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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:
35K50Systems of parabolic equations, boundary value problems (MSC2000)
92D25Population dynamics (general)
35B10Periodic solutions of PDE
35K57Reaction-diffusion equations
References:
[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 · doi:10.1016/j.nonrwa.2005.11.007
[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 · doi:10.1017/S0308210500002742
[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 · doi:10.1006/jmaa.2001.7555
[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 · doi:10.1007/s002850100079
[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 · doi:10.1007/s002850050105
[11]Xiao, D.; Ruan, S.: Global dynamics of a ratio-dependent predator – prey system, J. math. Biol. 43, 268-290 (2001) · Zbl 1007.34031 · doi:10.1007/s002850100097
[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 · doi:10.1016/j.cam.2003.06.005
[13]Walter, W.: Differential inequalities and maximum principles: theory, new methods and applications, Nonlinear anal. Appl. 30, 4695-4711 (1997) · Zbl 0893.35014 · doi:10.1016/S0362-546X(96)00259-3
[14]Smith, L. H.: Dynamics of competition, Lecture notes in mathematics 1714 (1999) · Zbl 1002.92564
[15]Pao, C. V.: Nonlinear parabolic and elliptic equations, (2005)