A new class of integrable Lotka--Volterra systems

A parameter-dependent class of Hamiltonian (generalized) Lotka--Volterra systems is considered. We prove that this class contains Liouville integrable as well as superintegrable cases according to particular choices of the parameters. We determine sufficient conditions which result in integrable behavior, while we numerically explore the complementary cases, where these analytically derived conditions are not satisfied.


Introduction
The Lotka-Volterra system has been introduced independently by Lotka [14] and Volterra [19] as a predator-prey model. Since then, many generalizations have been considered with applications to several scientific disciplines. These systems in general display rich dynamical behavior that varies according to the parameters that define each individual system. For example, there are Hamiltonian and non-Hamiltonian Lotka-Volterra systems, as well as integrable, not integrable and chaotic ones. From the point of view of integrability, various kinds of generalized Lotka-Volterra systems have been extensively studied in the literature, e.g. [1,4,5,6,9,10,11,15,16,18]. A numerical study of a 4-dimensional non-integrable Lotka-Volterra system can be found in [17].
In this paper, we study a parametric family of (generalized) Lotka-Volterra systems of the formẋ This family includes some particular interesting cases. The case of r i = 0 and a i = 1, can be considered as a finite dimensional reduction of a Bogoyavlenskii lattice [2,3] with fixed boundary conditions. The integrability of this case and its corresponding Kahan discretization has been studied in detail in [12]. In [13], the Liouville integrability and superintegrability of the more general cases, with r i = 0 and arbitrary a i ∈ R, was proved and explicit solutions were given for the corresponding continuous and discrete systems.
Here, our aim is to study the integrable and dynamical aspects of (1), with arbitrary parameters a i and r i in R.
As it is shown, for any choice of the parameters, system (1) is Hamiltonian (Section 3). A first approach to trace integrable cases is the following. We consider the integrals of the r i = 0 case as they appear in [13], and we demand to be in involution with the Hamiltonian function of (1). This restriction leads to a system for the parameters a i and r i . Solutions of this system provide necessary and sufficient conditions which ensure the pairwise involutivity of all the integrals (including the Hamiltonian). This procedure provides several Liouville integrable cases. By considering a permutation symmetry of the system more integrable cases appear as well as superintegrable cases according to particular choices of the parameters. These results appear to Sections 4-5.
In Section 6, we numerically explore the behavior of (1) with n = 4 for the cases where integrability is not proven by the analytical arguments of the previous sections. To this end we perform a series of numerical simulations for various different parameters which determine the system (1). Integrability or non-integrability is manifested by the Poincaré surfaces of section as well as the evolution of the largest Lyapunov exponent for various initial conditions at gradually increasing energies. We have strong indications that more integrable cases exist, however, we find non-integrable cases as well. Notable non-integrable examples are found for the 4-dimensional Lotka-Volterra system (1) with bounded trajectories in phase space, whose orbits demonstrate a particularly rich complexity.

A class of Lotka-Volterra systems
Generalized Lotka-Volterra or just Lotka-Volterra systems are systems of the forṁ where A = (A ij ) is any arbitrary n × n matrix, known as the community matrix and r = (r 1 , . . . , r n ) is a vector in R n . In this paper, we are going to study a particular class of Lotka-Volterra systems, with community matrix . . a n −a 1 0 a 3 . . . a n −a 1 −a 2 0 . . . a n . . .
and parameters a 1 , . . . , a n ∈ R. In this case, system (2) can be written as (1), or equivalently, asẋ where P is the antisymmetric matrix The special case of (1) with r = 0 has been extensively studied in [12,13], where the Liouville and superintegrability of the corresponding systems were proved and explicit solutions given. Here, our aim is to investigate the integrability of particular cases with r = 0. In due course we mainly restrict our attention to the case that n is even.

