Mathematical analysis of the effect of cuckoo bird's incubation period in population dynamics

Although caring and raising children is the duty of all parents, some animals relieve themselves of this labored work by relying on others to raise their young. After the host has made a nest and laid eggs, the parasite locates the nest and lays a few eggs in the host's nest. Several possible interactions occur between the host and parasite in brood parasitism: the parasite does not destroy the host's eggs and the host does not abandon her nest with eggs. Here, we establish a mathematical framework for studying an ecological model on brood parasitism with stage structure of the parasite, where time delay represents the incubation period of the parasite. For the proposed model, we studied the impact of incubation time delay on the coexistence equilibrium. By using corresponding characteristic equations, the local stability behavior of each of the feasible equilibria of the model and existence of Hopf bifurcations at the coexistence equilibrium are established. Further, the global properties of some equilibria are established using Lyapunov-LaSalle invariance principle. The incubation time delay was found to be crucial for establishing stability behavior. When the incubation time delay increased and crossed some critical values, Hopf bifurcation may occur at the coexistence equilibrium. Further, by using normal form and center manifold theories, formulas were established to determine the stability and direction of the Hopf bifurcation. Numerical simulations were conducted to support the theoretical analysis. Our mathematical model could help lay a foundation for further studies on brood parasitism by the cuckoo bird.