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Impulsive state feedback control of a predator-prey model. (English) Zbl 1134.49024
Summary: The dynamics of a predator--prey model with impulsive state feedback control, which is described by an autonomous system with impulses, is studied. The sufficient conditions of existence and stability of semi-trivial solution and positive period-1 solution are obtained by using the Poincaré map and analogue of the Poincaré criterion. The qualitative analysis shows that the positive period-1 solution bifurcates from the semi-trivial solution through a fold bifurcation. The bifurcation diagrams of periodic solutions are obtained by using the Poincaré map, and it is shown that a chaotic solution is generated via a cascade of period-doubling bifurcations.

49N75Pursuit and evasion games in calculus of variations
Full Text: DOI
[1] D.D. Bainov, P.S. Simeonov, Impulsive differential equations: periodic solutions and applications, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 66, Longman Scientific, New York, 1993. · Zbl 0815.34001
[2] Bainov, D. D.; Dishliev, A. B.; Stamova, I. M.: Lipschitz quasistability of impulsive differential -- difference equations with variable impulsive perturbations. J. comput. Appl. math. 70, 267-277 (1996) · Zbl 0854.34073
[3] Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects. Math. comput. Modelling 26, 59-72 (1997) · Zbl 1185.34014
[4] Berezansky, L.; Braverman, E.: Linearized oscillation theory for a nonlinear delay impulsive equation. J. comput. Appl. math. 161, 477-495 (2003) · Zbl 1045.34039
[5] D’onofrio, A.: Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. biosci. 179, 57-72 (2002) · Zbl 0991.92025
[6] J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences, vol. 42, Springer, New York, 1983. · Zbl 0515.34001
[7] Kamel, O. M.; Soliman, A. S.: On the optimization of the generalized coplanar hohmann impulsive transfer adopting energy change concept. Acta astronautica 56, 431-438 (2005)
[8] Y.A. Kuznetsov, Elements of applied bifurcation theory, Applied Mathematical Sciences, vol. 112, Springer, New York, 1995. · Zbl 0829.58029
[9] Lakmeche, A.; Arino, O.: Bifurcation of nontrival periodic solutions of impulsive differential equations arising chemotherapeutic treatment. Dynamics of continuous, discrete and impulsive system 7, 265-287 (2000) · Zbl 1011.34031
[10] Liu, X. Z.; Rohlf, K.: Impulsive control of a Lotka -- Volterra system. IMA J. Math. control inf. 15, 269-284 (1998) · Zbl 0949.93069
[11] Liu, X. N.; Chen, L. S.: Complex dynamics of Holling type $\Pi $ lotaka -- Volterra predator -- prey system with impulsive perturbations on the predator. Chaos, solitons and fractals 16, 311-320 (2003) · Zbl 1085.34529
[12] Lsksmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations. (1989)
[13] Lu, Z. H.; Chi, X. B.; Chen, L. S.: The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission. Math. comput. Modelling 36, 1039-1057 (2002) · Zbl 1023.92026
[14] Roberts, M. G.; Kao, R. R.: The dynamics of an infections disease in a population with birth pulses. Math. biosci. 149, 23-36 (1998) · Zbl 0928.92027
[15] Shulgin, B.; Stone, L.; Agur, Z.: Theoretical examination of pulse vaccination policy in the SIR epidemic model. Math. comput. Modelling 31, 207-215 (2000) · Zbl 1043.92527
[16] Simeonov, P. E.; Bainov, D. D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Internat. J. Systems sci. 19, 2562-2585 (1988) · Zbl 0669.34044
[17] Tang, S. Y.; Chen, L. S.: Multiple attractors in stage-structured population models with birth pulses. Bull. math. Biol. 65, 479-495 (2003)
[18] Tang, S. Y.; Chen, L. S.: Density-dependent birth rate, birth pulses and their population dynamic consequences. J. math. Biol. 44, 185-199 (2002) · Zbl 0990.92033
[19] Van Lenteren, J. C.: Integrated pest management in protected crops. Integrated pest management, 311-320 (1995)
[20] T. Yang, Impulsive Control Theory, Springer, Berlin Heidelberg, 2001, pp. 307 -- 333.