Turing–Hopf bifurcation analysis and normal form derivation in a diffusive Predator–Prey model with time delay and indirect prey-taxis

In this paper, we consider a diffusive Predator–Prey system with hunting delay and indirect prey-taxis under homogeneous Neumann boundary conditions. Due to the presence of indirect prey-taxis, the system is described by three partial differential equations, which complicates the analysis. First, we analyze the corresponding characteristic equation of the linearized system at the positive constant steady state to obtain the conditions for Turing bifurcation, Hopf bifurcation and Turing–Hopf bifurcation. Next, to understand and classify the spatiotemporal dynamics near the Turing–Hopf bifurcation point, we derive an algorithm for calculating the normal form of the Turing–Hopf bifurcation for this system. This allows us to analytically determine the dynamical classifications near the Turing–Hopf bifurcation point using the obtained third-order truncated normal form. Finally, we conduct numerical simulations to illustrate the theoretical results. These simulations reveal the existence of a stable positive constant steady state, stable spatially homogeneous periodic solution and the transition from an unstable spatially inhomogeneous periodic solution to a stable spatially homogeneous periodic solution.