Complex Dynamics in a General Diffusive Predator–Prey Model with Predator Maturation Delay

We formulate and analyze a general diffusive predator–prey system with predator maturation delay. Global asymptotic stability of the predator-free equilibrium and uniform persistence results are obtained under different conditions on model parameters. We then use Leray–Schauder degree theory to establish the existence of the spatial heterogeneous steady state. Moreover, we prove the global existence of nonconstant positive steady states bifurcated from the positive constant steady state. Taking the time delay as the bifurcation parameter, we conduct local and global Hopf bifurcation analysis and prove the boundedness of global Hopf branches. Rigorous analyses for global Hopf bifurcation and branches are challenging but important in understanding global transitions of dynamics.