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

  • Waleed AzizEmail author


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.


Three dimensional Lotka–Volterra system Darboux integrability Linearizability First integral Inverse Jacobi multiplier Monodromy 

Mathematics Subject Classification

Primary 37K10 Secondary 34A05 



We would like to thank the referees for their valuable comments, suggestions and efforts towards improving the original manuscript.


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Authors and Affiliations

  1. 1.Mathematics Department, College of ScienceSalahaddin University-ErbilErbilIraq
  2. 2.Department of Computer Engineering, College of Engineering and Computer ScienceLebanese French UniversityErbilIraq

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