Linear Darboux polynomials for Lotka-Volterra systems, trees and superintegrable families

We present a method to construct superintegrable $n$-component Lotka-Volterra systems with $3n-2$ parameters. We apply the method to Lotka-Volterra systems with $n$ components for $1<n<6$, and present several $n$-dimensional superintegrable families. The Lotka-Volterra systems are in one-to-one correspondence with trees on $n$ vertices.


Introduction
The original 2-dimensional Lotka-Volterra (LV) system, whereẋ denotes the derivative with respect to time, was derived as a model to describe the interaction between predator and prey fish [18,25,10]. Sternberg [22,Chapter 11] gives a dynamical systems perspective and an explanation why fishing decreases the number of predators. The 2-dimensional system (1) has been generalised to n-dimensional systems of the forṁ where b is a real vector, and A is a real matrix, and these have been studied extensively. For references on various aspects of LV systems, including integrability as well as their history, see [1,2,3,4,6,7,8,10,12,14,16,17,19]. Prelle and Singer wrote a very influential paper [20] proving that if a polynomial ODE has an elementary integral, then it has a logarithmic integral. Note that in the mathematical physics literature the matrix A is often assumed to be skew symmetric. This is not assumed here. A vector field on an n-dimensional manifold is called superintegrable if it admits n − 1 functionally independent constants of motion (i.e. first integrals), cf. [24]. In this paper we construct superintegrable n-component Lotka-Volterra systems with 3n − 2 parameters.
Darboux polynomials (DPs) are building blocks of rational integrals and their generalizations [11,13]. Given an ordinary differential equation (ODE) where x(t) and f are n-dimensional vectors, a Darboux polynomial P (x) is defined by the existence of a polynomial C(x) s.t.
In section 2, we provide a method to obtain m integrals for an n-dimensional homogeneous quadratic ODE, from m + n Darboux polynomials. In section 3, we give conditions on b and A which are equivalent to P i,k = αx i + βx k being a DP for (2). In section 4, we look at the intersection of the above two classes, i.e. at homogeneous Lotka-Volterra systems, and use the described method and mentioned DPs to construct some superintegrable systems in dimensions 2, 3, and 4. In section 5, we explain how these superintegrable n-dimensional LV systems are in one-to-one correspondence with trees on n vertices. Such a tree has n − 1 edges, and each of these edges corresponds to an integral. If an edge exists between vertices i and k, the corresponding integral can be written as a product of P i,k and powers of the variables x j , j = 1 . . . n. In section 6, we cover the superintegrable LV-systems which relate to the 3 non-isomorphic trees on 5 vertices. We also describe the factorisation of the exponents of the variables in terms of minors of the matrix A. In our final section we give some details for the superintegrable n-dimensional LV systems that relate to tall trees. In the appendix we explain how the Euler top relates to a special case of our superintegrable 3-dimensional LV system.

A rather general method
Let Hence cofactors C i form a linear space. Note that C 1 = C 2 if and only if P1 P2 is an integral. We also have P α1 and more generally It follows that integrals that arise in this way are factorisable.
If there are more functionally independent DPs than the dimension of this linear space, then there must be one or more integrals. The method we introduce here, produces m integrals for an n-dimensional homogeneous quadratic ODE, from n + m Darboux polynomials.
• Find n independent DPs for the ODE: The C i will be linear. Defining v to be the vector with components v i := ln(P i ), i = 1, . . . , n, the equation (6) can be written asv = Ax where A is some constant invertible matrix.
• Find m additional DPs for the ODE (m ≤ n−1 is a necessary condition for the integrals to be independent). Defining w to be the vector with components w i := ln(P i ), i = n+1, . . . , n+m, we getẇ = Bx Eliminating x, we again getẇ For n-component Lotka-Volterra (LV) systems, n Darboux polynomials are given by the components of the vector x, and we set v = x. From (9), by exponentiation of the logarithmic integrals I, we obtain m integrals of the form and |A| is the determinant of A.
Lemma 1. Consider a system witḣ The expression, with αβ = 0, is a DP if and only if, for some constant b and all j ∈ {i, k} c , and Proof. We first show that the conditions (13), (14) and (15) are sufficient, i.e. if they are satisfied, then P i,k defined by (12) is a DP for the ODE defined by (11). Equation (12) implies with (11) that and where (using (13)) Next we show the conditions are necessary, i.e. if P i,k defined by (12) is a DP for the ODE defined by (11) then (13), (14) and (15) hold. Equation (12) implies with (11) that First consider all terms that contain x j on the r.h.s., where j ∈ {i, k} c : This must vanish if we substitute We for all j ∈ {i, k} c . Now consider all remaining terms that do not contain any x j , with j ∈ {i, k} c , i.e.
Once again (21) must vanish if we substitute (19). Hence Of course several low-dimensional instances of Lemma 1 have appeared in papers by various authors over the years, cf. e.g. a 2D instance in equation (3.2) of [15], a 3D instance in Proposition 1#(3) of [5], and a 4D instance in equation (12)  The system admits the Darboux polynomials x 1 , x 2 , with cofactors a 1 x 1 + b 1 x 2 , a 2 x 2 + c 1 x 1 , and the Darboux polynomial (c 1 − a 1 ) x 1 + (a 2 − b 1 ) x 2 , with cofactor a 1 x 1 + a 2 x 2 . They give rise to matrices and hence to the integral

