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Stability and Neimark-Sacker bifurcation of a ratio-dependence predator-prey model. (English) Zbl 1369.39011
Summary: In this paper, stability and bifurcation of a two-dimensional ratio-dependence predator-prey model has been studied in the close first quadrant \(\mathbb R^2_+\). It is proved that the model undergoes a period-doubling bifurcation in a small neighborhood of a boundary equilibrium and moreover, Neimark-Sacker bifurcation occurs at a unique positive equilibrium. We study the Neimark-Sacker bifurcation at unique positive equilibrium by choosing \(b\) as a bifurcation parameter. Some numerical simulations are presented to illustrate the theoretical results.

MSC:
39A28 Bifurcation theory for difference equations
39A30 Stability theory for difference equations
92D25 Population dynamics (general)
65Q10 Numerical methods for difference equations
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[1] Ahmad, On the nonautonomous Lotka-Volterra competition equation, Proceedings of the American Mathematical Society 117 pp 199– (1993) · Zbl 0848.34033 · doi:10.1090/S0002-9939-1993-1143013-3
[2] Tang, On positive periodic solutions of Lotka-Volterra competition systems with deviating arguments, Proceedings of the American Mathematical Society 134 pp 2967– (2006) · Zbl 1101.34056 · doi:10.1090/S0002-9939-06-08320-1
[3] Zhou, Stable periodic solutions in a discrete periodic logistic equation, Applied Mathematics Letters 16 (2) pp 165– (2003) · Zbl 1049.39017 · doi:10.1016/S0893-9659(03)80027-7
[4] Liu, A note on the existence of periodic solution in discrete predator-prey models, Applied Mathematical Modelling 34 pp 2477– (2010) · Zbl 1195.39004 · doi:10.1016/j.apm.2009.11.012
[5] Fan, Permanence for a delayed discrete ratio-dependent predator-prey system with Holling type functional response, Journal of Mathematical Analysis and Applications 299 pp 357– (2004) · Zbl 1063.39013 · doi:10.1016/j.jmaa.2004.02.061
[6] Fan, Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system, Mathematical and Computer Modelling 35 pp 951– (2002) · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6
[7] Fazly, Periodic solutions for discrete time predator-prey system with monotone functional responses, Comptes Rendus de l’Académie des Sciences - Series I 345 pp 199– (2007)
[8] Hu, Four positive periodic solutions of a discrete time delayed predator-prey system with nonmonotonic functional response and harvesting, Computers & Mathematics with Applications 56 pp 3015– (2008) · Zbl 1165.34400 · doi:10.1016/j.camwa.2008.09.009
[9] Xia, Discrete-time analogues of predator-prey models with monotonic or nonmonotonic functional responses, Nonlinear Analysis: Real World Applications 8 pp 1079– (2007) · Zbl 1127.39038 · doi:10.1016/j.nonrwa.2006.06.007
[10] Yang, Permanence for a delayed discrete ratio-dependent predator-prey model with monotonic functional responses, Nonlinear Analysis: Real World Applications 10 pp 1068– (2009) · Zbl 1167.34359 · doi:10.1016/j.nonrwa.2007.11.022
[11] Khan, Dynamics of a modified Nicholson-Bailey host-parasitoid model, Ad. Differential Equations 2015 pp 23– (2015) · Zbl 1390.39014
[12] Khan, Behavior of an exponential system of difference equations, Discrete Dynamics in Nature and Society
[13] Khan, Qualitative behavior of two systems of higher-order difference equations, Mathematical Methods in the Applied Sciences 39 (11) pp 3058– (2016) · Zbl 1344.39003 · doi:10.1002/mma.3752
[14] Khan, Global dynamics of two systems of exponential difference equations by Lyapunov function, Ad. Differential equation 1 (2014) pp 297– (2014)
[15] Khan, Global dynamics of a competitive system of rational difference equations, Mathematical Methods in the Applied Sciences 38 (18) pp 4786– (2015) · Zbl 1346.39017 · doi:10.1002/mma.3392
[16] Qureshi, Local stability of an open-access anchovy fishery model, Computational Ecology and Software 5 (1) pp 48– (2015)
[17] Khan, Stability analysis of a discrete biological model, International Journal of Biomathematics 9 (2) pp 1– (2015)
[18] Kalabuŝić, Global dynamics of a competitive system of rational difference equations in the plane, Advances in Difference Equations (2009) · Zbl 1187.39024 · doi:10.1155/2009/132802
[19] Kalabuŝić, Dynamics of a two-dimensional system of rational difference equations of Leslie-Gower type, Advances in Difference Equations (2011) · Zbl 1384.37031
[20] Elsayed, Solutions and periodicity of a rational recursive sequences of order five, Bulletin of the Malaysian Mathematical Sciences Society 38 (1) pp 95– (2015) · Zbl 1308.39002 · doi:10.1007/s40840-014-0005-0
[21] Garić-Demirović, Global behavior of four competitive rational systems of difference equations in the plane, Discrete Dynamics in Nature and Society 2009
[22] Kalabuŝić, Multiple attractors for a competitive system of rational difference equations in the plane, Abstract and Applied Analysis (2011) · Zbl 1252.39020 · doi:10.1155/2011/295308
[23] Touafek, On the solutions of systems of rational difference equations, Mathematical and Computer Modelling 55 pp 1987– (2012) · Zbl 1255.39011 · doi:10.1016/j.mcm.2011.11.058
[24] Elsayed, Solutions of rational difference system of order two, Mathematical and Computer Modelling 55 pp 378– (2012) · Zbl 1255.39003 · doi:10.1016/j.mcm.2011.08.012
[25] Elsayed, Behavior and expression of the solutions of some rational difference equations, Journal of Computational Analysis and Applications 15 (1) pp 73– (2013)
[26] Hu, Stability and bifurcation analysis of a discrete predator-prey model with nonmonotonic functional response, Nonlinear Analysis: Real World Applications 12 (4) pp 2356– (2011) · Zbl 1215.92063 · doi:10.1016/j.nonrwa.2011.02.009
[27] Jing, Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals 27 (1) pp 259– (2006) · Zbl 1085.92045 · doi:10.1016/j.chaos.2005.03.040
[28] Zhang, Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Analysis: Real World Applications 11 (5) pp 4141– (2010) · Zbl 1206.34104 · doi:10.1016/j.nonrwa.2010.05.001
[29] Sen, Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecological Complexity 11 pp 12– (2012) · doi:10.1016/j.ecocom.2012.01.002
[30] Khan, Neimark-Sacker bifurcation of a two-dimensional discrete-time predator-prey model, Springer Plus (2016) · doi:10.1186/s40064-015-1618-y
[31] Guckenheimer, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector fields (1983) · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[32] Kuznetson, Elements of Applied Bifurcation Theory (2004) · doi:10.1007/978-1-4757-3978-7
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