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Zhang, Guodong; Zhu, Lulu; Chen, Boshan

Hopf bifurcation in a delayed differential-algebraic biological economic system. (English) Zbl 1221.34227

Nonlinear Anal., Real World Appl. 12, No. 3, 1708-1719 (2011).
This paper deals with a predator-prey differential system incorporating harvesting effort on the prey and a constant time delay which represents the reaction time of the preys. The model is completed with an algebraic equation which incorporates the economic effect of harvesting, according to the economic theory proposed by Gordon in 1954.
The main goal of the paper is to analyze the behaviour of the positive steady-state of this differential-algebraic system in terms of the delay considered as a varying parameter. Using the general theory of normal forms and the center manifold theory, the existence of a Hopf bifurcation is analyzed. The paper includes some numerical simulations to illustrate the analytical findings.
Reviewer: Eva Sanchez (Madrid)

Cited in 9 Documents

MSC:

34K60 Qualitative investigation and simulation of models involving functional-differential equations
34K18 Bifurcation theory of functional-differential equations
34K13 Periodic solutions to functional-differential equations
34K21 Stationary solutions of functional-differential equations
34K19 Invariant manifolds of functional-differential equations
34K17 Transformation and reduction of functional-differential equations and systems, normal forms

Keywords:

delayed differential-algebraic system; Hopf bifurcation; predator-prey economic system
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\textit{G. Zhang} et al., Nonlinear Anal., Real World Appl. 12, No. 3, 1708--1719 (2011; Zbl 1221.34227)
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References:

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