zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
The periodic solutions for general periodic impulsive population systems of functional differential equations and its applications. (English) Zbl 1221.34181
Summary: The general periodic impulsive population systems of functional differential equations are investigated. By using the method of Poincaré map and Horn’s fixed point theorem, we prove that the ultimate boundedness of all solutions implies the existence of periodic solutions. As applications of this result, the existence of positive periodic solutions for the general periodic impulsive Kolmogorov-type population dynamical systems is discussed. We further prove that as long as the system is permanent, there must exist at least one positive periodic solution. In addition, the permanence and existence of positive periodic solutions are discussed for the periodic impulsive single-species logistic models and the periodic impulsive n-species Lotka–Volterra competitive models with delays.
34K13Periodic solutions of functional differential equations
34K45Functional-differential equations with impulses
92D25Population dynamics (general)
[1]Ahmad, S.; Stamova, I. M.: Asymptotic stability of an N-dimensional impulsive competitive system, Nonlinear anal. RWA 8, 654-663 (2007) · Zbl 1152.34342 · doi:10.1016/j.nonrwa.2006.02.004
[2]Ballinger, G.; Liu, X.: Permanence of population growth models with impulsive effects, Math. comput. Modelling 26, 59-72 (1997) · Zbl 1185.34014 · doi:10.1016/S0895-7177(97)00240-9
[3]Braverman, E.; Mamdani, R.: Continuous versus pulse harvesting for population models in constant and variable environment, J. math. Biol. 57, 413-434 (2008) · Zbl 1143.92327 · doi:10.1007/s00285-008-0169-z
[4]Zhao, Z.; Yang, L.; Chen, L.: Impulsive perturbations of a predator–prey system with modified Leslie-gower and Holling type II, J. appl. Math. comput. 35, 119-134 (2001) · Zbl 1222.34057 · doi:10.1007/s12190-009-0346-2
[5]Akhmet, M. U.; Beklioglu, M.; Ergenc, T.; Tkachenko, V. I.: An impulsive ratio-dependent predator–prey system with diffusion, Nonlinear anal. RWA 7, 1255-1267 (2006) · Zbl 1114.35097 · doi:10.1016/j.nonrwa.2005.11.007
[6]Georgescu, P.; Morosanu, G.: Pest regulation by means of impulsive controls, Appl. math. Comput. 190, 790-803 (2007) · Zbl 1117.93006 · doi:10.1016/j.amc.2007.01.079
[7]Georgescu, P.; Zhang, H.: An impulsive controlled predator-pest model with disease in the pest, Nonlinear anal. RWA 11, 270-287 (2010) · Zbl 1192.34057 · doi:10.1016/j.nonrwa.2008.10.060
[8]X. Wang, Y. Tao, X. Song, Analysis of pest-epidemic model by releasing diseased pest with impulsive transmission, Nonlinear Dyn., (in press) (doi:10.1007/s11071-010-9882-4).
[9]Kocherha, O. I.; Nenya, O. I.; Tkachenko, V. I.: On positive periodic solutions of nonlinear impulsive functional differential equations, Nonlinear oscil. 11, 527-536 (2008)
[10]Li, Y.: Positive periodic solutions of nonlinear differential systems with impulses, Nonlinear anal. 68, 2389-2405 (2008) · Zbl 1162.34064 · doi:10.1016/j.na.2007.01.066
[11]Li, X.; Zhang, X.; Jiang, D.: A new existence theory for positive periodic solutions to functional differential equations with impulsive effects, Comput. math. Appl. 51, 1761-1772 (2006) · Zbl 1156.34053 · doi:10.1016/j.camwa.2006.02.007
[12]Liu, K.; Meng, X.; Chen, L.: A new stage structured predator–prey gomportz model with time delay and impulsive perturbations on the prey, Appl. math. Comput. 196, 705-719 (2008) · Zbl 1131.92064 · doi:10.1016/j.amc.2007.07.020
