| Title:
|
Existence and global attractivity of positive almost periodic solutions for a kind of fishing model with pure-delay (English) |
| Author:
|
Zhang, Tianwei |
| Author:
|
Liao, Yongzhi |
| Language:
|
English |
| Journal:
|
Kybernetika |
| ISSN:
|
0023-5954 (print) |
| ISSN:
|
1805-949X (online) |
| Volume:
|
53 |
| Issue:
|
4 |
| Year:
|
2017 |
| Pages:
|
612-629 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
By using some analytical techniques, modified inequalities and Mawhin's continuation theorem of coincidence degree theory, some sufficient conditions for the existence of at least one positive almost periodic solution of a kind of fishing model with delay are obtained. Further, the global attractivity of the positive almost periodic solution of this model is also considered. Finally, three examples are given to illustrate the main results of this paper. (English) |
| Keyword:
|
almost periodic solution |
| Keyword:
|
coincidence degree |
| Keyword:
|
fishing model |
| Keyword:
|
global attractivity |
| MSC:
|
34K13 |
| MSC:
|
92D25 |
| idZBL:
|
Zbl 06819627 |
| idMR:
|
MR3730255 |
| DOI:
|
10.14736/kyb-2017-4-0612 |
| . |
| Date available:
|
2017-11-12T09:56:07Z |
| Last updated:
|
2018-05-25 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146947 |
| . |
| Reference:
|
[1] Berezansky, L., Braverman, E., Idels, L.: On delay differential equations with Hills type growth rate and linear harvesting..Comput. Math. Appl. 49 (2005), 549-563. MR 2124386, 10.1016/j.camwa.2004.07.015 |
| Reference:
|
[2] Berezansky, L., Idels, L.: Stability of a time-varying fishing model with delay..Appl. Math. Lett. 21 (2008), 447-452. MR 2402835, 10.1016/j.aml.2007.03.027 |
| Reference:
|
[3] Dai, B. X., Su, H., Hu, D. W.: Periodic solution of a delayed ratio-dependent predator-prey model with monotonic functional response and impulse..Nonlinear Anal. TMA 70 (2009), 126-134. MR 2468223, 10.1016/j.na.2007.11.036 |
| Reference:
|
[4] Du, B., Hu, M., Lian, X.: Dynamical behavior for a stochastic predator-prey model with HV type functional response..Bull. Malays. Math. Sci. Soc. 40 (2016), 1, 487-503. MR 3592918, 10.1007/s40840-016-0325-3 |
| Reference:
|
[5] Du, B., Liu, Y., Batarfi, H. A.: Almost periodic solution for a neutral-type neural networks with distributed leakage delays on time scales..Neurocomputing 173 (2016), 921-929. 10.1016/j.neucom.2015.08.047 |
| Reference:
|
[6] Du, B., Liu, Y., Atiatallah, A. I.: Existence and asymptotic behavior results of periodic solution for discrete-ime neutral-type neural networks..J. Franklin Inst., Engrg. Appl. Math. 353 (2016), 448-461. MR 3448152, 10.1016/j.jfranklin.2015.11.013 |
| Reference:
|
[7] Egami, C.: Positive periodic solutions of nonautonomous delay competitive systems with weak Allee effect..Nonlinear Anal. RWA 10 (2009), 494-505. MR 2451726, 10.1016/j.nonrwa.2007.10.010 |
| Reference:
|
[8] Fan, Y. H., Wang, L. L.: Periodic solutions in a delayed predator-prey model with nonmonotonic functional response..Nonlinear Anal. RWA 10 (2009), 3275-3284. MR 2523287, 10.1016/j.nonrwa.2008.10.032 |
| Reference:
|
[9] Fink, A. M.: Almost Periodic Differential Equation..Spring-Verlag, Berlin, Heidleberg, New York, 1974. MR 0460799, 10.1007/bfb0070324 |
| Reference:
|
[10] Gaines, R., Mawhin, J.: Coincidence Degree and Nonlinear Differential Equations..Springer Verlag, Berlin 1977. MR 0637067, 10.1007/bfb0089537 |
| Reference:
|
[11] Gopalsamy, K.: Stability and Oscillations in Delay Differential Equations of Population Dynamics..Kluwer Acad. Publ., 1992. MR 1163190, 10.1007/978-94-015-7920-9 |
| Reference:
|
[12] He, C. Y.: Almost Periodic Differential Equations..Higher Education Publishing House, Beijing, 1992 (in Chinese). |
| Reference:
|
[13] Kot, M.: Elements of Mathematical Ecology..Cambr. Univ. Press, 2001. MR 2006645, 10.1017/cbo9780511608520 |
| Reference:
|
[14] Kuang, Y.: Delay Differential Equations With Applications in Population Dynamics..Academic Press, Inc., 1993. Zbl 0777.34002, MR 1218880, 10.1016/s0076-5392(08)x6164-8 |
| Reference:
|
[15] Liang, R. X., Shen, J. H.: Positive periodic solutions for impulsive predator-prey model with dispersion and time delays..Appl. Math. Comput. 217 (2010), 661-676. MR 2678579, 10.1016/j.amc.2010.06.003 |
| Reference:
|
[16] Liao, Y. Z., Zhang, T. W.: Almost periodic solutions of a discrete mutualism model with variable delays..Discrete Dynamics in Nature and Society Volume 2012, Article ID 742102, 27 pages. MR 3008486, 10.1155/2012/742102 |
