×

Global stability in a periodic delayed predator-prey system. (English) Zbl 1122.34048

The authors study existence and global attractivity of a positive periodic solution of the periodic delayed predator-prey system \[ \begin{aligned} y_1'(t)&=y_1(t)\left[r_1(t)-a_1(t)y_1(t)+\sum_{i=1}^nb_{1i}(t)y_1(t-\tau_i(t))-\sum_{j=1}^mc_{1j}(t) y_2(t-\rho_{j}(t))\right],\\ y_2'(t)&=y_2(t)\left[r_2(t)-a_2(t)y_2(t)+\sum_{j=1}^mb_{2j}(t)y_2(t-\eta_j(t))+\sum_{i=1}^nc_{2i}(t) y_1(t-\sigma_{i}(t))\right]. \end{aligned}\tag{1} \]
By using the method of coincidence degree and a Lyapunov functional, under some assumptions, easily verifiable sufficient conditions are obtained for the existence and global attractivity of a positive periodic solution of (1). Biological interpretations on the main results are also given.

MSC:

34K13 Periodic solutions to functional-differential equations
34K20 Stability theory of functional-differential equations
92D25 Population dynamics (general)
34K60 Qualitative investigation and simulation of models involving functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Halbach, U., Life table data and population dynamics of the rotifer Brachionous Calyciflorus palls as influenced by periodically oscillating temperature, (Effects of Temperature on Ectothermic Organisms (1973), Spring-Verlag: Spring-Verlag Heidelberg), 217-228
[2] Freedman, H. I.; Wu, J., Periodic solution of single species models with periodic delay, SIAM J. Math. Anal., 23, 689-701 (1992) · Zbl 0764.92016
[3] Fan, M.; Wang, K., Periodic solutions of single population model with hereditary effect, Appl. Math., 13, 58-61 (2000), (in Chinese) · Zbl 1008.92028
[4] Miler, R. K., On Voterra’s population equation, SIAM J. Appl. Math., 14, 446-452 (1996)
[5] Seifert, G., On delay-differential equation for single species population variations, Nonlinear Anal. TMA, 9, 1051-1059 (1987) · Zbl 0629.92019
[6] Freedman, H. I.; Xia, H., Periodic solution of single species models with delay, differential equations,dynamical systems and control science, Lecture Notes Pure Appl. Math., 152, 55-74 (1994) · Zbl 0794.34056
[7] Fujimoto, H., Dynamical behaviours for population growth equations with delays, Nonlinear Anal. TMA, 31, 549-558 (1998) · Zbl 0887.34071
[8] Chen, B. S.; Liu, Q. Y., On the stable periodic solutions of single species models with hereditary effect, Math. Appl., 12, 42-46 (1999), (in Chinese)
[9] Kuang, Y.; Smith, H. L., Global stability for infinite delay Lotka-Volterra type systems, J. Diff. Eq., 103, 221-246 (1993) · Zbl 0786.34077
[10] Zhang, J.; Chen, L., Periodic solutions of single-species nonautonomous diffusion models with continuous time delays, Math. Comput. Modell., 23, 17-27 (1996) · Zbl 0864.60058
[11] Liu, Z. J.; Wang, W. D., Persistence and periodic solutions of a nonautonomous predator-prey diffusion system with Holling III functional response and continuous delay, Discr. Cont. Dyn. Sys., Series B, 4, 653-662 (2004) · Zbl 1101.92052
[12] Cushing, J. M., Stable limit cycles of time-dependent multispecies interactions, Math. Biosci., 31, 259-273 (1976) · Zbl 0341.92011
[13] Cushing, J. M., Periodic time-dependent predator-prey systems, SAIM J. Appl. Math., 34, 82-95 (1997) · Zbl 0348.34031
[14] Ma, Z. E.; Wang, W. D., Asymptotic behavior of predator-prey system with time dependent coefficients, Appl. Anal., 34, 79-90 (1989) · Zbl 0658.34044
[15] Amine, Z.; Ortega, R., A periodic prey-predator system, J. Math. Anal. Appl., 185, 477-489 (1994) · Zbl 0808.34043
[16] Fang, H.; Cao, J. D., Global existence for positive periodic solutions to a class of predator-prey systems, J. Biomath., 15, 403-407 (2000) · Zbl 1056.92515
[17] Chen, F. D.; Shi, J. L.; Chen, X. X., Positive periodic solution of a predator-prey system with several delays, J. Biomath., 20, 51-57 (2005) · Zbl 1069.92024
[18] Wang, W. D.; Ma, Z. E., Harmless delays for uniform persistence, J. Math. Anal. Appl., 158, 256-268 (1991) · Zbl 0731.34085
[19] Chen, L. S., Mathematical Models and Methods in Ecology (1988), Science Press: Science Press Beijing
[20] Lu, S. P., Existence of positive periodic solutions for neutral population model with multiple delays, Appl. Math. Comput., 153, 885-902 (2004) · Zbl 1042.92026
[21] Lu, S. P., On the existence of positive periodic solutions for neutral function differential equation with multiple deviating arguments, J. Math. Anal. Appl., 280, 321-333 (2003) · Zbl 1034.34084
[22] Z.J. Liu, L.S. Chen, Periodic solution of neutral Lotka-Volterra system with periodic delays, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2005.12.029.; Z.J. Liu, L.S. Chen, Periodic solution of neutral Lotka-Volterra system with periodic delays, J. Math. Anal. Appl. (2006), doi:10.1016/j.jmaa.2005.12.029. · Zbl 1114.34050
[23] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
[24] Gopalsamy, K., Stability and Oscillation in Delay Differential Equations of Population Dynamics, vol. 74 (1992), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0752.34039
[25] Teng, Z. D.; Chen, L. S., Necessary and sufficient conditions for existence of positive periodic solutions of periodic predator-prey systems, Acta Math. Sci., Ser. A, 18, 402-406 (1998) · Zbl 0914.92019
[26] He, X. Z., Stability and delays in a predator-prey system, J. Math. Anal. Appl., 198, 355-370 (1996) · Zbl 0873.34062
[27] Wen, X. Z.; Wang, Z. C., Global attractivity of positive periodic solution of multispecies ecological delay system, Acta Math. Sci., Ser. A, 24, 641-653 (2004) · Zbl 1067.34072
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.