Dynamical behavior in a harvested differential-algebraic prey-predator model with discrete time delay and stage structure. (English) Zbl 1185.49043

Liu, Chao; Zhang, Qingling; Duan, Xiaodong

J. Franklin Inst. 346, No. 10, 1038-1059 (2009).

Summary: A differential-algebraic model system which considers a prey-predator system with stage structure for prey and harvest effort on predator is proposed. By using the differential-algebraic system theory and bifurcation theory, the dynamic behaviors of the proposed model system with and without discrete time delay are investigated. Local stability analysis of the model system without discrete time delay reveals that there is a phenomenon of singularity induced bifurcation due to variation of the economic interest of harvesting, and a state feedback controller is designed to stabilize the proposed model system at the interior equilibrium; on the other hand, the local stability of the model system with discrete time delay is also studied. The theoretical analysis shows that the discrete time delay has a destabilizing effect in the model of population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay exceeds a certain threshold. Numerical simulations are carried out to show the consistency with theoretical analysis.

MSC:

49N75	Pursuit and evasion games
34H05	Control problems involving ordinary differential equations
93D99	Stability of control systems
91A24	Positional games (pursuit and evasion, etc.)

Keywords:

differential-algebraic system; stage structure; discrete time delay; harvesting; bifurcation

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