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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