Complexity and chaos control in a discrete-time prey-predator model. (English) Zbl 1510.92158
Summary: We investigate the complex behavior and chaos control in a discrete-time prey-predator model. Taking into account the Leslie-Gower prey-predator model, we propose a discrete-time prey-predator system with predator partially dependent on prey and investigate the boundedness, existence and uniqueness of positive equilibrium and bifurcation analysis of the system by using center manifold theorem and bifurcation theory. Various feedback control strategies are implemented for controlling the bifurcation and chaos in the system. Numerical simulations are provided to illustrate theoretical discussion.
MSC:
| 92D25 | Population dynamics (general) |
Keywords:
prey-predator model; stability analysis; period-doubling bifurcation; Neimark-Sacker bifurcation; chaos controlReferences:
| [1] | Murdoch, W.; Briggs, C.; Nisbet, R., Consumer-resource dynamics (2003), Princeton University Press: Princeton University Press New York |
| [2] | Ricklefs, R. E.; Miller, G. L., Ecology (2000), Freeman: Freeman New York |
| [3] | Linda, J.; Allen, S., An introduction to mathematical biology (2007), Prentice Hall |
| [4] | Brauer, F.; Castillo-Chavez, C., Mathematical models in population biology and epidemiology (2000), Springer |
| [5] | Edelstein-Keshet, L., Mathematical models in biology (1988), McGraw-Hill · Zbl 0674.92001 |
| [6] | Elaydi, S., Discrete chaos: with applications in science and engineering (2008), Chapman and Hall/CRC · Zbl 1153.39002 |
| [7] | Leslie, P. H., Some further notes on the use of matrices in population mathematics, Biometrica, 35, 213-245 (1948) · Zbl 0034.23303 |
| [8] | Rosenzweig, M. L.; MacArthur, R. H., Graphical representation and stability conditions of predator-prey interactions, Am Naturalist, 97, 209-223 (1963) |
| [9] | Sun, G.-Q.; Zhang, J.; Song, L.-P.; Jin, Z.; Li, B.-L., Pattern formation of a spatial predator-prey system, Appl Math Comput, 218, 11151-11162 (2012) · Zbl 1278.92041 |
| [10] | Hu, J.-H.; Xue, Y.-K.; Sun, G.-Q.; Jin, Z.; Zhang, J., Global dynamics of a predator-prey system modeling by metaphysiological approach, Appl Math Comput, 283, 369-384 (2016) · Zbl 1410.92100 |
| [11] | Sun, G.-Q.; Wu, Z.-Y.; Wang, Z.; Jin, Z., Influence of isolation degree of spatial patterns on persistence of populations, Nonlinear Dyn, 83, 811-819 (2016) |
| [12] | Sun, G.-Q., Mathematical modeling of population dynamics with allee effect, Nonlinear Dyn, 85, 1, 1-12 (2016) |
| [13] | Li, L.; Wang, Z.-J., Global stability of periodic solutions for a discrete predator-prey system with functional response, Nonlinear Dyn, 72, 507-516 (2013) · Zbl 1269.92070 |
| [14] | Aziz-Alaoui, A.; Okiye, M. D., Boundedness and global stability for a predator-prey model with modified Leslie-Gower and holling-type II schemes, Appl Math Lett, 16, 7, 1069-1075 (2003) · Zbl 1063.34044 |
| [15] | Du, Y.; Peng, R.; Wang, M., Effect of a protection zone in the diffusive leslie predator-prey model, J Differ Equ, 246, 10, 3932-3956 (2009) · Zbl 1168.35444 |
| [16] | Zhu, Y.; Wang, K., Existence and global attractivity of positive periodic solutions for a predator-prey model with modified Leslie-Gower holling-type II schemes, J Math Anal Appl, 384, 400-408 (2011) · Zbl 1232.34077 |
| [17] | Ji, C.; Jiang, D.; Shi, N., Analysis of a predator-prey model with modified Leslie-Gower and holling-type II schemes with stochastic perturbation, J Math Anal Appl, 359, 482-498 (2009) · Zbl 1190.34064 |
| [18] | Ji, C.; Jiang, D.; Shi, N., A note on a predator-prey model with modified Leslie-Gower and holling-type II schemes with stochastic perturbation, J Math Anal Appl, 377, 1, 435-440 (2011) · Zbl 1216.34040 |
| [19] | Gupta, R. P.; Chandra, P., Bifurcation analysis of modified Leslie-Gower predator-prey model with Michaelis-Menten type prey harvesting, J Math Anal Appl, 398, 278-295 (2013) · Zbl 1259.34035 |
| [20] | Yang, X., Uniform persistence and periodic solutions for a discrete predator-prey system with delays, J Math Anal Appl, 316, 161-177 (2006) · Zbl 1107.39017 |
| [21] | Singh, H.; Dhar, J.; Bhatti, H. S., Discrete-time bifurcation behavior of a prey-predator system with generalized predator, Adv Differ Equ, 2015, 206 (2015) · Zbl 1422.92129 |
| [22] | Sun, G.-Q.; Zhang, G.; Jin, Z., Dynamic behavior of a discrete modified Ricker and Beverton-Holt model, Comput Math Appl, 57, 1400-1412 (2009) · Zbl 1186.34074 |
