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Multiple bifurcations and periodic “bubbling” in a delay population model. (English) Zbl 1065.92035
Summary: The flip bifurcation and periodic doubling bifurcations of a discrete population model without delay influence is firstly studied and the phenomenon of Feigenbaum’s cascade of periodic doublings is also observed. Secondly, we explore the Neimark-Sacker bifurcation in the delay population model (two-dimensional discrete dynamical systems) and the unique stable closed invariant curve which bifurcates from the nontrivial fixed point. Finally, a computer-assisted study for the delay population model is also delved into. Our computer simulation shows that the introduction of delay effects in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node $\to$ stable focus $\to$ lower-dimensional closed invariant curves (quasi-periodic solutions, limit cycles) or/and stable periodic solutions $\to$ chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling), the sudden change between two different attractors, etc.

##### MSC:
 92D25 Population dynamics (general) 39A10 Additive difference equations 37N25 Dynamical systems in biology
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##### References:
 [1] Aroson, D. G.; Chory, M. A.; Hall, G. R.; Mcgehee, R. P.: Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study. Commun math phys 83, 303-354 (1982) · Zbl 0499.70034 [2] Chow, S. N.; Hale, J. K.: Methods of bifurcations theory. (1996) [3] Devaney, R.: An introduction to chaotic dynamical systems. (1986) · Zbl 0632.58005 [4] Hale, J. K.; Koçak, H.: Dynamics and bifurcations. (1993) [5] Huang, L.; Peng, M. S.: Qualitative analysis of a discrete population model. Math acta scientia 19, No. 1, 45-52 (1999) · Zbl 0930.39013 [6] Yuri, A. Kuznetsov: Elements of applied bifurcation theory. (1998) · Zbl 0914.58025 [7] Peng, M. S.: Rich dynamics of discrete delay ecological models. Chaos, solitons and fractals 24, No. 5, 1279-1285 (2004) · Zbl 1082.39014 [8] Peng, M. S.; Bai, E. W.; Lonngren, K. E.: On the synchronization of delay discrete models. Chaos, solitons and fractals 22, No. 3, 573-576 (2004) · Zbl 1060.93543 [9] Peng, M. S.; Huang, L.: Asymptotic stability and oscillation in a discrete population model with nonlinearity. J. Beijing institute of tech. (Naturel edition, in chinese) 19, No. 2, 150-156 (1999) · Zbl 0946.92022 [10] Pounder, J. R.; Rogers, T. D.: The geometry of chaos: dynamics of a nonlinear second-order difference equation. Bull mathematical biol 42, 551-597 (1980) · Zbl 0439.39001 [11] Roger, T. D.; Clarke, B. L.: A continuous map with many periodic points. Appl math comput 8, 17-33 (1981) · Zbl 0475.39005 [12] Stone, L.: Period-doubling reversals and chaos in simple ecological models. Nature 363, 411-441 (1993) [13] Thompson, J. M. T.; Stewart, H. B.: Nonlinear dynamics and chaos: geometrical methods for engineers and scientists. (1986) · Zbl 0601.58001 [14] Vandermeer, J. H.: Periodic ’bubbling’ in a simple ecological models: pattern and chaos formation in aquartic model. Ecol model 95, 311-317 (1997)