Rough Center In A 3-Dimensional Lotka-Volterra System

This paper identifies rough center for a Lotka-Volterra system, a 3-dimensional quadratic polynomial differential system with four parameters h, n, lambda, mu. The known work shows the appearance of four limit cycles, but centre condition is not determined. In this paper, we verify the existence of at least four limit cycles in the positive equilibrium due to Hopf bifurcation by computing normal forms. Furthermore, by applying algorithms of computational commutative algebra we find Darboux polynomial and give a centre manifold in closed form globally, showing that the positive equilibrium of centre-focus is actually a rough center on a centre manifold.