×

zbMATH — the first resource for mathematics

Complexity of a delayed predator-prey model with impulsive harvest and Holling type II functional response. (English) Zbl 1168.34052
The authors study the problem
\[ \begin{aligned} \dot{y}_1(t)&= y_1(t)\left(r-ay_1(t-\tau)-\frac{\alpha_1 y_2(t)}{1+\beta_1 y_1(t)}\right),\\ \dot{y}_2(t)&= y_2(t)\left(\frac{\alpha_2 y_1(t)}{1+\beta_2 y_1(t)}-d\right), \quad t\not=nT,\;n=1,\dots,\\ \Delta y_1(nT)&= -py_1(nT^-), \quad y_i(t)=\varphi_i (t)>0,\;t<0,\;i=1,2. \end{aligned} \]
They obtain the following results. If \((1-p)e^{rT}<1\), then the system is extinct. If \((1-p)e^{rT}>1\) then the zero solution becomes unstable. Moreover if \(\alpha_2>d\beta_2\), and \((\alpha_2-d\beta_2)(rT+\ln (1-p))> \operatorname{ad} Te^{2rT+\ln (1-p)}\) then the system has at least one \(T\) periodic solution.

MSC:
34K45 Functional-differential equations with impulses
34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] DOI: 10.1137/0132006 · Zbl 0348.34031
[2] DOI: 10.1016/S0096-3003(96)00111-7 · Zbl 0876.92022
[3] Kuang Y., Fields Inst. Commun. 21 pp 325–
[4] DOI: 10.1142/9789812798725
[5] DOI: 10.1006/jmaa.1996.0471 · Zbl 0876.92021
[6] Freedman H. I., Deterministic Mathematical Models in Population Ecology (1980) · Zbl 0448.92023
[7] Kuang Y., Delay Differential Equations with Applicatios in Population Dynamics (1993)
[8] DOI: 10.1006/jmaa.1996.0093 · Zbl 0851.34030
[9] DOI: 10.1016/S0304-3800(00)00442-7
[10] DOI: 10.1006/jdeq.2000.3982 · Zbl 1003.34064
[11] DOI: 10.1016/S0960-0779(01)00063-7 · Zbl 1022.92042
[12] DOI: 10.1016/S0960-0779(03)00075-4 · Zbl 1068.92045
[13] DOI: 10.1016/S0022-247X(02)00718-7 · Zbl 1029.34042
[14] DOI: 10.1016/j.chaos.2005.09.077 · Zbl 1203.34071
[15] DOI: 10.1016/j.chaos.2006.03.068 · Zbl 1156.34029
[16] DOI: 10.1016/j.chaos.2005.12.045 · Zbl 1155.34361
[17] Holling C. S., Mem. Ent. Sec. Can. 45 pp 1–
[18] L. Chen, X. Song and Z. Lu, Mathematical Models and Methods in Ecology (Sichuan Science and Technology, Chendu, China, 2003) p. 108.
[19] DOI: 10.1006/tpbi.1993.1026 · Zbl 0782.92020
[20] DOI: 10.1006/jmaa.1995.1224 · Zbl 0853.34011
[21] DOI: 10.1016/0362-546X(95)00182-U · Zbl 0919.34016
[22] DOI: 10.1016/S0025-5564(97)10016-5 · Zbl 0928.92027
[23] DOI: 10.1016/S0895-7177(00)00040-6 · Zbl 1043.92527
[24] DOI: 10.1142/S0218127402004954 · Zbl 1051.93512
[25] DOI: 10.1016/S0960-0779(02)00408-3 · Zbl 1085.34529
[26] DOI: 10.1016/j.chaos.2004.05.047 · Zbl 1066.92041
[27] DOI: 10.1016/j.chaos.2004.05.044 · Zbl 1081.34041
[28] DOI: 10.1016/j.chaos.2005.01.021 · Zbl 1065.92050
[29] DOI: 10.1007/978-94-015-8893-5
[30] DOI: 10.1142/2892
[31] DOI: 10.1016/j.jmaa.2005.04.005 · Zbl 1110.34019
[32] DOI: 10.1016/j.jmaa.2005.06.014 · Zbl 1101.34051
[33] DOI: 10.1016/j.vaccine.2006.05.018
[34] DOI: 10.1016/S0895-7177(04)90519-5 · Zbl 1065.92066
[35] DOI: 10.1016/j.mcm.2003.12.011 · Zbl 1112.34052
[36] DOI: 10.1142/S0252959905000403 · Zbl 1096.34053
[37] DOI: 10.1016/j.cam.2004.04.010 · Zbl 1070.34089
[38] DOI: 10.1016/j.physd.2005.06.032 · Zbl 1087.34028
[39] DOI: 10.1016/S1468-1218(02)00084-6 · Zbl 1011.92052
[40] DOI: 10.1007/s002850100121 · Zbl 0990.92033
[41] Choisy M., Physica D 22 pp 26–
[42] DOI: 10.1016/j.chaos.2005.12.025 · Zbl 1195.92066
[43] Shen L., Nonlinear Anal.: Real World Appl.
[44] Bainov D., Impulsive Differential Equations: Periodic Solutions and Applications (1993) · Zbl 0815.34001
[45] DOI: 10.1142/0906
[46] R. E. Gaines and J. L. Mawhin, Coincidence Degree and Nonlinear Differential Equations (Springer-Verlag, 1973) pp. 3590–3593.
[47] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics (Academic, San Diego, 1993) pp. 63–90.
[48] Davies B., Exploring Chaos, Theory and Experiment (1999) · Zbl 0959.37001
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.