Integrability and Linearizability Problems of Three Dimensional Lotka–Volterra Equations of Rank-2

Abstract

This paper deals with both integrability and linearizability problems of three dimensional Lotka–Volterra systems at the origin with rank-2. We give necessary conditions of a system with quadratic nonlinearities with $$(1:3:-\,1)$$ -resonance. We proved the sufficiency of these conditions by showing the existence of two functionally independent first integrals via the Darboux method with inverse Jacobi multiplier together with some other techniques such as two equations define a linearizable node and the third equation is linearizable using the power series argument as well as the monodromy method.