×

On a periodic multi-species ecological model. (English) Zbl 1080.92059

Summary: A periodic predator-prey model with \(m\)-predators and \(n\)-preys is proposed, which can be seen as a modification of the traditional Lotka-Volterra model. By using a comparison theorem, the ultimately bounded region of the system is obtained. By using the comparison theorem and Brouwer fixed point theorem, sufficient conditions which guarantee the existence of positive periodic solutions of the system are obtained. Finally, by constructing a suitable Lyapunov function, some sufficient conditions are obtained for the existence of a unique globally attractive periodic solution of the system. The results obtained generalized the main results of J. D. Zhao and W. C. Chen in ibid. 147, No. 3, 881–892 (2004; Zbl 1029.92026).

MSC:

92D40 Ecology
34C25 Periodic solutions to ordinary differential equations
34C60 Qualitative investigation and simulation of ordinary differential equation models
37N25 Dynamical systems in biology

Citations:

Zbl 1029.92026
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gopalsamy, K., Global asymptotic stability in a periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27, 66-72 (1985) · Zbl 0588.92019
[2] Zhao, X. Q., The qualitative analysis of \(n\)-species Lotka-Volterra periodic competition systems, Math. comput. Modell., 15, 11, 3-8 (1991) · Zbl 0756.34048
[3] Tineo, A.; Alvarz, C., A different consideration about the globally asymptotically stable solution of the periodic \(n\)-competing species problem, J. Math. Anal. Appl., 159, 44-55 (1991) · Zbl 0729.92025
[4] Gopalsamy, K., Exchange of equilibria in two Lotka-Volterra competition models, J. Aust. Math. Soc. Ser. B, 24, 160-170 (1982) · Zbl 0498.92016
[5] Chattopadhyay, J., Effect of toxic substance on a two-species competitive system, Ecol. Model, 84, 287-289 (1996)
[6] Cui, J.; Chen, L., Asymptotic behavior of the solution for a class of time-dependent competitive system with feedback controls, Ann. Diff. Eqs., 9, 1, 11-17 (1993)
[7] Maynard-Smith, J., Models in Ecology (1974), Cambridge University: Cambridge University Cambridge · Zbl 0312.92001
[8] Song, X. Y.; Chen, L. S., Periodic solutions of a delay differential equation of plankton allelopathy, Acta Mathematica Scientia, 23, A, 8-13 (2003), (in Chinese) · Zbl 1036.34082
[9] Jin, Z.; Ma, Z. E., Periodic solutions for delay differential equations model of plankton allelopathy, Comput. Math. Appl.s, 44, 491-500 (2002) · Zbl 1094.34542
[10] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[11] Yang, P.; Xu, R., Global attractivity of the periodic Lotka-Volterra system, J. Math. Anal. Appl., 233, 1, 221-232 (1999) · Zbl 0973.92039
[12] Zhao, J. D.; Chen, W. C., Global asymptotic stability of a periodic ecological model, Appl. Math. Comp., 147, 3, 881-892 (2004) · Zbl 1029.92026
[13] Xia, Y.; Chen, F.; Cao, J.; Chen, A., Existence and global attractivity of an almost periodic ecological model, Appl. Math. Comput., 157, 449-475 (2004) · Zbl 1049.92038
[14] Ayala, F. J.; Gilpin, M. E.; Eherenfeld, J. G., Competition between species: Theoretical models and experimental tests, Theor. Population Biol., 4, 331-356 (1973)
[15] Gilpin, M. E.; Ayala, F. J., Global models of growth and competition, Proc. Nat. Acad. Sci. USA, 70, 3590-3593 (1973) · Zbl 0272.92016
[16] Fan, M.; Wang, K., Global periodic solutions of a generalized \(n\)-species Gilpin-Ayala competition model, Comp. Math. Appl., 40, 1141-1151 (2000) · Zbl 0954.92027
[17] Gilpin, M. E.; Ayala, F. J., Schoener’s model and Drosophila competition, Theor. Population Biol., 9, 12-14 (1976)
[18] Li, C. R.; Lu, S. J., The qualitative analysis of \(N\)-species periodic coefficient, nonliear relation, prey-competition systems, Appl. Math-JCU, 12, 2, 147-156 (1997), (in Chinese) · Zbl 0880.34042
[19] Chen, F. D.; Lin, S. J., Periodicity in a Logistic type system with several delays, Comp. Math. Appl., 48, 1-2, 35-44 (2004) · Zbl 1061.34050
[20] Chen, F. D., Periodicity in a food-limited population model with toxicants and state dependent delays, J. Math. Anal. Appl., 288, 1, 132-142 (2003)
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.