Dynamic analysis and chaos control of an environmental population model incorporating the wind flow in a stochastic and deterministic perspective

A novel prey-predator model by adding a wind-flow factor is presented. The topological types of the fixed points are calculated in the first stage. Four conditions are derived for the two calculated fixed points, each corresponding to a different classification. All positive solutions of the model are proved to be bounded, and there exists only one positive interior fixed point. Furthermore, the period-doubling and Neimark-Sacker bifurcations are obtained. Both types of bifurcations are verified by mathematical conditions and plots. In one-parameter bifurcation, the presence of both periodic behaviour and quasiperiodic oscillations indicates that the wind-induced dispersal affects on the prey-predator interaction. However, for the two-parameter plots, the chaotic, periodic, and quasiperiodic regions were obtained, highlighting the complex effects of the wind flow factor on the model dynamics. Specifically, in a stochastic environment, the conditions for local and global stability were derived. In this graphical abstract, the complete dynamics of the ecological population are presented. The discrete and stochastic forms of the continuous-time model were chosen to analyze the population dynamics when a particular wind flow affects it. In the discrete case, only two fixed points were obtained, and the classifications in terms of the stability of both fixed points were obtained through the Routh-Hurwitz criterion. It is observed that when both populations coexist, dynamic complexities arise, resulting in period-doubling and Neimark-Sacker-type bifurcations in the model. Further analysis of the coexisting population using two-parameter dynamics reveals more complex results, confirming chaotic windows, periodic windows, and well-known periodic structures such as shrimps and Arnold tongues. While observing the real dynamics, the model was formulated in stochastic form. The stochastic form of the model is observed to be locally and globally stable. In the end, the rich dynamics of the model under discussion and comprehensive numerical results were obtained.