A delayed predation model with cannibalism and Holling-II functional responses

This paper considers the gestation delay of the predator population and, based on the idea that cannibalism leads to an increased spawning rate of mature species, establishes a delayed predation model with stage structure and cannibalism. First, the positive invariant set of the system is provided, and the existence and local asymptotic stability of the equilibrium point of the system without delay are analyzed within the positive invariant set. Second, the cannibalism rate is used as the bifurcation parameter to analyze the Hopf bifurcation occurring at the positive equilibrium point of the system without delay. Afterward, the conditions for the occurrence of Hopf bifurcation at the unique positive equilibrium point of the system with delay are determined, and the relevant parameters that determine the bifurcation properties are obtained using the center manifold theorem and the normal form theory. Finally, numerical simulations are conducted to verify the reliability of the theoretical results.