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In the paper, stability and bifurcation behaviors of the Bazykin's predator-prey ecosystem with Holling type Ⅱ functional response are studied theoretically and numerically. Mathematical theory works mainly give some critical threshold conditions to guarantee the existence and stability of all possible equilibrium points, and the occurrence of Hopf bifurcation and Bogdanov-Takens bifurcation. Numerical simulation works mainly display that the Bazykin's predator-prey ecosystem has complex dynamic behaviors, which also directly proves that the theoretical results are effective and feasible. Furthermore, it is easy to see from numerical simulation results that some key parameters can seriously affect the dynamic behavior evolution process of the Bazykin's predator-prey ecosystem. Moreover, limit cycle is proposed in view of the supercritical Hopf bifurcation. Finally, it is expected that these results will contribute to the dynamical behaviors of predator-prey ecosystem.

In the paper, we proposed a Bazykin’s predator-prey system to explore the equilibrium point and Bogdanov-Takens bifurcation problems. Firstly, we derived some key parameter threshold conditions to ensure that the Bazykin’s predator-prey system had a multiple focus of multiplicity one, weak focus of order 2, cusps of codimension 2 and a degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3. Furthermore, the distinction of two types of codimension 2 cusps was also discussed, which showed that the threshold of the two types of cusps could exhibit a cusp, which was a special case of the mentioned degenerate Bogdanov-Takens singularity (focus or center case) of codimension 3. Secondly, we systematically calculated that the Bazykin’s predator-prey system could undergo two types of Bogdanov-Takens bifurcations of codimension 2 and a degenerate focus type Bogdanov-Takens bifurcation of codimension 3. Finally, some numerical examples were implemented to verify the correctness and feasibility of mathematical theory derivation, which also directly showed all possible equilibrium points and Bogdanov-Takens bifurcations of Bazykin’s predator-prey system. In a word, all the research results could play an important theoretical support role in the study of controlling cyanobacteria bloom.

In this paper, complex dynamical behaviors of a predator-prey system with the Beddington–DeAngelis functional response and the Allee-like effect on predator were studied by qualitative analysis and numerical simulations. Theoretical derivations have given some sufficient and threshold conditions to guarantee the occurrence of transcritical, saddle-node, pitchfork, and nondegenerate Hopf bifurcations. Computer simulations have verified the feasibility and effectiveness of the theoretical results. In short, we hope that these works could provide a theoretical basis for future research of complexity in more predator-prey ecosystems.