| Reference:
|
[17] Lin, X. L., Jiang, Y. L., Wang, X. Q.: Existence of periodic solutions in predator-prey with Watt-type functional response and impulsive effects..Nonlinear Anal. TMA 73 (2010), 1684-1697. MR 2661351, 10.1016/j.na.2010.05.003 |
| Reference:
|
[18] Lu, S.: Applications of topological degree associated condensing field to the existence of periodic solutions for neutral functional differential equations with nonlinear difference operator..Acta Mathematica Sinica, English Series, to appear. MR 3568082 |
| Reference:
|
[19] Lu, S., Zhong, T., Chen, L.: Periodic solutions for $p$-Laplacian Rayleigh equations with singularities..Boundary Value Problems 2016, 96 (2016). MR 3499658 |
| Reference:
|
[20] Lu, S., Chen, L.: The problem of existence of periodic solutions for neutral functional differential system with nonlinear difference operator..J. Math. Anal. Appl. 387 (2012), 1127-1136. MR 2853200, 10.1016/j.jmaa.2011.10.022 |
| Reference:
|
[21] Shu, J. Y., Zhang, T. W.: Multiplicity of almost periodic oscillations in a harvesting mutualism model with time delays..Dynam. Cont. Disc. Impul. Sys. B: Appl. Algor. 20 (2013), 463-483. MR 3135007 |
| Reference:
|
[22] Wang, K.: Existence and global asymptotic stability of positive periodic solution for a predator-prey system with mutual interference..Nonlinear Anal. RWA 10 (2009), 2774-2783. MR 2523240, 10.1016/j.nonrwa.2008.08.015 |
| Reference:
|
[23] Wang, X. P.: Stability and existence of periodic solutions for a time-varying fishing model with delay..Nonlinear Anal.: RWA 11 (2010), 3309-3315. MR 2683790, 10.1016/j.nonrwa.2009.11.023 |
| Reference:
|
[24] Wang, K.: Periodic solutions to a delayed predator-prey model with Hassell-Varley type functional response..Nonlinear Anal. RWA 12 (2011), 137-145. MR 2728669, 10.1016/j.nonrwa.2010.06.003 |
| Reference:
|
[25] Wang, Q., Ding, M. M., Wang, Z. J., Zhang, H. Y.: Existence and attractivity of a periodic solution for an $N$-species Gilpin-Ayala impulsive competition system..Nonlinear Anal. RWA 11 (2010), 2675-2685. MR 2661935, 10.1016/j.nonrwa.2008.08.015 |
| Reference:
|
[26] Zhang, T. W.: Multiplicity of positive almost periodic solutions in a delayed Hassell-Varleytype predator-prey model with harvesting on prey..Math. Meth. Appl. Sci. 37 (2013), 686-697. MR 3180630, 10.1002/mma.2826 |
| Reference:
|
[27] Zhang, T. W.: Almost periodic oscillations in a generalized Mackey-Glass model of respiratory dynamics with several delays..Int. J. Biomath. 7 (2014), 1450029 (22 pages). MR 3210478, 10.1142/s1793524514500296 |
| Reference:
|
[28] Zhang, T. W., Gan, X. R.: Existence and permanence of almost periodic solutions for Leslie-Gower predator-prey model with variable delays..Elect. J. Differ. Equa. 2013 (2013), 1-21. MR 3065058 |
| Reference:
|
[29] Zhang, T. W., Gan, X. R.: Almost periodic solutions for a discrete fishing model with feedback control and time delays..Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 150-163. MR 3142456, 10.1016/j.cnsns.2013.06.019 |
| Reference:
|
[30] Zhang, T. W., Li, Y. K.: Positive periodic solutions for a generalized impulsive $n$-species Gilpin-Ayala competition system with continuously distributed delays on time scales..Int. J. Biomath. 4 (2011), 23-34. MR 2795143, 10.1142/s1793524511001131 |
| Reference:
|
[31] Zhang, T. W., Li, Y. K., Ye, Y.: On the existence and stability of a unique almost periodic solution of Schoener's competition model with pure-delays and impulsive effects..Commun. Nonlinear Sci. Numer. Simul. 17 (2012), 1408-1422. MR 2843805, 10.1016/j.cnsns.2011.08.008 |
| Reference:
|
[32] Zhang, T. W., Li, Y. K., Ye, Y.: Persistence and almost periodic solutions for a discrete fishing model with feedback control..Commun. Nonlinear Sci. Numer. Simul. 16 (2011), 1564-1573. MR 2736833, 10.1016/j.cnsns.2010.06.033 |
| Reference:
|
[33] Zhang, G. D., Shen, Y., Chen, B. S.: Positive periodic solutions in a non-selective harvesting predator-prey model with multiple delays..J. Math. Anal. Appl. 395 (2012), 298-306. MR 2943624, 10.1016/j.jmaa.2012.05.045 |
| Reference:
|
[34] Zhu, Y. L., Wang, K.: Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower Holling-type II schemes..J. Math. Anal. Appl. 384 (2011), 400-408. MR 2825193, 10.1016/j.jmaa.2011.05.081 |
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