived algorithm, we investigate a delayed diffusive predator–prey system with taxis, fear effect, and square root functional response function. We employ stability theory and bifurcation theory to study the existence of codimension-two Turing-Hopf bifurcation and obtain the third-order truncated normal form of Turing-Hopf bifurcation using the derived algorithm. With the obtained third-order truncated normal form of Turing-Hopf bifurcation for this practical system, we can analytically determine the dynamical classifications near the Turing-Hopf bifurcation point. Finally, we perform numerical simulations to verify the theoretical analysis results.