Bifurcation and chaos in a discrete mutualism model with proportional harvesting rate

[۱]‎‌‎ الایدی، صابر، آشوب گسسته با کاربردهایی در علوم و مهندسی‌، مترجمین: محمدحسین اکرمی و رسول کاظمی‌، انتشارات دانشگاه یزد، (۱۳۹۵).
[2] ‎Kot, M., “Elements of mathematical ecology”. The Press Syndicate of the University of Cambridge, Cambridge University Press, Cambridge, UK (2001).‎‎
[3] Thompson, J. N., “The geographic mosaic of coevolution. Chicago”, IL: University of Chicago Press, (2005).‎ ‎‎
[4] Georgescu, P., Zhang‎, H., “Lyapunov functionals for two-species mutualisms”, Applied Mathematics and Computation 226 (2014), 754-764. ‎
[5] Goh, B.S., “Stability in models of mutualism”, The American Naturist, 113(1979), 261-275.‎
[6] León, C.V.D., “Lyapunov function for two-species cooperative systems”, Applied Mathematics and Computation, 219 (2012), 2493-2497.‎ ‎
[7] Ouncharoen, R.‎, Pinjai, S., Dumrongpokaphan, T., Lenbury, Y., “Global stability analysis of predator-prey model with harvesting and delay”, Thai Journal of Mathematics, 8(2012), 589-605.‎
[8] ‎ Atabaigi, A., “Multiple Bifurcations and Dynamics of a Discrete Time Predator-Prey System with Group Defense and Non-Monotonic Functional Response”‎, Differential Equations and Dynamical Systems, (2016) 1-26.
[9] Ahmad, R., “Global stability of two-species mutualism model with proportional harvesting”, Int. J. Adv. Appl. Sci. 4(2017), 74-7‎9. ‎‎
[10] Liu, X., Xiao, D., “Complex dynamic behaviors of a discrete-time predator-prey system”, Chaos Solitons Fractals 32 (2007), 80-94.‎‌‎
[11] ‎ Kuznetsov, Y.A., “Elements of applied bifurcation theory”, Third edition. Springer Science and Business Media, New York, 2016.
[12] Boeing, G., “Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction”, Systems, 4 (2016), 37. ‎
[13] Cencini, M.‎, Cecconi, F., Vulpiani, A., “Chaos From Simple models to complex systems”, World Scientific, 2010.
[14] ‎Alligood, K.T., Tim, D.S., Yorke, J., “Chaos”, Springer Berlin Heidelberg, (1997).‎‎‌‎
[15] He, J., Yu, S., Cai, J., “Numerical Analysis and Improved Algorithms for Lyapunov-Exponent Calculation of Discrete-Time Chaotic Systems”, International Journal of Bifurcation and Chaos, 26 (2016), 1650219‎.‎‌‎
[16] Gutiérrez, ‎ J. M., Iglesias, A., “Mathematica package for analysis and control of chaos in nonlinear systems”, Computers in Physics, 12 (1998), 608-619.‎
[17] ‎Lynch, S., “Dynamical Systems with Applications using MATLAB”, 2nd ed., Basel, Birkhäuser, 2014.
[18] Doyle, J. C., Francis, B., Tannenbaum, A., “Feedback Control Theory”, Macmillan Publishing Company, New York, (1992). ‎ ‎
[19] Hasegawa, Y., “Control Problems of Discrete-Time Dynamical Systems”, Springer, New York Dordrecht London, (2015). ‎‌‎