Hamiltonian formalism
We consider the log-canonical Poisson structure The rank of this Poisson structure, for x 1 . . . x n = 0, is n for even n, and n − 1 for odd n.
With respect to the k i parameters, the system is written aṡ For odd n, the matrix P is not invertible. Hence, the Hamiltonian structure of Prop. 3.1 does not include all the cases of (1) for arbitrary r ∈ R n . However, for any n we can restrict our analysis to the Hamiltonian systems (7), i.e. systems (1) with r = P k.
By setting u i = log x i , the Poisson bracket (6) becomes a constant one, that is {u i , u j } = P ij , and the Hamiltonian function H u = n i=1 (a i e u i +k i u i ). In these coordinates our system is expressed aṡ P ij a j e u j + r i = j>i a j e u j − j<i a j e u j + r i , with r = P k. Remark 3.2. The parameters a i , . . . , a n of (1) can be rescaled to a 1 c 1 , . . . , a n c n , by using the transformation x i → x i /c i , for c i > 0. This linear transformation preserves the Poisson bracket and gives rise to an equivalent Hamiltonian system with Hamiltonian For example, all the positive and negative a i parameters can be rescaled to 1 and −1 respectively. Hence, we can consider systems with a i ∈ {−1, 0, 1} without any loss of generality.
If a 1 a 2 . . . a n = 0, the functions v i define new coordinates on R n but generally this is not true. Furthermore, for any even n we define the functions for m = 1, . . . , n 2 . We also set H 0 is the Hamiltonian function in the case of r = k = 0. So, the generic Hamiltonian of (1) is written as In [13], it is proved that for any m, l ∈ {1, . . . , n 2 }, as well as the following theorem which establishes the Liouville integrability of the system in the case of r = 0. Here, our first goal is to determine the parameters a i and r, so that the more general system (1) inherits the same integrals as the r = 0 case which ensure Liouville integrability. In the following, we always assume that n is even.
(this identity has been proved in [13]) and Therefore, and by substituting in (10) we derive (9).
We can also recast the sum that appears in (9) Hence, from Lemma 4.2, the next proposition follows.
Solutions of the system S jm = 0, for m = 1, . . . , n 2 − 1 and j = 1, . . . , 2m, provide conditions on the parameters a i and k i ensuring that the functions F 1 , F 2 , . . . , F n 2 −1 are first integrals of the system. Moreover, according to (8), these integrals are pairwise in involution. Therefore, in the case where F m = 0, for all m = 1, . . . , n 2 − 1, these conditions on the parameters provide Liouville integrability. For example, in the particular case where a j = 0, for every j = 1, . . . , n, the corresponding system implies the unique solution k 1 = k 2 = · · · = k n−2 = 0.
In such a case, F 1 = · · · = F λ = 0. So, for any choice of parameters there are not enough F -type integrals to ensure the integrability of the system. However, Theorem 4.1 suggests that we could probably replace the first λ missing F -integrals by λ J-integrals. Hence next, we are going to determine the conditions on the parameters to ensure that {J m , H} = 0, for m = 1, . . . , λ 2 .
Proof. We consider m ∈ {1, . . . , λ}. From Theorem 4.1, it follows that {J m , H 0 } = 0. So, Also, Consequently, after some calculations we obtain Substituting this to (13), we derive (12).   We will close this section by proving the functional independence of the integrals. Proof. For k = 0, J 1 , . . . , J λ , F λ+1 , . . . , F n 2 −1 , H are functionally independent. This follows from Theorem 4.1, since in this case H coincides with H 0 . Hence, by continuity the same functions remain functionally independent for parameters k in a sufficiently small open neighborhood U of k = 0. Now, let us consider any k = (k 1 , . . . , k n ) ∈ R n . Then there is µ > 0 and k ′ = (k ′ 1 . . . , k ′ n ) ∈ U, such that k = µk ′ . Also, in view of Remark 3.2, we can rescale the a i parameters to µa i , by setting y i = x i /µ. The Hamiltonian function in the new y-coordinates then becomes the Hamiltonian of the corresponding system with parameters a i and k ′ i . Therefore, from the functional independence of J 1 , . . . , J λ , F λ+1 , . . . , F n 2 −1 , H ′ that we proved, the functional independence of J 1 , . . . , J λ , F λ+1 , . . . , F n 2 −1 , H y follows and consequently the functional independence of J 1 , . . . , J λ , F λ+1 , . . . , F n 2 −1 , H for all parameters a i and k i .

