In the paper, a predator-prey model with the Allee effect and harvesting effort was proposed to explore the interaction mechanism between prey and predator. Under the framework of mathematical theory deduction, some conditions for the occurrence of transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations were derived with harvesting effort and the Allee effect as key parameters. Under the framework of bifurcation dynamics numerical simulation, the evolution process of specific bifurcation dynamics behavior was gradually visualized to reveal the influence mechanism of the Allee effect and harvesting effort. The research results indicated that the Allee effect and harvesting effort not only seriously affected the bifurcation dynamics essential characteristics of the model (1.3), but also could promote the formation of constant steady state and periodic oscillation persistent survival mode of prey and predator. Furthermore, it is worth noting that appropriate harvesting effort was beneficial for the formation of a sustainable survival cycle between prey and predator. In summary, we hoped that the research findings could contribute to the comprehensive promotion of bifurcation dynamics studies in the predator-prey model.

From the perspective of ecological control, harvesting behavior plays a crucial role in the ecosystem natural cycle. This paper proposes a diffusive predator-prey system with predator harvesting to explore the impact of harvesting on predatory ecological relationships. First, the existence and boundedness of system solutions were investigated and the non-existence and existence of non-constant steady states were obtained. Second, the conditions for Turing instability were given to further investigate the Turing patterns. Based on these conditions, the amplitude equations at the threshold of instability were established using weakly nonlinear analysis. Finally, the existence, direction, and stability of Hopf bifurcation were proven. Furthermore, numerical simulations were used to confirm the correctness of the theoretical analysis and show that harvesting has a strong influence on the dynamical behaviors of the predator-prey systems. In summary, the results of this study contribute to promoting the research and development of predatory ecosystems.

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.