## Dynamical behavior in a stage-structured differential-algebraic prey-predator model with discrete time delay and harvesting.(English)Zbl 1176.34101

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, dynamic behavior of the proposed model system with and without discrete time delay is 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; Furthermore, local stability of the model system with discrete time delay is studied. It reveals that the discrete time delay has a destabilizing effect in the population dynamics, and a phenomenon of Hopf bifurcation occurs as the discrete time delay increases through a certain threshold. Finally, numerical simulations are carried out to show the consistency with theoretical analysis obtained in this paper.

### MSC:

 34K60 Qualitative investigation and simulation of models involving functional-differential equations 92D25 Population dynamics (general) 34K20 Stability theory of functional-differential equations 34K18 Bifurcation theory of functional-differential equations
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### References:

 [1] Arino, O.; Sanchez, E.; Fathallah, A., State-dependent delay differential equations in population dynamics: Modeling and analysis, (Fields Institute Communications, vol. 29 (2001), American Mathematical Society: American Mathematical Society Providence, RI), 19-36 · Zbl 1002.34073 [2] Xu, R.; Chaplin, M. A.; Davidson, F. A., Persistence and stability of a stage-structured predator-prey model with time delays, Applied Mathematics and Computation, 150, 259-277 (2004) · Zbl 1064.92049 [3] Gourley, S. A.; Kuang, Y., A stage structured predator-prey model and its dependence on maturation delay and death rate, Journal of Mathematical Biology, 49, 188-200 (2004) · Zbl 1055.92043 [4] Zhang, X.; Chen, L.; Neumann, U. A., The stage-structured predator-prey model and optimal harvesting policy, Mathematical Biosciences, 168, 201-210 (2000) · Zbl 0961.92037 [5] Bandyopadhyay, M.; Banerjee, S., A stage-structured prey-predator model with discrete time delay, Applied Mathematics and Computation, 182, 1385-1398 (2006) · Zbl 1102.92044 [6] Myerscough, M. R.; Gray, B. F.; Hogarth, W. L.; Norbury, J., An analysis of an ordinary differential equations model for a two species predator-prey system with harvesting and stocking, Journal of Mathematical Biology, 30, 389-411 (1992) · Zbl 0749.92022 [7] Song, X.; Chen, L., Optimal harvesting and stability for a two-species competitive system with stage structure, Mathematical Biosciences, 170, 173-186 (2001) · Zbl 1028.34049 [8] Song, X.; Chen, L., Optimal harvesting and stability for a predator-prey system with stage structure, Acta Mathematica Applicate Sinicia, English Series, 18, 3, 423-430 (2002) · Zbl 1054.34125 [9] Kar, T. K., Selective harvesting in a prey-predator fishery with time delay, Mathematical and Computer Modelling, 38, 449-458 (2003) · Zbl 1045.92046 [10] Kar, T. K.; Pahari, U. K., Modelling and analysis of a prey-predator system with stage-structure and harvesting, Nonlinear Analysis: Real World Applications, 8, 601-609 (2007) · Zbl 1152.34374 [11] Gordon, H. S., The economic theory of a common property resource: The fishery, Journal of Political Economy, 62, 2, 124-142 (1954) [12] Zhang, Y.; Zhang, Q. L., Chaotic control based on descriptor bioeconomic systems, Control and Decision, 22, 4, 445-452 (2007) [13] Zhang, Y.; Zhang, Q. L.; Zhao, L. C., Bifurcations and control in singular biological economical model with stage structure, Journal of Systems Engineering, 22, 3, 232-238 (2007) [14] Zhang, X., Bifurcations of a class of singular biological economic models, Chaos, Solitons and Fractals (2007) [15] Dai, L., Singular Control System (1989), Springer: Springer New York [16] Marszalek, W.; Trzaska, Z. W., Singularity-induced bifurcations in electrical power system, IEEE Transactions on Power Systems, 20, 1, 302-310 (2005) [17] Ayasun, S.; Nwankpa, C. O.; Kwatny, H. G., Computation of singular and singularity induced bifurcation points of differential-algebraic power system model, IEEE Transactions on Circuits System. I, 51, 8, 1525-1537 (2004) · Zbl 1374.34027 [18] Yue, M.; Schlueter, R., Bifurcation subsystem and its application in power system analysis, IEEE Transaction on Power System, 19, 4, 1885-1893 (2004) [19] Venkatasubramanian, V.; Schaettler, H.; Zaborszky, J., Local bifurcations and feasibility regions in differential-algebraic systems, IEEE Transactions on Automatic Control, 40, 12, 1992-2013 (1995) · Zbl 0843.34045 [20] Venkatasubramanian, V., Singularity induced bifurcation and the van den Pol oscillator, IEEE Transactions on Circuits System. I, 41, 765-769 (1994) · Zbl 0872.34037 [21] Beardmore, R. E., The singularity-induced bifurcation and its kronecker normal form, SIAM Journal of Matrix Analysis and Application, 23, 1, 126-137 (2001) · Zbl 1002.34028 [22] Yang, L. J.; Tang, Y., An improved version of the singularity induced-bifurcation theorem, IEEE Transactions on Automatic Control, 49, 6, 1483-1486 (2001) · Zbl 1031.93097 [23] Kot, M., Elements of Mathematical Biology (2001), Cambridge University Press: Cambridge University Press Cambridge [24] Freedman, H.; Rao, V. S.H., The trade-off between mutual interference and time lags in predator-prey systems, Bulletin of Mathematical Biology, 45, 991-1004 (1983) · Zbl 0535.92024 [25] Hale, J. K., Theory of Functional Differential Equations (1997), Springer: Springer New York · Zbl 0189.39904 [26] Clark, C. W., Mathematical Bioeconomics: The Optimal Management of Renewable Resource (1990), John Wiley and Sons: John Wiley and Sons New York · Zbl 0712.90018
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