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.
Normalizable, integrable and linearizable saddle points in the lotka-volterra system
AbstractWe consider the Lotka-Volterra Equations
$$\dot x = x(1 + ax + by), \dot y = y( - \lambda + cx + dy),$$
 with λ a non negative number. Our aim is to understand the mechanisms which lead to the origin being linearizable, integrable or normalizable. In the case of integrability and linearizability, there is a natural dichotomy. When the system has an invariant line other than the axes, then the system is integrable and we give necessary and sufficient conditions for linearizability in this case. When there is no such line, then the conditions for linearizability and integrability are the same. In this case we show that the monodromy groups of the separatrices play a key role. In particular for λ=p/q withp+q ≤12 and λ=n/2,2/n withn∈N the origin is linearizable if and only if the monodromy groups can be shown to be linearizable by elementary arguments. We give 4 classes of these conditions, and their duals, in terms of the parameters of the system, and conjecture that these, together with two exceptional cases of Darboux linearizability, are the only integrability mechanisms for rational values of λ. The work on normalizability is more tentative. We give some sufficient conditions for this via monodromy groups, and give a complete classification when λ=0. We also investigate in detail the case λ=1, witha+c=0. Much of our ideas here are based on recent work on the unfolding of the Ecalle-Voronin modulus of analytic classification [13]. In particular we give examples of “half-normalizable” systems as well as an experimental example of a “transcritical bifurcation” of the functional moduli associated to the critical point.
Liouvillian first integrals for the planar Lotka-Volterra system
We complete the classication of all Lotka-Volterra systemsx=x(ax+by+c),y=y(Ax+By+C), having a Liouvillian first integral. In our classification we take into account the first integrals coming from the existence of exponential factors.