Dynamics analysis of autonomous and nonautonomous predator-prey models with nonlinear harvesting

In the paper, autonomous and nonautonomous predator-prey models with nonlinear harvesting and Beddington-DeAngelis functional response were proposed. The mathematical goal was to explore the evolution process of specific bifurcation dynamics, the existence, and attractiveness of positive periodic solutions. The ecological objective was to ascertain the population growth coexistence modes and their underlying driving mechanisms from a specific perspective of dynamic evolution. Regarding the autonomous predator-prey model, mathematical theoretical work has investigated the existence and local stability of all equilibrium points, as well as the occurrence of specific bifurcation dynamics. Regarding the nonautonomous predator-prey model, the boundedness of all solutions, the possibility and global attractiveness of a positive periodic solution were theoretically derived in detail. The numerical simulation work not only verified the feasibility of the theoretical derivation work, but also dynamically showed that the autonomous model had transcritical bifurcation, saddle-node bifurcation, Hopf bifurcation, and Bogdanov-Takens bifurcation, while the nonautonomous model had attractive periodic solutions. It was worth emphasizing that predator and prey had steady state constant growth coexistence mode and steady state periodic oscillation growth coexistence mode. It must also be pointed out that their intrinsic driving mechanisms were mainly the specific bifurcation dynamics evolution mechanism in autonomous model and seasonal disturbance of key ecological environment parameters in the nonautonomous model. In summary, it was expected that these research results would contribute to the rapid development of nonlinear dynamics in predator-prey models.