Limit cycles and singular point quantities for a 3D Lotka–Volterra system

Four limit cycles are constructed for a three dimensional Lotka–Volterra system. This gives a good example to the cyclicity of 3D Lotka–Volterra systems. A recursion formula for computation of the singular point quantities is given for the corresponding Hopf bifurcation equation. What is worth mentioning is that the expressions of focal values are simpler, and the formula is readily done with using computer symbol operation system such as Mathematica due to its linearity.