[13]Liu, X.; Chen, L.: Global attractivity of positive periodic solutions for nonlinear impulsive systems, Nonlinear anal. 65, 1843-1857 (2006) · Zbl 1111.34010 · doi:10.1016/j.na.2005.10.041
[14]Yan, J.: Existence of positive periodic solutions of impulsive functional differential equations with two parameters, J. math. Anal. appl. 327, 854-868 (2007) · Zbl 1114.34052 · doi:10.1016/j.jmaa.2006.04.018
[15]Zeng, Z.: Existence and multiplicity of positive periodic solutions for a class of higher-dimension functional differential equations with impulses, Comput. math. Appl. 58, 1911-1920 (2009) · Zbl 1189.34141 · doi:10.1016/j.camwa.2009.07.086
[16]Zhang, N.; Dai, B.; Qian, X.: Periodic solutions for a class of higher-dimension functional differential equations with impulses, Nonlinear anal. 68, 629-638 (2008) · Zbl 1134.34045 · doi:10.1016/j.na.2006.11.024
[17]Zhang, S.; Tan, D.: Permanence in a food chain system with impulsive perturbations, Chaos solitons fractals 40, 392-400 (2009) · Zbl 1197.34013 · doi:10.1016/j.chaos.2007.07.074
[18]Zhao, Z.; Chen, L.: Dynamic analysis of lactic acid fermentation with impulsive input, J. math. Chem. 47, 1189-1208 (2010) · Zbl 1190.92010 · doi:10.1007/s10910-008-9494-0
[19]Shen, J.; Li, J.; Wang, Q.: Boundedness and periodicity in impulsive ordinary and functional differential equations, Nonlinear anal. 65, 1986-2002 (2006) · Zbl 1120.34005 · doi:10.1016/j.na.2005.11.006
[20]Baek, H.: Dynamic complexities of a three-species beddington–deangelis system with impulsive control strategy, Acta appl. Math. 110, 23-38 (2010) · Zbl 1194.34087 · doi:10.1007/s10440-008-9378-0
[21]Baek, H.: Species extinction and permanence of an impulsively controlled two-prey one-predator system with seasonal effects, Biosystems 98, 7-18 (2009)
[22]Bunimovich-Mendrazitsky, S.; Byrne, H.; Stone, L.: Mathematical model of pulsed immunotherapy for superficial bladder cancer, Bull. math. Biol. 70, 2055-2076 (2008) · Zbl 1147.92013 · doi:10.1007/s11538-008-9344-z
[23]He, M.; Chen, F.: Dynamic behaviors of the impulsive periodic multi-species predator–prey system, Comput. math. Appl. 57, 248-265 (2009) · Zbl 1165.34308 · doi:10.1016/j.camwa.2008.09.041
[24]Hou, J.; Teng, Z.; Gao, S.: Permanence and global stability for nonautonomous N-species Lotka–valterra competitive system with impulses, Nonlinear anal. RWA 11, 1882-1896 (2010) · Zbl 1200.34051 · doi:10.1016/j.nonrwa.2009.04.012
[25]Liu, Z.; Hui, J.; Wu, J.: Permanence and partial extinction in an impulsive delay competitive system with the effect of toxic substances, J. math. Chem. 46, 1213-1231 (2009) · Zbl 1197.92046 · doi:10.1007/s10910-008-9513-1
[26]Meng, X.; Chen, L.; Li, Q.: The dynamics of an impulsive delay predator–prey model with variable coefficients, Appl. math. Comput. 198, 361-374 (2008) · Zbl 1133.92029 · doi:10.1016/j.amc.2007.08.075
[27]Pei, Y.; Zeng, G.; Chen, L.: Species extinction and permanence in a prey–predator model with two-type functional responses and impulsive biological control, Nonlinear dyn. 52, 71-81 (2008) · Zbl 1169.92048 · doi:10.1007/s11071-007-9258-6
[28]Pei, Y.; Liu, S.; Li, C.: Complex dynamics of an impulsive control system in which predator species share a common prey, J. nonlinear sci. 19, 249-266 (2009) · Zbl 1172.92039 · doi:10.1007/s00332-008-9034-x
