×

Existence of multiple periodic solutions for delay Lotka-Volterra competition patch systems with harvesting. (English) Zbl 1168.34349

Summary: We establish some new and interesting sufficient conditions on the existence of multiple positive periodic solutions for a delay Lotka-Volterra competition patch system with harvesting. Our method is based on Mawhin’s coincidence degree and some novel techniques for defining the operator \(N(u,\lambda )\) and obtaining a priori bounds.

MSC:

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

References:

[1] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press Boston · Zbl 0777.34002
[2] Cushing, J. M., Integrodifferential Equations and Delay Models in Population Dynamics, Lecture Notes in Biomathematics, Vol. 20 (1977), Springer-Verlag: Springer-Verlag New York · Zbl 0363.92014
[3] Gopalsamy, K., Stability and oscillation in delay differential equations of population dynamics. Stability and oscillation in delay differential equations of population dynamics, Mathematics and its Applications 74 (1992), Kluwer Academic Publishers Group: Kluwer Academic Publishers Group Dordrecht · Zbl 0752.34039
[4] Gyori, I.; Ladas, G., Oscillation Theory of Delay Differential Equations (1991), Oxford Science Publications: Oxford Science Publications Oxford · Zbl 0780.34048
[5] May, R. M., Stability and Complex in Model Ecosystem (1974), Princeton University Press: Princeton University Press Princeton
[6] Zhang, Z. Q.; Wang, Z. C., Periodic solutions for a two-species nonautonomous competition Lotka-Volterra system with time delay, J. Math. Anal. Appl., 265, 38-48 (2002) · Zbl 1003.34060
[7] Bereketoglu, H.; Györi, I., Global asymptotic stability in a nonautonomous Lotka-Volterra type system with infinite delay, J. Math. Anal. Appl., 210, 279-291 (1997) · Zbl 0880.34072
[8] Zhang, J. G.; Chen, L. S.; Chen, X. D., Persistence and global stability for two-species nonautonomous competition Lotka-Volterra system with time delay, Nonlin. Anal., 37, 1019-1028 (1999) · Zbl 0949.34060
[9] Fan, M.; Wang, K.; Jiang, D. Q., Existence and global attractivity of positive periodic solutions of periodic n-species Lotka-Volterra competition systems with several deviating arguments, Math. Biosci., 106, 47-61 (1999) · Zbl 0964.34059
[10] Zeng, G. Z.; Chen, L. S.; Chen, J. F., Persistence and periodic orbits for two-species nonautonomous diffusion Lotka-Volterra models, Math. Comput. Model., 20, 69-80 (1994) · Zbl 0827.34040
[11] Wang, W.; Fergola, P.; Tenneriello, C., Global attractivity of periodic solutions of population models, J. Math. Anal. Appl., 211, 498-511 (1997) · Zbl 0879.92027
[12] Li, Y. K., Periodic solutions for delay Lotka-Volterra competition system, J. Math. Anal. Appl., 246, 230-244 (2000) · Zbl 0972.34057
[13] Clark, C. W., Mathematical Bioeconomics. Mathematical Bioeconomics, The optimal management of renewable resources (1990), John Wiley and Sons: John Wiley and Sons New York - Toronto · Zbl 0712.90018
[14] Brauer, F.; Soudack, A. C., Coexistence of properties of some predator-prey systems under constant rate harvesting and stocking, J. Math. Biol., 12, 101-114 (1981) · Zbl 0482.92015
[15] Brauer, F.; Soudack, A. C., On constant effort harvesting and stocking in a class of predator-prey systems, J. Theor. Biol., 95, 247-252 (1982)
[16] Xiao, D. M.; Ruan, S. G., Bogdanov-Takens bifurcations in predator-prey systems with constant rate harvesting, Fields Inst. Commun., 21, 493-506 (1999) · Zbl 0917.34029
[17] Chen, Y., Multiple periodic solutions of delayed predator-prey systems with type IV functional responses, Nonlin. Anal: Real World Appl., 5, 45-53 (2004) · Zbl 1066.92050
[18] Gaines, R. E.; Mawhin, J., Coincidence Degree and Nonlinear Differential Equations (1977), Springer-Verlag: Springer-Verlag Berlin · Zbl 0326.34021
[19] Xia, Y. H.; Cao, J. D.; Cheng, S. S., Multiple periodic solutions of a delayed stage-structured predator-prey model with non-monotone functional responses, Appl. Math. Modell. (2006)
[20] Feng, F.; Chen, S., Four periodic solutions of a generalized delayed predator-prey system, Appl. Math. Comput., 181, 932-939 (2006) · Zbl 1112.34048
[21] Zhang, Z. Q.; Tian, T. S., Multiple positive periodic solutions for a generalized predator-prey system with exploited terms, Nonlin. Anal: Real World Appl. (2006)
[22] Huusko, A.; Hyvarinen, P., A high harvest rate induces a tendency to generation cycling in a freshwater fish population, J. Anim. Ecol., 74, 525-531 (2005)
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.