Multiple bifurcations and periodic “bubbling” in a delay population model

In this paper, 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 explored the Neimark–Sacker bifurcation in the delay population model (two-dimension 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 effect in a nonlinear difference equation derived from the logistic map leads to much richer dynamic behavior, such as stable node→stable focus→an lower-dimensional closed invariant curve (quasi-periodic solution, limit cycle) or/and stable periodic solutions→chaotic attractor by cascading bubbles (the combination of potential period doubling and reverse period-doubling) and the sudden change between two different attractors, etc.