Novel organizational patterns of stability phases in a single-species population model: chiral tree, spikes adding-doubling complexification cascade

We numerically investigate the complex dynamic behavior of a single-species population model with stage structure, considering the influence of the birth rate and birth pulse. Lyapunov stability diagram, isoperiodic diagram, and isospike diagram are employed to reveal the organizational patterns of complex dynamic behaviors. The isospike diagram is proven to be particularly effective in accurately representing the internal fine structure of stability oscillations. This enabled the discovery of certain novel organizational patterns, previously observed only in continuous systems, including the chiral tree, shrimp cascade covered by multiple spikes, and adding-doubling complexification cascade. Additionally, profiting from the powerful calculating ability of CPU+GPU heterogeneous parallel computing, we assess the effect of initial value perturbation on the topology of stability phases. The results indicate that the organizational patterns of the stability phases are primarily influenced by fine-tuning the control parameters rather than by the perturbation of initial values. The identification of these novel organizational patterns in our study provides valuable insights into complex behaviors in discrete systems. Furthermore, our findings open up exciting possibilities for investigating previously unobserved phenomena.