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Multiple periodic solutions of a ratio-dependent predator-prey model. (English) Zbl 1197.34065
Summary: A delayed ratio-dependent predator-prey model with non-monotone functional response is investigated in this paper. Some new and interesting sufficient conditions are obtained for the global existence of multiple positive periodic solutions of the ratio-dependent model. Our method is based on Mawhin’s coincidence degree and some estimation techniques for the a priori bounds of unknown solutions to the equation $Lx = \lambda $Nx. An example is represented to illustrate the feasibility of our main result. Editorial remark: There are doubts about a proper peer-reviewing procedure of this journal. The editor-in-chief has retired, but, according to a statement of the publisher, articles accepted under his guidance are published without additional control.

34C25Periodic solutions of ODE
92D25Population dynamics (general)
Full Text: DOI
[1] Kuang, Y.: Delay differential equations with applications dynamics, Series of mathematics in science and engineering 191 (1993) · Zbl 0777.34002
[2] Gopalsamy, K.: Stability and oscillations in delay differential equations of population dynamics, (1993) · Zbl 0752.34039
[3] Beretta, E.; Kuang, Y.: Convergence results in a well-known delayed predator -- prey system, J math anal appl 204, 840-853 (1996) · Zbl 0876.92021 · doi:10.1006/jmaa.1996.0471
[4] Kuang, Y.: Rich dynamics of gause-type ratio-dependent predator -- prey systems, Fields inst commun 21, 325-337 (1999) · Zbl 0920.92032
[5] Akcakaya, H. R.: Population cycles of mannals: evidence for a ratio-dependent predation pyhothesis, Ecol monogr 62, 119-142 (1992)
[6] Arditi, R.; Ginzburg, L. R.: Coupling in predator -- prey dynamics: ratio-dependence, J theoret biol 139, 311-326 (1989)
[7] Arditi, R.; Ginzburg, L. R.; Akcakaya, H. R.: Variation i plankton densities among lakes: a case for ratio-dependent models, Am naturalist 138, 1287-1296 (1991)
[8] Arditi, R.; Perrin, N.; Saiah, H.: Functional response and heterogeneities: an experimental test with cladocerans, Oikos 60, 69-75 (1991)
[9] Arditi, R.; Saiah, H.: Empirical evidences of the role of heterogeneity in ratio-dependent consumption, Ecology 73, 1544-1551 (1992)
[10] Gutierrez, A. P.: The physiological basis of ratio-dependent predator -- prey theory: a metabolic pool model of Nicholson’s blowflies as an example, Ecology 73, 817-827 (1992)
[11] Berryman, A. A.: The origins and evolution of predator -- prey theory, Ecology 75, 1530-1535 (1992)
[12] Lundberg, P.; Fryxell, J. M.: Expected population density versus productivity in ratio-dependent and prey-dependent models, Am naturalist 145, 153-161 (1995)
[13] Beretta, E.; Kuang, Y.: Global analysis in some delayed ratio-dependent predator -- prey systems, Nonlinear anal TMA 32, 381-408 (1998) · Zbl 0946.34061 · doi:10.1016/S0362-546X(97)00491-4
[14] Kuang, Y.; Beretta, E.: Global qualitative analysis of a ratio-dependent predator -- prey systems, J math biol 36, 389-406 (1998) · Zbl 0895.92032 · doi:10.1007/s002850050105
[15] Hsu, S. B.; Hwang, T. W.; Kuang, Y.: Global analysis of michaelis -- menten type ratio-dependent predator -- prey system, J math biol 42, 489-506 (2003) · Zbl 0984.92035 · doi:10.1007/s002850100079
[16] Jost, C.; Arino, O.; Arditi, R.: About deterministic extinction in ratio-dependent predator -- prey models, Bull math biol 61, 19-32 (1999) · Zbl 1323.92173
[17] Fan, M.; Wang, Q.; Zou, X. F.: Dynamics of a nonautonomous ratio-dependent predator -- prey system, Proc R soc edin A 133, 97-118 (2003) · Zbl 1032.34044 · doi:10.1017/S0308210500002304
[18] Bush, A. W.; Cook, A. E.: The effect of time delay and growth rate inhibition in the bacterial treatment of wastewater, J theoret biol 63, 385-395 (1976)
[19] Andrews, J. F.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates, Biotechnol bioeng 10, 707-732 (1986)
