Integrability and Center Analysis of a Family of Lotka–Volterra Three‐Species Biological System

ABSTRACT The integrability problem of Lotka–Volterra three‐species biological system is investigated. We compute all invariant algebraic surfaces and exponential factors of the system. Based on Darboux method, we show the nonexistence of polynomial, rational, and Darboux first integrals for some specific values of parameters. We also prove that the system has no formal first integrals when some parameters are strictly positive. In contrast, the system is completely integrable when two of the involved parameters in the system are zero. We also obtain necessary and sufficient conditions for an equilibrium point of the system to be a center on the center manifold for system.