The dynamical behaviors of a Leslie type predator–prey system are explored when the functional response is increasing for both predator and prey. Qualitative and quantitative analysis methods based on stability theory, bifurcation theory and numerical simulation are adopted. It is showed that the system is dissipative and permanent, and its solutions are bounded. Global stability of the unique positive equilibrium is investigated by constructing Dulac function and applying Poincaré–Bendixson theorem. The bifurcation behaviors are further explored and the number of limit cycles is determined. By calculating the first Lyapunov number and the first two focus values, it is proved that the positive equilibrium is not a center but a weak focus of multiplicity at most two, so the system undergoes Hopf bifurcation and Bautin bifurcation. The normal form of Bautin bifurcation is also obtained by introducing the complex system. Moreover, numerical simulations are run to demonstrate the validity of theoretical results.

当捕食者和猎物的功能反应都在增加时，探索了莱斯利型捕食者 - 猎物系统的动力学行为。采用基于稳定性理论、分岔理论和数值模拟的定性和定量分析方法。结果表明，该系统是耗散永久的，其解是有界的。通过构造杜拉克函数并应用庞加莱-本迪克森定理，研究了唯一正平衡的全局稳定性。进一步探索分岔行为并确定极限环数。通过计算第一个李雅普诺夫数和前两个焦点值，证明了正平衡不是中心而是重数最多为两个的弱焦点，因此系统经历了Hopf分岔和Bautin分岔。Bautin分岔的范式也是通过引入复系统得到的。此外，运行数值模拟以证明理论结果的有效性。