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)
Dynamic behaviors of the periodic predator-prey model with modified Leslie-Gower Holling-type II schemes and impulsive effect. (English) Zbl 1142.34031
The authors investigate a T-periodic predator-prey model with modified Leslie-Gower and Holling-type II terms which, moreover, is governed by periodically placed impulses. They first derive results on existence and uniquness of a positive periodic solution for one of the two species if the second one vanishes. Then, using Floquet theory for linear periodic impulsive equations, the authors establish local stability conditions for the above-mentioned solutions, and finally, applying the bifurcation theorem of Rabinowitz to these solutions, they prove a result on the existence of a strictly positive periodic solution for both species. An example concludes the paper.

34C60Qualitative investigation and simulation of models (ODE)
92D25Population dynamics (general)
34A37Differential equations with impulses
34C23Bifurcation (ODE)
34C25Periodic solutions of ODE
Full Text: DOI
[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] Aziz-Alaoui, M. A.: Study of a Leslie -- gower-type tritrophic population, Chaos solitons fractals 14, 1275-1293 (2002) · Zbl 1031.92027 · doi:10.1016/S0960-0779(02)00079-6
[3] Aziz-Alaoui, M. A.; Okiye, M. Daher: Boundedness and global stability for a predator -- prey model with modified Leslie -- gower and Holling-type II schemes, Appl. math. Lett. 16, 1069-1075 (2003) · Zbl 1063.34044 · doi:10.1016/S0893-9659(03)90096-6
[4] Bainov, D. D.; Simeonov, P. S.: Impulsive differential equations: periodic solutions and applications, (1993) · Zbl 0815.34001
[5] D’onofrio, A.: Pulse vaccination strategy in the SIR epidemic model: global asymptotic stable eradication in presence of vaccine failures, Math. comput. Modell. 36, 473-489 (2002) · Zbl 1025.92011 · doi:10.1016/S0895-7177(02)00177-2
[6] Funasaki, E.; Kot, M.: Invasion and chaos in a periodically pulsed mass-action chemostat, Theor. popul. Biol. 44, 203-424 (1993) · Zbl 0782.92020 · doi:10.1006/tpbi.1993.1026
[7] Korobeinikov, A.: A Lyapunov function for Leslie -- gower predator -- prey models, Appl. math. Lett. 14, 697-699 (2002) · Zbl 0999.92036 · doi:10.1016/S0893-9659(01)80029-X
[8] Lakmeche, A.; Arino, O.: Nonlinear mathematical model of pulsed therapy of heterogeneous tumor, Nonlinear anal. 2, 455-465 (2001) · Zbl 0982.92016 · doi:10.1016/S1468-1218(01)00003-7
[9] Lakshmikantham, V.; Bainov, D. D.; Simeonov, P. S.: Theory of impulsive differential equations, (1989) · Zbl 0718.34011
[10] Leslie, P. H.: Some further notes on the use of matrices in population mathematics, Biometrica 35, 213-245 (1948) · Zbl 0034.23303
[11] Leslie, P. H.; Gower, J. C.: The properties of a stochastic model for the predator -- prey type of interaction between two species, Biometrica 47, 219-234 (1960) · Zbl 0103.12502
[12] Letellier, C.; Aguirré, L.; Maquet, J.; Aziz-Alaoui, M. A.: Should all the species of a food chain be counted to investigate the global dynamics, Chaos solitons fractals 13, 1099-1113 (2002) · Zbl 1004.92039 · doi:10.1016/S0960-0779(01)00116-3
[13] Letellier, C.; Aziz-Alaoui, M. A.: Analysis of the dynamics of a realistic ecological model, Chaos solitons fractals 13, 95-107 (2002) · Zbl 0977.92029 · doi:10.1016/S0960-0779(00)00239-3
[14] Liu, X. N.; Chen, L. S.: Complex dynamics of Holling type II Lotka -- Volterra predator -- prey system with impulsive perturbations on the predator, Chaos solitons fractals 16, 311-320 (2003) · Zbl 1085.34529 · doi:10.1016/S0960-0779(02)00408-3
[15] López-Gómez, J.; Ortega, R.; Tineo, A.: The periodic predator -- prey Lotka -- Volterra model, Adv. diff. Equations 1, 403-423 (1996) · Zbl 0849.34026
[16] Paneyya, J. C.: A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competition environment, Bull. math. Biol. 58, 425-447 (1996) · Zbl 0859.92014 · doi:10.1007/BF02460591
[17] Pielou, E. C.: An introduction to mathematical ecology, (1969) · Zbl 0259.92001
[18] Rabinowitz, P. H.: Some global results for nonlinear eigenvalue problems, J. funct. Anal. 7, 487-513 (1971) · Zbl 0212.16504 · doi:10.1016/0022-1236(71)90030-9
[19] Rinaldi, S.; Muratori, S.; Kuznetsov, Y. A.: Multiple, attracto, catastrophes, and chaos in seasonally perturbed predator -- prey communities, Bull. math. Biol. 55, 15-36 (1993) · Zbl 0756.92026
[20] Roberts, M. G.; Kao, R. R.: The dynamics of an infectious disease in a population with birth pulse, Math. biosci. 149, 23-36 (1998) · Zbl 0928.92027 · doi:10.1016/S0025-5564(97)10016-5
[21] Shulgin, B.; Stone, L.; Agur, Z.: Theoretical examination of pulse vaccination policy in the SIR epidemic model, Math. comput. Model. 31, No. 4/5, 207-215 (2000) · Zbl 1043.92527 · doi:10.1016/S0895-7177(00)00040-6
[22] Tang, S. Y.; Chen, L. S.: The periodic predator -- prey Lotka -- Volterra model with impulsive effect, J. mech. Med. biol. 2, 1-30 (2002)
[23] Tang, S. Y.; Chen, L. S.: Multiple attractors in stage-structured population models with birth pulses, Bull. math. Biol. 65, 479-495 (2003)
[24] Upadhyay, R. K.; Rai, V.: Why chaos is rarely observed in natural populations, Chaos solitons fractals 8, 1933-1939 (1997)