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Periodic solutions and homoclinic bifurcation of a predator-prey system with two types of harvesting. (English) Zbl 1281.92069
Summary: In this paper, a predator-prey model with both constant rate harvesting and state dependent impulsive harvesting is analyzed. By using differential equation geometry theory and the method of successor functions, the existence, uniqueness and stability of the order one periodic solution have been studied. Sufficient conditions which guarantee the nonexistence of order \(k\) \((k\geq 2)\) periodic solution are given. We also present that the system exhibits the phenomenon of homoclinic bifurcation under some parametric conditions. Finally, some numerical simulations and biological explanations are given.

MSC:
92D40 Ecology
34K13 Periodic solutions to functional-differential equations
34C23 Bifurcation theory for ordinary differential equations
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[1] Xiao, D.X.; Ruan, S.G., Bogdanov-Takens bifurcations in predator-prey system with constant rate harvesting, Fields Inst. Commun., 21, 493-506, (1999) · Zbl 0917.34029
[2] Brauer, F.; Soudack, A.C., Stability regions and transition phenomena for harvested predator-prey systems, J. Math. Biol., 7, 319-337, (1979) · Zbl 0397.92019
[3] Brauer, F., Destabilization of predator-prey systems under enrichment, Int. J. Control, 23, 541-552, (1976) · Zbl 0319.92012
[4] Brauer, F.; Soudack, A.C.; Jarosch, H.S., Stabilization, and destabilization of predator-prey systems under harvesting and nutrient enrichment, Int. J. Control, 23, 553-573, (1976) · Zbl 0317.92003
[5] Brauer, F.; Soudack, A.C., Stability regions in predator-prey systems with constant rate prey harvesting, J. Math. Biol., 8, 55-71, (1979) · Zbl 0406.92020
[6] Dai, G.R.; Tang, M.X., Coexistence region and global dynamics of a harvested predator-prey system, SIAM J. Appl. Math., 58, 193-210, (1998) · Zbl 0916.34034
[7] Dai, G.R.; Xu, C., Constant rate predator harvested predator-prey system with Holling-type I functional response, Acta Math. Sci., 14, 34-144, (1994)
[8] Chen, L.J.; Chen, F.D., Global analysis of a harvested predator-prey model incorporating a constant prey refuge, Int. J. Biomath., 3, 205-223, (2010) · Zbl 1342.92160
[9] Pei, Y.Z.; Li, C.G.; Chen, L.S., Continuous and impulsive harvesting strategies in a stage-structured predator-prey model with time delay, Math. Comput. Simul., 10, 2994-3008, (2009) · Zbl 1172.92038
[10] Liu, Z.J.; Tan, R.H., Impulsive harvesting and stocking in a monod-Haldane functional response predator-prey system, Chaos Solitons Fractals, 34, 454-464, (2007) · Zbl 1127.92045
[11] Negi, K.; Gakkhar, S., Dynamics in a beddington-deangelis prey-predator system with impulsive harvesting, Ecol. Model., 206, 421-430, (2007)
[12] Tang, S.Y.; Chen, L.S., The effect of seasonal harvesting on stage-structured population models, J. Math. Biol., 48, 357-374, (2004) · Zbl 1058.92051
[13] Zhang, X.A.; Chen, L.S.; Neumann, A.U., The stage-structured predator-prey model and optimal harvesting policy, Math. Biosci., 168, 201-210, (2000) · Zbl 1252.70020
[14] Zeng, G.Z.; Chen, L.S.; Sun, L.H., Existence of periodic solution of order one of planar impulsive autonomous system, J. Comput. Appl. Math., 186, 466-481, (2006) · Zbl 1088.34040
[15] Jiang, G.R.; Lu, Q.S.; Qian, L.N., Complex dynamics of a Holling type II prey-predator system with state feedback control, Chaos Solitons Fractals, 31, 448-461, (2007) · Zbl 1203.34071
[16] Nie, L.F.; Peng, J.G.; Teng, Z.D.; Hu, L., Existence and stability of periodic solution of a Lotka-Volterra predator-prey model with state-dependent impulsive effects, J. Comput. Appl. Math., 224, 544-555, (2009) · Zbl 1162.34007
[17] Zhu, C.R.; Zhang, W.N., Linearly independent homoclinic bifurcations parameterized by a small function, J. Differ. Equ., 240, 38-57, (2007) · Zbl 1138.34024
[18] Baras, E.; Lagardère, J.P., Fish telemetry in aquaculture: review and perspectives, Aquac. Int., 3, 77-102, (1995)
[19] Bègout, A.M.L.; Lagardère, J.P., Weather related variability. consequences of the swimming activity of a marine fish, C. R. Acad. Sci., Sér. 3 Sci. Vie, 321, 641-648, (1998)
[20] Stèphane, G.C.; Philippe, R.; Christian, F.; Benjamin, D.M.; David, A.D., Acoustical monitoring of fish density, behavior, and growth rate in a tank, Aquaculture, 251, 314-323, (2006)
[21] Chen, L.S., Pest control and geometric theory of semi-continuous dynamical system, J. Beihua Univ., 12, 1-9, (2011)
[22] Chen, G.Q., New approach to prove the nonexistence of limit cycle and its application, Acta Math. Sin., 20, 281-284, (1977) · Zbl 0417.34056
[23] Qu, Y.; Wei, J.J., Bifurcation analysis in a time-delay model for prey-predator growth with stage-structure, Nonlinear Dyn., 49, 285-294, (2007) · Zbl 1176.92056
[24] Wang, J.N.; Jiang, W.H., Bifurcation and chaos of a delayed predator-prey model with dormancy of predators, Nonlinear Dyn., 69, 1541-1558, (2012) · Zbl 1263.34063
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