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Permanence and periodic solutions for a diffusive ratio-dependent predator-prey system. (English) Zbl 1168.35377

Summary: This paper considers a diffusive predator-prey model, in which there is a ratio-dependent functional response with Holling III type. We establish some sufficient conditions for the ultimate boundedness of solutions and permanence of this system. The existence of a unique globally stable periodic solution is also presented.

MSC:

35K50 Systems of parabolic equations, boundary value problems (MSC2000)
92D25 Population dynamics (general)
35B10 Periodic solutions to PDEs
35K57 Reaction-diffusion equations
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[1] Berryman, A. A., The origins and evolution of predator-prey theory, Ecology, 75, 1530-1535 (1992)
[2] Holling, C. S., The functional response of predator to prey density and its role in mimicry and population regulation, Men. Ent. Sec. Can., 45, 1-60 (1965)
[3] Arditi, R.; Ginzburg, L. R., Coupling in predator-prey dynamics: ratio dependence, J. Theor. Biol., 139, 311-326 (1989)
[4] Arditi, R.; Ackakaya, H. R., Underestimation of mutual interference of predators, Oecologia, 83, 358-361 (1990)
[5] Arditi, R.; Ginzburg, L. R.; Ackakaya, H. R., Variation in plankton densities among lakes – a case for ratio-dependent predation models, Am. Nat., 138, 1287-1296 (1991)
[6] Akhmet, M. U.; Beklioglu, M.; Ergence, T.; Tkachenko, V. I., An impulsive ratio-dependent predator-prey system with diffusion, Nonlinear Anal.: Real World Appl., 7, 1255-1267 (2006) · Zbl 1114.35097
[7] Pang, P. Y.H.; Wang, M., Qualitative analysis of a ratio-dependent predator-prey with diffusion, Proc. Roy. Soc. Edin. A, 133, 919-942 (2003) · Zbl 1059.92056
[8] Fan, M.; Wang, K., Periodicity in a delayed ratio-dependent predator-prey system, J. Math. Anal. Appl., 262, 179-190 (2001) · Zbl 0994.34058
[9] 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
[10] Kuang, Y.; Beretta, E., Global qualitative analysis of a ratio-dependent predator-prey system, J. Math. Biol., 36, 389-406 (1998) · Zbl 0895.92032
[11] Xiao, D.; Ruan, S., Global dynamics of a ratio-dependent predator-prey system, J. Math. Biol., 43, 268-290 (2001) · Zbl 1007.34031
[12] Wang, L.-L.; Li, W.-T., Periodic solutions and permanence for a delayed nonautonomous ratio-dependent predator-prey model with Holling type functional response, J. Comput. Appl. Math., 162, 341-357 (2004) · Zbl 1076.34085
[13] Walter, W., Differential inequalities and maximum principles: theory, new methods and applications, Nonlinear Anal. Appl., 30, 4695-4711 (1997) · Zbl 0893.35014
[14] Smith, L. H., Dynamics of Competition, Lecture Notes in Mathematics, vol. 1714 (1999), Springer: Springer Berlin, pp. 192-240 · Zbl 1002.92564
[15] Pao, C. V., Nonlinear Parabolic and Elliptic Equations (2005), McGraw-Hill: McGraw-Hill New York · Zbl 1063.35020
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