×

Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response. (English) Zbl 1076.34085

Summary: By using the continuation theorem of the coincidence degree theory, the existence of positive periodic solutions for the delayed ratio-dependent predator-prey model with Holling type III functional response
\[ \begin{aligned} x'(t) &= x(t) \Biggl[a(t)-b(t) \int_{-\infty}^t k(t-s)x(s)\,ds \Biggr]- \frac {c(t)x^2(t)y(t)} {m^2y^2(t)+ x^2(t)},\\ y'(t) &= y(t) \biggl[\frac {e(t)x^2(t-\tau)} {m^2y^2 (t-\tau)+ x^2(t-\tau)}- d(t) \biggr], \end{aligned} \]
is established, where \(a(t)\), \(b(t)\), \(c(t)\), \(e(t)\) and \(d(t)\) are all positive periodic continuous functions with period \(\omega>0\), \(m>0\) and \(k(s)\) is a measurable function with period \(\omega\), and \(\tau\) is a nonnegative constant. The permanence of the system is also considered. In particular, if \(k(s)= \delta_0(s)\), where \(\delta_0(s)\) is the Dirac delta function at \(s=0\), our results show that the permanence of the above system is equivalent to the existence of a positive periodic solution.

MSC:

34K13 Periodic solutions to functional-differential equations
92D25 Population dynamics (general)
34K25 Asymptotic theory of functional-differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Arditi, R.; Perrin, N.; Saiah, H., Functional response and heterogeneitiesan experiment test with cladocerans, OIKOS, 60, 69-75 (1991)
[2] Beretta, E.; Kuang, Y., Global analysis in some delayed ratio-dependent predator-prey systems, Nonlinear Anal. TMA., 32, 381-408 (1998) · Zbl 0946.34061
[3] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 73, 1530-1535 (1992)
[4] Y.H. Fan, W.T. Li, Permanence in delayed ratio-dependent predator-prey models with monotonic functional response, submitted for publication.; Y.H. Fan, W.T. Li, Permanence in delayed ratio-dependent predator-prey models with monotonic functional response, submitted for publication. · Zbl 1152.34368
[5] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl., 262, 179-190 (2001) · Zbl 0994.34058
[6] Freedman, H. I., Deterministic Mathematical Models in Population Ecology (1980), Marcel-Dekker: Marcel-Dekker New York · Zbl 0448.92023
[7] Freedman, H. I.; Ruan, S. G., Uniform persistence in functional differential equations, J. Differential Equations, 115, 173-192 (1995) · Zbl 0814.34064
[8] Gaines, R. E.; Mawhin, J. L., Coincidence Degree and Nonlinear Differential Equations (1977), Springer: Springer Berlin · Zbl 0326.34021
[9] Hanski, I., The functional response of predatorworries bout scale, TREE, 6, 141-142 (1991)
[10] Hsu, S. B.; Hwang, T. W.; Kuang, Y., Global analysis of the Michaelis-Menten type ratio-dependent predator-prey system, J. Math. Biol., 42, 489-506 (2001) · Zbl 0984.92035
[11] H.F. Huo, W.T. Li, Periodic solutions of a ratio-dependent food chain model with delays, Taiwanese J. Math., in press.; H.F. Huo, W.T. Li, Periodic solutions of a ratio-dependent food chain model with delays, Taiwanese J. Math., in press. · Zbl 1064.34045
[12] Huo, H. F.; Li, W. T., Periodic solution of a periodic two-species competition model with delays, Internat. J. Appl. Math., 12, 13-21 (2003) · Zbl 1043.34074
[13] Jost, C.; Arino, O.; Arditi, R., About deterministic extinction in ratio-dependent predator-prey models, Bull. Math. Biol., 61, 19-32 (1999) · Zbl 1323.92173
[14] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[15] Li, Y. K., Periodic solutions of a periodic delay predator-prey system, Proc. Amer. Math. Soc., 127, 1331-1335 (1999) · Zbl 0917.34057
[16] W.T. Li, Y.H. Fan, S.G. Ruan, Periodic solutions in a delayed predator-prey model with nonmonotonic functional response, submitted for publication.; W.T. Li, Y.H. Fan, S.G. Ruan, Periodic solutions in a delayed predator-prey model with nonmonotonic functional response, submitted for publication.
[17] Li, Y. K.; Kuang, Y., Periodic solutions of periodic delay Lotka-Volterra equations and systems, J. Math. Anal. Appl., 255, 260-280 (2001) · Zbl 1024.34062
[18] May, R. M., Complexity and Stability in Model Ecosystems (1973), Princeton University Press: Princeton University Press Princeton, NJ
[19] Murry, J. D., Mathematical Biology (1989), Springer: Springer Berlin, New York
[20] Smith, H. L., Cooperative systems of differential equation with concave nonlinearities, Non-linear Anal., 10, 1037-1052 (1986) · Zbl 0612.34035
[21] Smith, H. L.; Kuang, Y., Periodic solutions of delay differential equations of threshold-type delay, (Graef, J.; Hale, J. K., Oscillation and Dynamicals in Delay Equations. Oscillation and Dynamicals in Delay Equations, Contemporary Mathematics, Vol. 129 (1992), American Mathematical Society: American Mathematical Society Providence, RI), 153-176 · Zbl 0762.34044
[22] Tang, B. R.; Kuang, Y., Existence, uniqueness and asymptotic stability of periodic solutions of periodic functional differential systems, Tohoku Math. J., 49, 217-239 (1997) · Zbl 0883.34074
[23] Wang, L. L.; Li, W. T., Existence of periodic solutions of a delayed predator-prey system with functional response, Internat. J. Math. Sci., 1, 55-63 (2002) · Zbl 1075.34067
[24] Wang, L. L.; Li, W. T., Existence and global stability of positive periodic solutions of a predator-prey system with delays, Appl. Math. Comput., 146, 167-185 (2003) · Zbl 1029.92025
[25] Wang, W. D.; Ma, Z. E., Harmless delays for uniform persistence, J. Math. Anal. Appl., 158, 256-268 (1991) · Zbl 0731.34085
[26] Zhao, T.; Kuang, Y.; Smith, H. L., Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems, Nonlinear Anal. TMA, 28, 1373-1394 (1997) · Zbl 0872.34047
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.