[29]Simons, R. R. L.; Gourley, S. A.: Extinction criteria in stage-structured population models with impulsive culling, SIAM J. Appl. math. 66, 1853-1870 (2006) · Zbl 1120.34064 · doi:10.1137/050637777
[30]Song, X.; Hao, M.; Meng, X.: A stage-structured predator–prey model with disturbing pulse and time delays, Appl. math. Model. 33, 211-223 (2009) · Zbl 1167.34372 · doi:10.1016/j.apm.2007.10.020
[31]Song, X.; Xiang, Z.: The prey-dependent consumption two-prey one-predator models with stage structure for the predator and impulsive effects, J. theoret. Biol. 242, 683-698 (2006)
[32]Stamova, I. M.: Impulsive control for stability of n-species Lotka–Volterra cooperation models with finite delays, Appl. math. Lett. 23, 1003-1007 (2010) · Zbl 1200.34103 · doi:10.1016/j.aml.2010.04.026
[33]Terry, A. J.: Impulsive culling of a structured population on two patches, J. math. Biol. 61, 843-875 (2010) · Zbl 1205.92078 · doi:10.1007/s00285-009-0325-0
[34]Wang, X.; Wang, W.; Lin, X.: Dynamics of a periodic watt-type predator–prey system with impulsive effect, Chaos solitons fractals 39, 1270-1282 (2009) · Zbl 1197.34064 · doi:10.1016/j.chaos.2007.06.031
[35]Zhang, H.; Georgescu, P.; Chen, L.: On the impulsive controllability and bifurcation of a predator-pest model of IPM, Biosystems 93, 151-171 (2008)
[36]Zhang, L.; Teng, Z.; Jiang, H.: Permanence for general nonautonomous impulsive population systems of functional differential equations and its applications, Acta appl. Math. 110, 1169-1197 (2010) · Zbl 1186.92052 · doi:10.1007/s10440-009-9500-y
[37]Dong, L.; Chen, L.: A periodic predator–prey system with impulsive perturbation, J. comput. Appl. math. 223, 578-584 (2009) · Zbl 1159.34327 · doi:10.1016/j.cam.2008.02.015
[38]Lin, X.; Jiang, Y.; Wang, X.: Existence of periodic solutions in predator–prey with watt-type functional response and impulsive effects, Nonlinear anal. 73, 1684-1697 (2010) · Zbl 1203.34073 · doi:10.1016/j.na.2010.05.003
[39]Liu, X.; Takeuchi, Y.: Periodicity and global dynamics of an impulsive delay lasota-wazewska model, J. math. Anal. appl. 327, 326-341 (2007) · Zbl 1116.34063 · doi:10.1016/j.jmaa.2006.04.026
[40]Liu, Z.; Wu, J.; Chen, Y.; Haque, M.: Impulsive perturbations in a periodic delay differential equation model of plankton allelopathy, Nonlinear anal. RWA 11, 432-445 (2010) · Zbl 1190.34084 · doi:10.1016/j.nonrwa.2008.11.017
[41]Saker, S. H.; Alzabut, J. O.: Existence of periodic solutions, global attractivity and oscillation of impulsive delay population model, Nonlinear anal. RWA 8, 1029-1039 (2007) · Zbl 1124.34054 · doi:10.1016/j.nonrwa.2006.06.001
[42]Sun, S.; Chen, L.: Existence of positive periodic solution of an impulsive delay logistic model, Appl. math. Comput. 184, 617-623 (2007) · Zbl 1125.34063 · doi:10.1016/j.amc.2006.06.060
[43]Wang, H.: Dispersal permanence of periodic predator–prey model with lvlev-type functional response and impulsive effects, Appl. math. Model. 34, 3713-3725 (2010) · Zbl 1201.34077 · doi:10.1016/j.apm.2010.02.009
[44]Wang, Q.; Ding, M.; Wang, Z.; Zhang, H.: Existence and attractivity of a periodic solution for N-species gilpin-ayala impulsive competition system, Nonlinear anal. RWA 11, 2675-2685 (2010) · Zbl 1197.34082 · doi:10.1016/j.nonrwa.2009.09.015
[45]Fu, X.; Yan, B.; Liu, Y.: Introduction to impulsive differential systems, (2005)
[46]Horn, W. A.: Some fixed point theorems for compact maps and flows in Banach spaces, Trans. amer. Math. soc. 149, 391-404 (1970) · Zbl 0201.46203 · doi:10.2307/1995402