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Codimension-two bifurcations of fixed points in a class of discrete prey-predator systems. (English) Zbl 1229.92072
Summary: The dynamic behaviour of a Lotka-Volterra system, described by a planar map, is analytically and numerically investigated. We derive analytical conditions for stability and bifurcation of the fixed points of the system and compute analytically the normal form coefficients for the codimension 1 bifurcation points (flip and Neimark-Sacker), and so establish sub- or supercriticality of these bifurcation points. Furthermore, by using numerical continuation methods, we compute bifurcation curves of fixed points and cycles with periods up to 16 under variation of one and two parameters, and compute all codimension 1 and codimension 2 bifurcations on the corresponding curves. For the bifurcation points, we compute the corresponding normal form coefficients. These quantities enable us to compute curves of codimension 1 bifurcations that branch off from the detected codimension 2 bifurcation points. These curves form stability boundaries of various types of cycles which emerge around codimension 1 and 2 bifurcation points. Numerical simulations confirm our results and reveal further complex dynamical behaviours.

39A28Bifurcation theory (difference equations)
37N25Dynamical systems in biology
65C20Models (numerical methods)
39A60Applications of difference equations
Full Text: DOI
[1] A. A. Berryman, “The origins and evolution of predator-prey theory,” Ecology, vol. 73, no. 5, pp. 1530-1535, 1992.
[2] M. Fan and K. Wang, “Periodic solutions of a discrete time nonautonomous ratio-dependent predator-prey system,” Mathematical and Computer Modelling, vol. 35, no. 9-10, pp. 951-961, 2002. · Zbl 1050.39022 · doi:10.1016/S0895-7177(02)00062-6
[3] H. Fang and J. D. Cao, “Global existence for positive periodic solutions to a class of predator-prey systems,” Journal of Biomathematics, vol. 15, no. 4, pp. 403-407, 2000. · Zbl 1056.92515
[4] H. I. Freedman, Deterministic Mathematical Models in Population Ecology, vol. 57 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1980. · Zbl 0448.92023
[5] B. S. Goh, Management and Analysis of Biological Populations, Elsevier, Amsterdam, The Netherlands, 1980.
[6] J. D. Murray, Mathematical Biology, vol. 19 of Biomathematics, Springer, Berlin, Germany, 2nd edition, 1993. · Zbl 0779.92001
[7] A. D. Bazykin, Nonlinear Dynamics of Interacting Populations, vol. 11 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, A. I. Khibnik and B. Krauskopf, Eds., World Scientific, River Edge, NJ, USA, 1998.
[8] V. I. Arnold, Geometrical Methods in the Theory of Ordinary Differential Equations, vol. 250 of Grundlehren der Mathematischen Wissenschaften, Springer, New York, NY, USA, 1983. · Zbl 0507.34003
[9] Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, vol. 112 of Applied Mathematical Sciences, Springer, New York, NY, USA, 3rd edition, 2004. · Zbl 1082.37002
[10] H. W. Broer, K. Saleh, V. Naudot, and R. Roussarie, “Dynamics of a predator-prey model with non-monotonic response function,” Discrete and Continuous Dynamical Systems. Series A, vol. 18, no. 2-3, pp. 221-251, 2007. · Zbl 1129.92061 · doi:10.3934/dcds.2007.18.221
[11] E. J. Doedel, R. A. Champneys, T. F. Fairgrieve, Yu. A. Kuznetsov, B. Sandstede, and X. J. Wang, “AUTO2000: Continuation and Bifurcation Sotware for Ordinary Differential Equations (with HomCont), Users’ Guide,” Concordia University, Montreal, Canada 1997-2000, http://indy.cs.concordia.ca.
[12] W. Govaerts and Yu. A. Kuznetsov, “Matcont: a Matlab software project for the numerical continuation and bifurcation study of continuous and discrete parameterized dynamical systems,” http://sourceforge.net/.
[13] R. P. Agarwal, Difference Equations and Inequalities: Theory, Methods, and Applicationa, vol. 228 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 2nd edition, 2000. · Zbl 1160.34313 · doi:10.4171/ZAA/964 · eudml:55311
[14] W. Govaerts, R. K. Ghaziani, Yu. A. Kuznetsov, and H. G. E. Meijer, “Numerical methods for two-parameter local bifurcation analysis of maps,” SIAM Journal on Scientific Computing, vol. 29, no. 6, pp. 2644-2667, 2007. · Zbl 1155.65397 · doi:10.1137/060653858
[15] M. Danca, S. Codreanu, and B. Bakó, “Detailed analysis of a nonlinear prey-predator model,” Journal of Biological Physics, vol. 23, no. 1, pp. 11-20, 1997.
[16] K. Murakami, “Stability and bifurcation in a discrete-time predator-prey model,” Journal of Difference Equations and Applications, vol. 13, no. 10, pp. 911-925, 2007. · Zbl 1127.39020 · doi:10.1080/10236190701365888
[17] S. Li and W. Zhang, “Bifurcations of a discrete prey-predator model with Holling type II functional response,” Discrete and Continuous Dynamical Systems. Series B, vol. 14, no. 1, pp. 159-176, 2010. · Zbl 1200.37043 · doi:10.3934/dcdsb.2010.14.159
[18] E. L. Allgower and K. Georg, Numerical Continuation Methods: An Introduction, vol. 13 of Springer Series in Computational Mathematics, Springer, Berlin, Germany, 1990. · Zbl 0723.65032
[19] R. L. Kraft, “Chaos, Cantor sets, and hyperbolicity for the logistic maps,” The American Mathematical Monthly, vol. 106, no. 5, pp. 400-408, 1999. · Zbl 0992.37029 · doi:10.2307/2589144