Bifurcation of limit cycles for 3D Lotka-Volterra competitive systems
The paper analyzes the number of limit cycles in a three-dimensional competitive Lotka-Volterra system. The authors refer to earlier results on Hopf bifurcation in such systems, such as obtained by M. L. Zeeman, J. Hofbauer and J. W.-H. So, and moreover, by Q. Wang, Y. Liu, H. Chen. To calculate the focal values, they use a recursion algorithm that avoids the usual center manifold procedure at an equilibrium with a pair of purely imaginary eigenvalues. For concrete calculations, they treat examples with four parameters and apply the computer algebra system \textit{Mathematica} to determine systems whose first two focal values vanish. This implies the existence of perturbed systems with three limit cycles in six cases of Zeeman's classification. In three of these cases, moreover, the Poincaré-Bendixson theorem yields a fourth limit cycle.
scientific article; zbMATH DE number 5899552