Complex coexistence induced by delays in a diffusive intraguild predation system with stage structure and spatial memory

We study Hopf bifurcations induced by spatial memory and stage structure in a diffusive intraguild predation system, aiming to reveal their effects on the spatiotemporal dynamics of three interacting species. First, by applying the geometric approach, we obtain the stability criteria and Hopf bifurcation conditions through an analysis of the linearized system with delay-dependent coefficients. Next, we develop a computational algorithm for the normal form of Hopf bifurcation induced by maturation delay in a system with three state variables and diffusion terms containing a delay. Finally, we combine numerical simulations to demonstrate that variations in periods of maturation and spatial memory can generate rich and complex coexistence dynamics, including spatially homogeneous periodic solutions, spatially heterogeneous periodic solutions with mode $-5$, and multiple stability switches.