| [23] | Din, Q.; Donchev, T., Global character of a host-parasite model, Chaos Soliton Fract, 54, 1-7 (2013) · Zbl 1341.37058 |
| [24] | Din, Q., Global stability of a population model, Chaos Soliton Fract, 59, 119-128 (2014) · Zbl 1348.92123 |
| [25] | Din, Q., Global behavior of a plant-herbivore model, Adv Differ Equ, 2015, 119 (2015) · Zbl 1422.92112 |
| [26] | Din, Q., Global behavior of a host-parasitoid model under the constant refuge effect, Appl Math Model, 40, 4, 2815-2826 (2016) · Zbl 1452.92032 |
| [27] | Balreira, E. C.; Elaydi, S.; Luís, R., Local stability implies global stability for the planar ricker competition model, Discrete Contin Dyn Syst Ser-B, 19, 2, 323-351 (2014) · Zbl 1282.39016 |
| [28] | Balreira, E. C.; Elaydi, S.; Luís, R., Global dynamics of triangular maps, Nonlinear Anal-Theory, 104, 75-83 (2014) · Zbl 1345.37025 |
| [29] | Din, Q., Dynamics of a discrete Lotka-Volterra model, Adv Differ Equ, 2013, 95 (2013) · Zbl 1380.39004 |
| [30] | Din, Q.; Elsayed, E. M., Stability analysis of a discrete ecological model, Comput Ecol Softw, 4, 2, 89-103 (2014) |
| [31] | Din, Q.; Ibrahim, T. F.; Khan, K. A., Behavior of a competitive system of second-order difference equations, Sci World J, 9 (2014) |
| [32] | Din, Q., Stability analysis of a biological network, Netw Biol, 4, 3, 123-129 (2014) |
| [33] | Din, Q.; Khan, K. A.; Nosheen, A., Stability analysis of a system of exponential difference equations, Discrete Dyn Nat Soc, 11 (2014) · Zbl 1419.39037 |
| [34] | Din, Q., Asymptotic behavior of an anti-competitive system of second-order difference equations, J Egyptian Math Soc, 24, 37-43 (2016) · Zbl 1332.39014 |
| [35] | Din, Q.; Khan, M. A.; Saeed, U., Qualitative behaviour of generalised beddington model, Z Naturforsch A, 71, 2, 145-155 (2016) |
| [36] | Din, Q., Qualitative behavior of a discrete SIR epidemic model, Int J Biomath, 9, 6, 15 (2016) · Zbl 1346.92068 |
| [37] | Din, Q., Global stability of beddington model, Qual Theor Dyn Syst, 25 (2016) |
| [38] | Din, Q., Global stability and Neimark-Sacker bifurcation of a host-parasitoid model, Int J Syst Sci, 9 (2016) |
| [39] | He, Z.; Lai, X., Bifurcation and chaotic behavior of a discrete-time predator-prey system, Nonlinear Anal RWA, 12, 403-417 (2011) · Zbl 1202.93038 |
| [40] | Liu, X.; Xiao, D., Complex dynamic behaviors of a discrete-time predator-prey system, Chaos Soliton Fract, 32, 80-94 (2007) · Zbl 1130.92056 |
| [41] | Jing, Z.; Yang, J., Bifurcation and chaos in discrete-time predator-prey system, Chaos Soliton Fract, 27, 259-277 (2006) · Zbl 1085.92045 |
| [42] | Chen, G.; Dong, X., From chaos to order: perspectives, methodologies, and applications (1998), World Scientific: World Scientific Singapore · Zbl 0908.93005 |
| [43] | Elaydi, S. N., An introduction to difference equations (2005), Springer-Verlag: Springer-Verlag New York · Zbl 1071.39001 |
| [44] | Lynch, S., Dynamical systems with applications using mathematica (2007), Birkhäuser: Birkhäuser Boston · Zbl 1138.37001 |
| [45] | Wang, T.; Wang, X.; Wang, M., Chaotic control of hénon map with feedback and nonfeedback methods, Commun Nonlinear Sci Numer Simulat, 16, 3367-3374 (2011) · Zbl 1221.93119 |
| [46] | Michel, A. N.; Hou, L.; Liu, D., Stability of dynamical systems: continuous, discontinuous, and discrete systems (2007), Boston: Boston Birkhäuser |
| [47] | Ott, E.; Grebogi, C.; Yorke, J. A., Controlling chaos, Phys Rev Letters, 64, 11, 1196-1199 (1990) · Zbl 0964.37501 |
| [48] | Romeiras, F. J.; Grebogi, C.; Ott, E.; Dayawansa, W. P., Controlling chaotic dynamical systems, Physica D, 58, 165-192 (1992) · Zbl 1194.37140 |
| [49] | Ogata, K., Modern control engineering (1997), PrenticeHall: PrenticeHall Englewood, New Jersey |
| [50] | Guckenheimer, J.; Holmes, P., Nonlinear oscillations, dynamical systems, and bifurcations of vector fields (1983), Springer-Verlag, New York · Zbl 0515.34001 |
| [51] | Robinson, C., Dynamical systems: stability, symbolic dynamics and chaos (1999), Boca Raton: Boca Raton New York · Zbl 0914.58021 |
| [52] | Wiggins, S., Introduction to applied nonlinear dynamical systems and chaos (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1027.37002 |
| [53] | Wan, Y. H., Computation of the stability condition for the hopf bifurcation of diffeomorphism on \(r^2\), SIAM J Appl Math, 34, 167-175 (1978) · Zbl 0389.58008 |
| [54] | Kuznetsov, Y. A., Elements of applied bifurcation theory (1997), Springer-Verlag: Springer-Verlag New York |
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