Complex dynamics and stability analysis in a discrete hybrid bioeconomic system with double time delays

In this paper, a discrete hybrid three-species food chain system is proposed, where commercial harvesting on top predator is considered. Two time delays are introduced to represent gestation delay for prey and predator population, respectively. In absence of time delay, sufficient conditions associated with economic interest and step size are derived to show system undergoes flip bifurcation. In presence of double time delays, existence of Neimark-Sacker bifurcation and local stability switch are discussed due to variations of time delays. Furthermore, by utilizing new normal form of delayed discrete hybrid system and center manifold theorem, direction and stability of Neimark-Sacker bifurcation are studied. Numerical simulations are performed not only to validate theoretical analysis, but also exhibit cascades of period-doubling bifurcation, chaotic behavior and stable closed invariant curve.