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Existence of positive periodic solution of periodic time-dependent predator-prey system with impulsive effects. (English) Zbl 1058.34051

The following impulsive problem is considered

N ˙ 1 =N 1 (b 1 -c 11 N 1 -c 12 N 2 ),N ˙ 2 =N 2 (-b 2 +c 21 N 1 -c 22 N 2 ),ΔN 1 =c k N 1 ,ΔN 2 =d k N 2 ,t=τ k ,

where b i (t),c ij (t) are continuous, T-periodic positive functions,

c k+q =c k ,d k+q =d k ,τ k+q =τ k +T,1+c k >0,1+d k >0·

Under some additional assumptions, using bifurcation theory, the authors prove the existence of a positive periodic solution for this system and discuss some biological applications.

MSC:
34C25Periodic solutions of ODE
34A37Differential equations with impulses
92D25Population dynamics (general)
34C23Bifurcation (ODE)
References:
[1]Amine, Z., Ortega, R.: A periodic prey-predator system. J. Math. Anal. Appl., 185, 477–489 (1994) · Zbl 0808.34043 · doi:10.1006/jmaa.1994.1262
[2]López-Gómez, J., Ortega, R., Tineo, A.: The periodic predator-prey Lotka–Volterra model. Advances in Differential Equations, 1(3), 403–423 (1996)
[3]Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems. J. Functional Analysis, 7, 487–513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[4]Crandall, M. G., Rabinowitz, P. H.: Bifurcation from simple eigenvalues. J. Funct. Anal., 8, 321–340 (1971) · Zbl 0219.46015 · doi:10.1016/0022-1236(71)90015-2
[5]Crandall, M. G., Rabinowitz, P. H.: Bifurcation, perturbation of simple eigenvalues, and linearized stability. Arch. Rat. Mech. Anal., 52, 161–180 (1973) · Zbl 0275.47044 · doi:10.1007/BF00282325
[6]Bainov, D., Simeonov, P.: Impulsive differential equations: Periodic solution and applications. Longman, England, (1993)
[7]Shulgin, B., Stone, L. et al.: Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol., 60, 1–26 (1998) · Zbl 0941.92026 · doi:10.1016/S0092-8240(98)90005-2
[8]Lakmeche, A., Arino, O.: Bifurcation of nontrivial periodic solution of impulsive differential equations arising chemotherapeutic treatment. Dynamics of Continuous, Discrete and Impulsive System, 7, 265–287 (2000)
[9]Liu, X.: Impulsive stabilization and applications to population growth models. J. Math., 25(1), 381–395 (1995)
[10]Liu, X., Zhang, S.: A cell population model described by impulsive PDEs-existence and numerical approximation. Comput. Math. Appl., 36(8), 1–11 (1998) · Zbl 0962.35181 · doi:10.1016/S0898-1221(98)00178-3
[11]Liu, X., Rohof, K.: Impulsive control of a Lotka–Volterra system. IMA Journal of Mathematical Control & Information, 15, 269–284 (1998) · Zbl 0949.93069 · doi:10.1093/imamci/15.3.269
[12]Funasaki, E., Kot, M.: Invasion and chaos in a periodically pulsed Mass-Action Chemostat. Theoretical Population Biology, 44, 203–224 (1993) · Zbl 0782.92020 · doi:10.1006/tpbi.1993.1026
[13]Halanay, A.: Differential equations: stability, oscillations, time lags, Academic Press, New York, 1996
[14]Krasnosel’Skii, M. A.: Topological methods in the theory of nonlinear integral equations, Macmillan, New York, 1964
[15]Vainberg, M. V., Trenogrn, V. A.: The methods of Lyapunov and Schmidt in the theorey of nonlinear equations and their further development. Russian Math. Surveys, 17(2), 1–60 (1962) · doi:10.1070/RM1962v017n02ABEH001127