n = 3
The system   relates to matrix The following are 2 additional Darboux polynomials: Thus we have and we find 2 = n − 1 integrals  The matrix has the property that A i,j = A i+1,j for all i ∈ {1, 2, 3} and j ∈ {i, i + 1} c . The associated Lotka-Volterra system is The system (27) has 7 Darboux polynomials. The obvious ones are P i = x i , i = 1, 2, 3, 4, with cofactors C i = n j=1 A i,j x j . The other three, obtained from Lemma 1, are: with cofactors The coefficient matrix from these cofactors is The rather general method, introduced in section 2, gives rise to the following 3 = n−1 functionally independent integrals: where I 1 is determined by and I 3 is determined by

4.3.2
Next we consider the matrix It has the property that A i,j = A k,j for all (i, k) ∈ {(1, 2), (2, 3), (2, 4)} and j ∈ {i, k} c . The corresponding Lotka-Volterra system reads The additional Darboux polynomials are The coefficient matrix from these cofactors is We label the special pairs of indices of rows of A as follows, The same label can be used to enumerate the functionally independent integrals, The special pairs of indices of rows of A can be interpreted as edges of a tree, which we will do in the next section.

Connection to trees
To each of the above n-component Lotka-Volterra systems we associate a free (unrooted) tree T on n vertices as follows. The tree has an edge between vertex i and vertex k if the condition that A i,j = A k,j for all j ∈ {i, k} c is satisfied. Thus, the systems (23), (25), (26) and (28) relate to the trees depicted in Figure 1. Vice versa, a tree T on n (ordered) vertices has n − 1 (ordered) edges. We associated to T a matrix A as follows. We start with an n×n diagonal matrix A, with A i,i = a i . Then for each edge of T we fix two off-diagonal entries of A as follows. For the m-th edge of the graph T , e m = (i, k) with i < k, we set A i,k = b m and A k,i = c m . In [23] we show that the remaining entries of the matrix A are uniquely determined by the condition that A i,j = A k,j when (i, k) is an edge of T and j ∈ {i, k} c . The matrix A has 3n − 2 free parameters and defines a Lotka-Volterra systeṁ with n − 1 integrals. In [23], we prove their functional independence, Theorem 2.
Theorem 2. Each tree on n vertices gives rise to a Lotka-Volterra system with 3n − 2 parameters, which admits n − 1 functionally independent integrals.
One can think of the parameters a i , b j , c k as weights in a complete digraph D (allowing both loops and multiple edges) which is associated to T . The matrix A is then nothing but the adjacency matrix of D. The connection between Lotka-Volterra systems and graphs, via the adjacency matrix of the graph, has been made before [2,9,7,12], but in the context of undirected or directed graphs, and (mainly) anti-symmetric (and hence Hamiltonian) Lotka-Volterra systems. The general setting of complete digraphs seems to be new. Note that the number of trees is given by the sequence [21, A000055].

Superintegrable 5-component Lotka-Volterra systems
There are 3 non-isomorphic trees on 5 vertices, see Figure 2. Following the procedure in the previous subsection, the trees in Figure 2 give rise to matrices (A) and hence to Lotka-Volterra systems, each with 13 free parameters, and Using the methods explained in sections 2 and 3, we can construct 4 functionally independent integrals for each of these systems. As in section 4, the exponents in the integrals exhibit interesting factorisation properties. Below we provide the integrals for systems (32), (33) and (34), expressing each exponent as a product of differences of parameters and a minor of A. We let A I;J denote the matrix A with rows i ∈ I and columns j ∈ J deleted. Its determinant |A I;J | is called a minor of A.

A hierarchy of superintegrable Lotka-Volterra systems
Consider the tall tree on n vertices depicted in Figure 3. It gives rise to the n × n matrix: of which matrices (23), (25),(26), and the left matrix in (31), are special cases taking n = 2, 3, 4 and 5 respectively.
For arbitrary n, the tall tree provides us with the Lotka-Volterra system: . .
One can show, cf. [23], that the exponents factorise and that the integrals K i are functionally independent (which implies superintegrability). Introducing the notation we find, for all i ∈ N n−1 1 , j ∈ N n 1 , This formula provides a more efficient way to calculate the exponents in the integrals I i than using the definition of Z, which involves matrix multiplication, inversion and taking the determinant of an n × n matrix. The special case a i = 0 (i = 1, . . . , n), b i = −c i+1 (i = 1, . . . , n − 1) was studied in [17].
Concluding remark. In this paper we have studied superintegrable Lotka-Volterra systems without imposing any additional structure. We intend to investigate the role of measure-preservation and symplectic structure on Lotka-Volterra equations in future work.
It is now easy to generalise system (39) whilst keeping the same number of Darboux polynomials (six). Indeed, we would take