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

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.

MSC:
34K60Qualitative investigation and simulation of models
34K18Bifurcation theory of functional differential equations
34K13Periodic solutions of functional differential equations
34K21Stationary solutions of functional-differential equations
34K19Invariant manifolds (functional-differential equations)
34K17Transformation and reduction of functional-differential equations and systems; normal forms
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