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Bifurcations of a two-dimensional discrete-time predator-prey model. (English) Zbl 07020831
Summary: We study the local dynamics and bifurcations of a two-dimensional discrete-time predator-prey model in the closed first quadrant $$\mathbb{R}_{+}^{2}$$. It is proved that the model has two boundary equilibria: $$O(0,0)$$, $$A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$$ and a unique positive equilibrium $$B (\frac{1}{\alpha _{2}},\frac{ \alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$$ under some restriction to the parameter. We study the local dynamics along their topological types by imposing the method of linearization. It is proved that a fold bifurcation occurs about the boundary equilibria: $$O(0,0)$$, $$A (\frac{\alpha _{1}-1}{\alpha _{1}},0 )$$ and a period-doubling bifurcation in a small neighborhood of the unique positive equilibrium $$B (\frac{1}{\alpha _{2}},\frac{\alpha _{1} \alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$$. It is also proved that the model undergoes a Neimark-Sacker bifurcation in a small neighborhood of the unique positive equilibrium $$B (\frac{1}{ \alpha _{2}},\frac{\alpha _{1}\alpha _{2}-\alpha _{1}-\alpha _{2}}{\alpha _{2}} )$$ and meanwhile a stable invariant closed curve appears. From the viewpoint of biology, the stable closed curve corresponds to the periodic or quasi-periodic oscillations between predator and prey populations. Numerical simulations are presented to verify not only the theoretical results but also to exhibit the complex dynamical behavior such as the period-2, -4, -11, -13, -15 and -22 orbits. Further, we compute the maximum Lyapunov exponents and the fractal dimension numerically to justify the chaotic behaviors of the discrete-time model. Finally, the feedback control method is applied to stabilize chaos existing in the discrete-time model.

##### MSC:
 39A10 Additive difference equations 40A05 Convergence and divergence of series and sequences 92D25 Population dynamics (general) 70K50 Bifurcations and instability for nonlinear problems in mechanics 35B35 Stability in context of PDEs
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##### References:
 [1] Volterra, V.: Leçons sur la Théorie Mathématique de la Lutte pour la Vie. Gauthier-Villars, Paris (1931) · Zbl 0002.04202 [2] Martelli, M.: Discrete Dynamical Systems and Chaos. Longman, New York (1992) · Zbl 0819.34001 [3] Landa, P.S.: Self oscillatory models of some natural and technical processes. In: Kreuzer, E., Schmidt, G. (eds.) Mathematical Research, vol. 72, p. 23 · Zbl 0800.00019 [4] Freedman, H. I., A model of predator-prey dynamics as modified by the action of a parasite, Math. Biosci., 99, 143-155, (1990) · Zbl 0698.92024 [5] May, R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (2001) · Zbl 1044.92047 [6] Volterra, V., Fluctuations in the abundance of a species considered mathematically, Nature, 118, 558-560, (1926) · JFM 52.0453.03 [7] Alebraheem, J.; Hasan, Y. A., Dynamics of a two predator-one prey system, Comput. Appl. Math., 33, 67-780, (2014) · Zbl 1310.93015 [8] Sinha, S.; Misra, O.; Dhar, J., Modelling a predator-prey system with infected prey in polluted environment, Appl. Math. Model., 34, 1861-1872, (2010) · Zbl 1193.34071 [9] Chen, Y.; Changming, S., Stability and Hopf bifurcation analysis in a prey-predator system with stage-structure for prey and time delay, Chaos Solitons Fractals, 38, 1104-1114, (2008) · Zbl 1152.34370 [10] Gakkhar, S.; Singh, A., Complex dynamics in a prey-predator system with multiple delays, Commun. Nonlinear Sci. Numer. Simul., 17, 914-929, (2012) · Zbl 1243.92051 [11] Yan, J.; Li, C.; Chen, X.; Ren, L., Dynamic complexities in 2-dimensional discrete-time predator-prey systems with Allee effect in the prey, Discrete Dyn. Nat. Soc., 2016, (2016) · Zbl 1376.92055 [12] Zhao, J.; Yan, Y., Stability and bifurcation analysis of a discrete predator-prey system with modified Holling-Tanner functional response, Adv. Differ. Equ., 2018, (2018) [13] Fang, Q.; Li, X., Complex dynamics of a discrete predator-prey system with a strong Allee effect on the prey and a ratio-dependent functional response, Adv. Differ. Equ., 2018, (2018) [14] Kangalgi, F.; Kartal, S., Stability and bifurcation analysis in a host-parasitoid model with Hassell growth function, Adv. Differ. Equ., 2018, (2018) [15] Li, L.; Shen, J., Bifurcations and dynamics of a predator-prey model with double Allee effects and time delays, Int. J. Bifurc. Chaos, 28, 1-14, (2018) · Zbl 1404.34092 [16] Zhao, M.; Li, C.; Wang, J., Complex dynamic behaviors of a discrete-time predator-prey system, J. Appl. Anal. Comput., 7, 478-500, (2017) [17] Cheng, L.; Cao, H., Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee effect, Commun. Nonlinear Sci. Numer. Simul., 38, 288-302, (2016) [18] Liu, W.; Cai, D.; Shi, J., Dynamic behaviors of a discrete-time predator-prey bioeconomic system, Adv. Differ. Equ., 2018, (2018) [19] Liu, X.; Chu, Y.; Liu, Y., Bifurcation and chaos in a host-parasitoid model with a lower bound for the host, Adv. Differ. Equ., 2018, (2018) · Zbl 1445.37065 [20] Sohel Rana, S. M., Chaotic dynamics and control of discrete ratio-dependent predator-prey system, Discrete Dyn. Nat. Soc., 2017, (2017) · Zbl 1370.92146 [21] Zhao, M.; Du, Y., Stability of a discrete-time predator-prey system with Allee effect, Nonlinear Analy. Diff. Equ., 4, 225-233, (2016) [22] Liu, X.; Xiao, D., Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Solitons Fractals, 32, 80-94, (2007) · Zbl 1130.92056 [23] Khan, A. Q.; Ma, J.; Xiao, D., Bifurcations of a two-dimensional discrete time plant-herbivore system, Commun. Nonlinear Sci. Numer. Simul., 39, 185-198, (2016) [24] Khan, A. Q.; Ma, J.; Xiao, D., Global dynamics and bifurcation analysis of a host-parasitoid model with strong Allee effect, J. Biol. Dyn., 11, 121-146, (2017) [25] Khan, A. Q., Stability and Neimark-Sacker bifurcation of a ratio-dependence predator-prey model, Math. Methods Appl. Sci., 40, 3833-4232, (2017) · Zbl 1369.39011 [26] Hu, Z.; Teng, Z.; Zhang, L., Stability and bifurcation analysis of a discrete predator-prey model with non-monotonic functional response, Nonlinear Anal., Real World Appl., 12, 2356-2377, (2011) · Zbl 1215.92063 [27] Jing, Z.; Yang, J., Bifurcation and chaos in discrete-time predator-prey system, Chaos Solitons Fractals, 27, 259-277, (2006) · Zbl 1085.92045 [28] Zhang, C. H.; Yan, X. P.; Cui, G. H., Hopf bifurcations in a predator-prey system with a discrete delay and a distributed delay, Nonlinear Anal., Real World Appl., 11, 4141-4153, (2010) · Zbl 1206.34104 [29] Sen, M.; Banerjee, M.; Morozov, A., Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect, Ecol. Complex., 11, 12-27, (2012) [30] Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields. Springer, New York (1983) · Zbl 0515.34001 [31] Kuznetsov, Y.A.: Elements of Applied Bifurcation Theory, 3rd edn. Springer, New York (2004) · Zbl 1082.37002 [32] Khan, A. Q., Supercritical Neimark-Sacker bifurcation of a discrete-time Nicholson-Bailey model, Math. Methods Appl. Sci., 41, 4841-4852, (2018) · Zbl 1394.39013 [33] Cartwright, J. H.E., Nonlinear stiffness Lyapunov exponents and attractor dimension, Phys. Lett. A, 264, 298-304, (1999) · Zbl 0949.37014 [34] Kaplan, J. L.; Yorke, J. A., Preturbulence: a regime observed in a fluid flow model of Lorenz, Commun. Math. Phys., 67, 93-108, (1979) · Zbl 0443.76059 [35] Elaydi, S.N.: An Introduction to Difference Equations. Springer, New York (1996) · Zbl 0840.39002 [36] Lynch, S.: Dynamical Systems with Applications Using Mathematica. Birkhäuser, Boston (2007) · Zbl 1138.37001
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