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Dynamics of a nonautonomous semiratio-dependent predator-prey system with nonmonotonic functional responses. (English) Zbl 1106.92067
Summary: A nonautonomous semi-ratio-dependent predator-prey system with nonmonotonic functional responses is investigated. For the general nonautonomous case, positive invariance, permanence, and global asymptotic stability for the system are studied. For the periodic (almost periodic) case, sufficient conditions for existence, uniqueness, and stability of a positive periodic (almost periodic) solution are obtained.
MSC:
92D40Ecology
34C25Periodic solutions of ODE
34D23Global stability of ODE
92D25Population dynamics (general)
34D05Asymptotic stability of ODE
34C60Qualitative investigation and simulation of models (ODE)
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Full Text: DOI EuDML
References:
[1] J. F. Andrews, “A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates,” Biotechnology and Bioengineering, vol. 10, no. 6, pp. 707-723, 1986. · doi:10.1002/bit.260100602
[2] I. Barbalat, “Systems d’equations differentielle d’oscillations nonlineaires,” Revue Roumaine de Mathématiques Pures et Appliquées, vol. 4, pp. 267-270, 1959. · Zbl 0090.06601
[3] A. W. Bush and A. E. Cook, “The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater,” Journal of Theoretical Biology, vol. 63, no. 2, pp. 385-395, 1976. · doi:10.1016/0022-5193(76)90041-2
[4] J. B. Collings, “The effects of the functional response on the bifurcation behavior of a mite predator-prey interaction model,” Journal of Mathematical Biology, vol. 36, no. 2, pp. 149-168, 1997. · Zbl 0890.92021 · doi:10.1007/s002850050095
[5] Y.-H. Fan, W.-T. Li, and L.-L. Wang, “Periodic solutions of delayed ratio-dependent predator-prey models with monotonic or nonmonotonic functional responses,” Nonlinear Analysis. Real World Applications, vol. 5, no. 2, pp. 247-263, 2004. · Zbl 1069.34098 · doi:10.1016/S1468-1218(03)00036-1
[6] M. Fan, Q. Wang, and X. Zou, “Dynamics of a non-autonomous ratio-dependent predator-prey system,” Proceedings of the Royal Society of Edinburgh. Section A. Mathematics, vol. 133, no. 1, pp. 97-118, 2003. · Zbl 1032.34044 · doi:10.1017/S0308210500002304
[7] R. E. Gaines and J. L. Mawhin, Coincidence Degree, and Nonlinear Differential Equations, Springer, Berlin, 1977. · Zbl 0339.47031
[8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, vol. 74 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, 1992. · Zbl 0752.34039
[9] H.-F. Huo and W.-T. Li, “Existence and global stability of positive periodic solutions of a discrete delay competition system,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 38, pp. 2401-2413, 2003. · Zbl 1026.92033 · doi:10.1155/S0161171203210231 · eudml:50430
[10] H.-F. Huo and W.-T. Li, “Periodic solutions of a periodic two-species competition model with delays,” International Journal of Applied Mathematics, vol. 12, no. 1, pp. 13-21, 2003. · Zbl 1043.34074
[11] H.-F. Huo and W.-T. Li, “Periodic solution of a delayed predator-prey system with Michaelis-Menten type functional response,” Journal of Computational and Applied Mathematics, vol. 166, no. 2, pp. 453-463, 2004. · Zbl 1047.34081 · doi:10.1016/j.cam.2003.08.042
[12] H.-F. Huo and W.-T. Li, “Periodic solutions of a ratio-dependent food chain model with delays,” Taiwanese Journal of Mathematics, vol. 8, no. 2, pp. 211-222, 2004. · Zbl 1064.34045
[13] H.-F. Huo and W.-T. Li, “Positive periodic solutions of a class of delay differential system with feedback control,” Applied Mathematics and Computation, vol. 148, no. 1, pp. 35-46, 2004. · Zbl 1057.34093 · doi:10.1016/S0096-3003(02)00824-X
[14] H.-F. Huo, W.-T. Li, and R. P. Agarwal, “Optimal harvesting and stability for two species stage-structured system with cannibalism,” International Journal of Applied Mathematics, vol. 6, no. 1, pp. 59-79, 2001. · Zbl 1023.92034
[15] H.-F. Huo, W.-T. Li, and S. S. Cheng, “Periodic solutions of two-species diffusion models with continuous time delays,” Demonstratio Mathematica, vol. 35, no. 2, pp. 433-446, 2002. · Zbl 1013.92035
[16] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, vol. 191 of Mathematics in Science and Engineering, Academic Press, Massachusetts, 1993. · Zbl 0777.34002
[17] P. H. Leslie, “Some further notes on the use of matrices in population mathematics,” Biometrika, vol. 35, no. 3-4, pp. 213-245, 1948. · Zbl 0034.23303 · doi:10.1093/biomet/35.3-4.213
[18] P. H. Leslie, “A stochastic model for studying the properties of certain biological systems by numerical methods,” Biometrika, vol. 45, no. 1-2, pp. 16-31, 1958. · Zbl 0089.15803 · doi:10.1093/biomet/45.1-2.16
[19] Y. K. Li and Y. Kuang, “Periodic solutions of periodic delay Lotka-Volterra equations and systems,” Journal of Mathematical Analysis and Applications, vol. 255, no. 1, pp. 260-280, 2001. · Zbl 1024.34062 · doi:10.1006/jmaa.2000.7248
[20] E. C. Pielou, Mathematical Ecology, John Wiley & Sons, New York, 1977. · Zbl 0259.92001
[21] S. Ruan and D. Xiao, “Global analysis in a predator-prey system with nonmonotonic functional response,” SIAM Journal on Applied Mathematics, vol. 61, no. 4, pp. 1445-1472, 2001. · Zbl 0986.34045 · doi:10.1137/S0036139999361896
[22] W. Sokol and J. A. Howell, “Kinetics of phenol oxidation by washed cells,” Biotechnology and Bioengineering, vol. 23, pp. 2039-2049, 1980. · doi:10.1002/bit.260230909
[23] Q. Wang, M. Fan, and K. Wang, “Dynamics of a class of nonautonomous semi-ratio-dependent predator-prey systems with functional responses,” Journal of Mathematical Analysis and Applications, vol. 278, no. 2, pp. 443-471, 2003. · Zbl 1029.34042 · doi:10.1016/S0022-247X(02)00718-7
[24] L.-L. Wang and W.-T. Li, “Existence of periodic solutions of a delayed predator-prey system with functional response,” International Journal of Mathematical Sciences, vol. 1, no. 1-2, pp. 55-63, 2002. · Zbl 1075.34067
[25] L.-L. Wang and W.-T. Li, “Existence and global stability of positive periodic solutions of a predator-prey system with delays,” Applied Mathematics and Computation, vol. 146, no. 1, pp. 167-185, 2003. · Zbl 1029.92025 · doi:10.1016/S0096-3003(02)00534-9
[26] L.-L. Wang and W.-T. Li, “Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response,” Journal of Computational and Applied Mathematics, vol. 162, no. 2, pp. 341-357, 2004. · Zbl 1076.34085 · doi:10.1016/j.cam.2003.06.005
[27] D. Xiao and S. Ruan, “Multiple bifurcations in a delayed predator-prey system with nonmonotonic functional response,” Journal of Differential Equations, vol. 176, no. 2, pp. 494-510, 2001. · Zbl 1003.34064 · doi:10.1006/jdeq.2000.3982
[28] T. Yoshizawa, Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions, Springer, New York, 1975. · Zbl 0304.34051