Polynomial, rational and analytic first integrals for a family of 3-dimensional Lotka-Volterra systems

Abstract

Abstract. We extend the study of the integrability done by Leach and Miritzis (J. Nonlinear Math. Phys. 13 (2006), 535-548) on the classical model of competition between three species studied by May and Leonard (SIAM J. Appl. Math. 29 (1975), 243-256), to all real values of the parameters. Additionally our results provide all polynomial, rational and analytic first integrals of this extended model. We also classify all the invariant algebraic surfaces of these models.

First integrals of Darboux type for a family of 3-dimensional Lotka–Volterra systems

“…In [8] the authors showed for the case a + b = −1, system (2) has also a first integral. We also note that the existence of first integrals for system (2) imply the existence of invariants for system (1).…”

“…Our main results on the polynomial integrability of system (2) were obtained in [8] and are: For proving our main result concerning the existence of first integrals of Darboux type we shall use the invariant algebraic surfaces of system (2). This is the basis of the Darboux theory of integrability.…”

“…In the next result obtained in [8] the authors characterize all the irreducible Darboux polynomials of system (2) with non-zero cofactor. Taking into account that system (2) is homogeneous, the study of the Darboux polynomials of system (2) with nonzero cofactor can be reduced to the study of the homogeneous Darboux polynomials with nonzero cofactor, for more details see [14].…”

On the dynamics of a class of Kolmogorov systems

“…when such differential systems have first integrals (see for instance [1,2,4,5,6,7,8,17,18,22]),or • in their periodic orbits (see for example [9,10,11,13,16,20,24,25,26]). …”

Darboux invariants for planar polynomial differential systems having an invariant conic

“…system (1) with m = 2, having two real or complex invariant straight lines taking into account their multiplicity was given in [2], and extensions to dimension 3 are given in [11]. Now we do the characterization of all polynomial differential systems in R 2 having an invariant conic and a Darboux invariant.…”