In this paper small amplitude limit cycles and the local bifurcation of critical periods for a quartic Kolmogrov system at the single positive equilibrium point (1, 1) are investigated. Firstly, through the computation of the singular point values, the conditions of the critical point (1, 1) to be a center and to be the highest degree fine singular point are derived respectively. Then, we prove that the maximum number of small amplitude limit cycles bifurcating from the equilibrium point (1, 1) is 7. Furthermore, through the computation of the period constants, the conditions of the critical point (1, 1) to be a weak center of finite order are obtained. Finally, we give respectively that the number of local critical periods bifurcating from the equilibrium point (1, 1) under the center conditions. It is the first example of a quartic Kolmogorov system with seven limit cycles and a quartic Kolmogorov system with four local critical periods created from a single positive equilibrium point.

中文翻译：

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四次Kolmogorov系统的小振幅极限环和临界周期的局部分叉

本文研究了单个正平衡点（1，1）上四次Kolmogrov系统的小幅度极限环和临界周期的局部分叉。首先，通过计算奇异点值，分别得出临界点（1，1）为中心和最高程度的精细奇异点的条件。然后，我们证明从平衡点（1，1）分叉的小幅度极限循环的最大数目为7。此外，通过计算周期常数，临界点（1，1）的条件为a。获得了有限阶的弱中心。最后，我们分别给出中心条件下从平衡点（1，1）分叉的局部临界期的数量。