On the integrability of some three-dimensional Lotka-Volterra equations with rank-$1$ resonances
We investigate the local integrability in C3 of some three-dimensional Lotka–Volterra equations at the origin with (p : q : r)-resonance, x˙ = P = x(p + ax + by + cz), y˙ = Q = y(q + dx + ey + fz), z˙ = R = z(r + gx + hy + kz). Recent work on this problem has centered on the case where the resonance is of “rank-2”. That is, there are two independent linear dependencies of p, q and r over Q. Here, we consider some situations where there is only one such dependency. In particular, we give necessary and sufficient conditions for integrability for the case of (i, −i, λ)-resonance with λ /∈ iR (after a scaling, this is just the case p + q = 0 with q/r /∈ R), and also the case of (i − 1, −i − 1, 2)-resonance (a subcase of p + q + r = 0) under the additional assumption that a = k = 0. Our necessary and sufficient conditions for integrability are given via the search for two independent first integrals of the form x αy βz γ(1 + O(x, y, z). However, a new feature in the case of rank-1 resonance is that there is a distinguished choice of analytic first integral, and hence it makes sense to seek conditions for just one (analytic) first integral to exist. We give necessary and sufficient conditions for just one first integral for the two families of systems mentioned above, except that for the second family some of the cases of sufficiency have been left as conjectural.
Liouvillian integration of the Lotka-Volterra system
Similar systems of equations have been studied by Volterra in his mathematical approach of the competition of species and this is the reason why this name has been given to such systems. In fact, LV (A, B, C) can be chosen as a normal form for most of factored quadratic systems; the study of its first integrals of degree 0 is thus of great mathematical interest. Given a homogeneous vector field, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space (or curves in the corresponding projective plane), such that the trajectories of the vector field are completely contained in a leaf. A first integral of degree 0 is then a function on the set of all leaves of the previous foliation. In the present paper, we give all values of the triple (A, B, C) of non-zero parameters for which LV (A, B, C) has a homogeneous liouvillian first integral of degree 0. We also discuss the corresponding problem of liouvillian integration for quadratic factored vector fields that cannot be put in Lotka-Volterra normal form. Our proof essentially relies on combinatorics and elementary algebraic geometry, especially in proving that some conditions are necessary.
Meromorphic function sharing a small function with a homogeneous differential polynomial
Abstract In this paper, we consider a Brück-type result for a homogeneous differential polynomial generated by a meromorphic function.