# zbMATH — the first resource for mathematics

Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state dependent impulsive effects. (English) Zbl 1162.34007
An impulsive system of Lotka-Volterra-predator-prey type, according to biological and chemical control strategy for pest is constructed
\begin{aligned} & \left.\begin{aligned} & \frac{dx}{dt}=x(t)[b_1-a_{11} x(t)-a_{12} y(t)]\\ & \frac{dy}{dt}=y(t)[-b_2+a_{21}x(t)]\end{aligned}\right\}\;x\neq h_1h_2,\\ & \left.\begin{aligned} & \Delta x(t)=0\\ & \Delta y(t)=y(t^+)-y(t)=\alpha\end{aligned}\right\}\quad x=h_1\\ & \left.\begin{aligned} & \Delta x(t)=x(t^+)-x(t)=-px(t)\\ & \Delta y(t)=y(t^+)-y(t)=-qy(t)\end{aligned}\right\}\quad x=h_2\end{aligned}\tag{1}
Sufficient conditions for the existence of
– stable semi-trivial solution of (1)
– order-1 periodic solution of (1)
– positive locally orbitally stable solution of (1)
– positive order-1 periodic solution
are founded.

##### MSC:
 34A37 Ordinary differential equations with impulses 34C25 Periodic solutions to ordinary differential equations 92D25 Population dynamics (general)
Full Text:
##### References:
 [1] Lakshmikantham, V.; Bainov, D.D.; Simeonov, P.S., Theory of impulsive differential equations, (1989), World Scientific Singapore · Zbl 0719.34002 [2] Saito, Y., Permanence and global stability for general lotka – volterra predator – prey with distributed delays, Nonlinear anal., 47, 6157-6168, (2001) · Zbl 1042.34581 [3] Takeuchi, Y., Global dynamical properties of lotka – volterra systems, (1996), World Scientific Singapore · Zbl 0844.34006 [4] Bainov, D.D.; Simeonov, P.S., Impulsive differential equations: periodic solutions and applications, vol. 66, (1993), Longman · Zbl 0793.34011 [5] Tang, S.; Cheke, R.A., State-dependent impulsive models of integrated pest mangagement (IPM) strategies and their dynamic consequences, J. math. biol., 50, 257-292, (2005) · Zbl 1080.92067 [6] Corless, R.M.; Gonnet, G.H.; Hare, D.E.G.; Jeffrey, D.J.; Knuth, D.E., On the Lambert $$W$$ function, Adv. comput. math., 5, 329-359, (1996) · Zbl 0863.65008 [7] Waldvogel, J., The period in the Volterra-Lotka predator – prey modle, SIAM J. numer. anal., 20, 1264-1272, (1983) · Zbl 0533.65051 [8] C. Carathéodory, Theory of Function of a Complex Variable, Chelsea, 1954 [9] Stamova, I.M.; Stamov, G.T., Lyapunov – razumikhin method for impulsive functional differential equations and applications to the population dynamics, J. comput. appl. math., 130, 163-171, (2001) · Zbl 1022.34070 [10] Tang, S.; Chen, L., Density-dependent birth rate, birth pulses and their population dynamic consequences, J. math. biol., 44, 185-199, (2002) · Zbl 0990.92033 [11] Jiang, G.; Lu, Q., Impulsive state feedback control of a predator – prey model, J. comput. appl. math., 200, 193-207, (2007) · Zbl 1134.49024 [12] Redheffer, R., Lotka – volterra systems with constant interaction coefficients, Nonlinear anal., 46, 1151-1164, (2001) · Zbl 1003.34039 [13] Zeng, G.; Chen, L., Existence of periodic solution of order one of planar impulsive autonomous system, J. comput. appl. math., 186, 466-481, (2006) · Zbl 1088.34040 [14] Jiang, G.; Lu, Q., Complex dynamics of a Holling type II prey – predator system with state feedback control, Chaos, sol. fractal., 31, 448-461, (2007) · Zbl 1203.34071 [15] Gopalsamy, K., Stability and oscillations in delay different equations of population dynamics, (1992), Kluwer Academic Dordrecht, Norwell, MA · Zbl 0752.34039 [16] D’Onofrio, A., Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. comput. modelling, 36, 473-489, (2002) · Zbl 1025.92011 [17] Kuang, Y., Delay differential equations, with applications in population dynamics, (1993), Academic Press NewYork · Zbl 0777.34002 [18] Simeonov, P.S.; Bainov, D.D., Orbital stability of periodic solutions of autonomous systems with impulse effect, Int. J. syst. SCI., 19, 2561-2585, (1988) · Zbl 0669.34044 [19] D’Onofrio, A., Stability properties of pulse vaccination strategy in SEIR epidemic model, Math. biosci., 179, 57-72, (2002) · Zbl 0991.92025 [20] Ballinger, G.; Liu, X., Permanence of population growth models with impulsive effects, Math. comput. modelling, 26, 59-72, (1997) · Zbl 1185.34014 [21] Liu, X.Z.; Rohlf, K., Impulsive control of lotka – volterra system, IMA J. math. contr. inform., 15, 269-284, (1998) · Zbl 0949.93069 [22] Berezansky, L.; Braverman, E., Linearized oscillation theory for nonlinear delay impulsive equation, J. comput. appl. math., 161, 477-495, (2003) · Zbl 1045.34039 [23] Wang, F.; Pang, G.; Chen, L., Qualitative analysis and applications of a kind of state-dependent impulsive differential equations, J. comput. appl. math., (2007) [24] Hirstova, S.G.; Bainov, D.D., Existence of periodic solutions of nonlinear systems of differential equations with impulsive effect, J. math. annl. appl., 125, 192-202, (1985) [25] Liu, X., Stability results for impulsive differential systems with application to population growth models, Dyn. stab. syst., 9, 163-174, (1994) · Zbl 0808.34056 [26] Liu, B.; Teng, Z.; Chen, L., Analysis of a predator – prey model with Holling II functional response concerning impulsive control strategy, J. comput. appl. math., 193, 347-362, (2006) · Zbl 1089.92060 [27] Gao, S.; Chen, L.; Teng, Z., Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull math biol., (2006) [28] 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 [29] Zhang, T.; Teng, Z., Extinction and permanence for a pulse vaccination delayed SEIRS epidemic model, Chaos, soliton. fractal, (2007) [30] Ruan, S.; Xiao, D., Global analysis in a predator – prey system with nonmonotonic functional response, SIAM J. appl. math., 61, 1445-1472, (2001) · Zbl 0986.34045 [31] Wang, W.; Shen, J.; Nieto, J.J., Permanence and periodic solution of predator – prey system with Holling type functional response and impulses, Discrete dyn. nat. soc., (2007) · Zbl 1146.37370 [32] Gao, S.; Teng, Z.; Nieto, J.J.; Torres, A., Analysis of an SIR epidemic model with pulse vaccination and distributed time delay, J. biomed. biotechnology, 2007, (2007), Article ID 64870 [33] Ahmad, S.; Stamova, I.M., Asymptotic stability of competitive systems with delays and impulsive perturbations, J. math. anal. appl., (2007) · Zbl 1153.34044 [34] Lenteren, J.C.V., Integrated pest management in protected crops, (), 311-320
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.