Inhomogeneous Hopf bifurcation in a diffusive predator-prey system with indirect predator-taxis

In this paper, our focus is on the study of Hopf bifurcation in a diffusive predator–prey system that incorporates indirect predator-taxis. We commence by examining the stability of the unique positive constant steady state and the occurrence of Hopf bifurcation within this system. Following this, we utilize the center manifold theorem and the normal form theory to devise an algorithm for calculating the normal form of the Hopf bifurcation in this system. This algorithm enables us to determine the direction and stability of the resulting periodic solution from the Hopf bifurcation. Through numerical simulations, we demonstrate the effectiveness of our algorithm, thereby confirming the validity of our theoretical analysis. Additionally, we observe the emergence of stable spatially inhomogeneous periodic solutions resulting from the Hopf bifurcation.