[20] Sokol, W.; Howell, J. A.: Kinetics of phenol oxidation by washed cells, Biotechnol bioeng 23, 2039-2049 (1980)
[21] Gains, R. E.; Mawhin, J. L.: Coincidence degree and nonlinear differential equations, (1977)
[22] Fan, Y.; Li, W. T.; Wang, L.: Periodic solutions of delayed ratio-dependent predator -- prey models with monotonic or nonmonotonic functional responses, Nonlinear anal RWA 5, 247-263 (2004) · Zbl 1069.34098 · doi:10.1016/S1468-1218(03)00036-1
[23] Wang, L. L.; Li, W. T.: Existence and global stability of positive periodic solutions of a predator -- prey system with delays, Appl math comput 146, 167-185 (2003) · Zbl 1029.92025 · doi:10.1016/S0096-3003(02)00534-9
[24] Chen, Y.: Multiple periodic solution of delayed predator -- prey systems with type IV functional responses, Nonlinear anal RWA 5, 45-53 (2004) · Zbl 1066.92050 · doi:10.1016/S1468-1218(03)00014-2
[25] Fan, M.; Kuang, Y.: Dynamics of non-autonomous predator prey system with the beddington -- deangelis functional response, J math anal appl 295, 15-39 (2004) · Zbl 1051.34033 · doi:10.1016/j.jmaa.2004.02.038
[26] Fan, M.; Wong, P. J. Y.; Agarwal, R. P.: Periodicity and stability in periodic n-species Lotka -- Volterra competition system with feedback controls and deviating arguments, Acta math sin 19, 801-822 (2003) · Zbl 1047.34080 · doi:10.1007/s10114-003-0311-1
[27] Fan, M.; Wong, P. J. Y.; Agarwal, R. P.: Periodicity in a class of nonautonomous scalar equations with deviating arguments and applications to population models, Dyn syst 19, 279-301 (2004) · Zbl 1062.34073 · doi:10.1080/14689360412331279867
[28] Ruan, S.; Xiao, D.: Global analysis in a predator -- prey system with nonmonotonic functional response, SIAM J appl math 61, 1445-1472 (2001) · Zbl 0986.34045 · doi:10.1137/S0036139999361896
[29] Xiao, D.; Ruan, S.: Multiple bifurcations in a delayed predator -- prey system with nonmonotonic functional response, J diff eq 176, 494-510 (2001) · Zbl 1003.34064 · doi:10.1006/jdeq.2000.3982
[30] Xiao, D.; Ruan, S.: Codimension two bifurcations in a delayed predator -- prey system with group defense, Int J bifurc chaos 11, No. 8, 2123-2132 (2001) · Zbl 1091.92504 · doi:10.1142/S021812740100336X
[31] Ma, W. X.; Maruno, K. I.: Complexiton solutions of the Toda lattice equation, Phys A 343, 219-237 (2004)
[32] Xia, Y. H.; Cao, J.: Almost periodic solutions for an ecological model with infinite delays, Proc Edinburgh math soc 50, No. 1, 229-249 (2007) · Zbl 1130.34044 · doi:10.1017/S0013091504001233
[33] Xia, Y. H.; Cao, J.; Zhang, H.; Chen, F. D.: Almost periodic solutions of n-species competitive system with feedback controls, J math anal appl 294, No. 2, 503-522 (2004) · Zbl 1053.34040 · doi:10.1016/j.jmaa.2004.02.025
[34] Xia, Y. H.; Cao, J.: Almost periodicity in an ecological model with M-predators and N-preys by ”pure-delay type” system, Nonlinear dyn 39, No. 3, 275-304 (2005) · Zbl 1093.92061 · doi:10.1007/s11071-005-4006-2
[35] Xia, Y. H.; Cao, J.; Lin, M.: Discrete-time analogues of predator -- prey models with monotonic or nonmonotonic functional responses, Nonlinear anal. 8, No. 4, 1079-1095 (2007) · Zbl 1127.39038 · doi:10.1016/j.nonrwa.2006.06.007
[36] Xia, Y. H.; Cao, J.: Global attractivity of a periodic ecological model with m-predators and n-preys by ”pure-delay type” system, Comput math appl 52, No. 6 -- 7, 829-852 (2006) · Zbl 1135.34038 · doi:10.1016/j.camwa.2006.06.002
[37] Xia, Y. H.; Cao, J.; Cheng, S. S.: Periodicity in a Lotka Volterra mutualism system with several delays, Appl math modelling 31, No. 9, 1960-1969 (2007) · Zbl 1167.34343 · doi:10.1016/j.apm.2006.08.013
[38] Xia, Y. H.; Cao, J.; Cheng, S. S.: Multiple periodic solutions of a delayed stage-structured predator -- prey model with nonmonotone functional responses, Appl math modelling 31, No. 9, 1947-1959 (2007) · Zbl 1167.34342 · doi:10.1016/j.apm.2006.08.012