Symmetry and superintegrability
In [12,13], a second set of first integrals in involution has been introduced for the case of r = 0. By considering this set of integrals we can derive more integrable cases of our system. The main observation to accomplish this is that system (7) remains invariant under the transformation x i → x n+1−i and the reparametrization a i → −a n+1−i , k i → −k n+1−i , for i = 1, . . . , n. Let us now consider the involution ι(x 1 , x 2 , . . . , x n ) → (x n , x n−1 , . . . , x 1 ) and the functionsJ m = J • ι ,Ĩ m = I • ι ,F m =ṽ 2mĨm , whereṽ i := a n+1−i x n+1−i + a n+2−i x n+2−i + · · · + a n x n , for i = 1, . . . , n. Then, by Theorem 4.6 and the described symmetry of the system we derive the next theorem.
6 Numerical results for n = 4 The purpose of this section is to explore numerically the behavior of 4-dimensional Lotka-Volterra systems of the form (1) and investigate their integrability in cases that are not described in the previous sections. In the rest of the paper we will restrict to the case of a 1 = a 2 = a 3 = a 4 = 1 and we vary only the k i values. We perform a series of numerical calculations for the systeṁ with different k 1 , . . . , k 4 values, which are complementary to the two integrable cases described by Theorem 5.2. We numerically integrate the system's equations of motion together with its variational equations to compute the value of the largest Lyapunov exponent λ. The variational equations of the system (14) are where δx = (δx 1 , δx 2 , δx 3 , δx 4 ) is a vector which evolves on the tangent space of the system (14) and ∇ 2 H denotes the Hessian matrix of the Hamiltonian function H calculated along the reference orbit x(t) of the system (14). In particular, we used the classical Runge-Kutta forth-order scheme with time-step τ = 10 −4 for the numerical integration of the systems (14) and (15), which conserved the energy E = H(x) of the system (14) with accuracy of more than 8 significant figures during integration times of the order of a few thousand. The indicator which controls of the relative energy error is where E 0 is the initial energy of the system and E(t) the actual energy during the numerical integration. For k i < 0, i = 1, . . . 4, the point x 0 = (−k 1 , −k 2 , −k 3 , −k 4 ) is an elliptic fixed point of the system. Furthermore, in this case H(x) admits a global minimum at x 0 and all the orbits of the system are bounded.
We start our numerical study with examples of bounded motion, which correspond to negative values for all k i . In Fig.1 some Poincaré surfaces of section x 2 = 1, x 1 > 1 are shown for different k i < 0 values at E = 6. However, at this energy level all of them exhibit regular behavior. These Poincaré surfaces of section are constructed for a grid of initial conditions on the x 3 , x 4 plane, with x 2 = 1 and x 1 found numerically by Newton's method requiring that H(x) = E. We find a rich morphology consisting of periodic and quasiperiodic trajectories, island chains as well as separatrices. Each fixed point on the Poincaré surface represents a periodic orbit, while the ellipse-like curves correspond to quasiperiodic trajectories lying on tori. Fig.2 presents different trajectories projected on the x 2 , x 3 , x 4 plane for the system with (k 1 , k 2 , k 3 , k 4 ) = (−1, −4, −2, −1) which corresponds to Fig.1(d). The first three panels of Fig.2 correspond to E = 6 and the last one to E = 20.
We find qualitatively similar behavior to the examples of Fig.1 for k 1 = · · · = k 4 = −1, as Fig.3 indicates. In the Poincaré surface of section x 2 = 1, x 1 > 1 of Fig.3(a) is E = 4.2 and there is no evidence of chaotic behavior. We verify this result in Fig.4(a) by computing the largest Lyapunov exponent λ, which approximately decays as 1/t for randomly chosen initial conditions. Similarly with the well-known Hénon-Heiles model [8], chaotic dynamics    in the Lotka-Volterra system (14) for k i < 0 (or k i = −1) emerges for larger values of the energy. In the rest of the panels of Fig.3, where the total energy E is gradually increased, we observe a gradual transformation of fixed points and ellipses-like curves, while at energies of the order of E = 30 (Fig.3(d)) the chaotic motion is not only evident but also prevails over the ordered motion. The maximal Lyapunov exponent at this energy, which is plotted in Fig.4(b), converges to a positive value λ ≃ 0.01. As we have seen in example 5.3, the only integrable cases for n = 4, a = (1, 1, 1, 1) predicted by Theorem 5.2 are for k = (0, 0, k 3 , k 4 ), k 3 , k 4 ∈ R or k = (k 1 , k 2 , 0, 0), k 1 , k 2 ∈ R. We choose (k 1 , k 2 , k 3 , k 4 ) = (0, 0, −1, −1), for which the quantity (x 1 +x 2 )x 4 /x 3 is preserved beside the Hamiltonian. Fig.5(a) displays the evolution of the four variables log x i in time for a random choice of initial conditions. It turns out that x 2 decays asymptotically to zero, approximately like e −0.63t , while the rest variables x 1 , x 3 , x 4 asymptotically approach a periodic orbit, as is illustrated in Fig.6(a). However, a similar behavior appears in other cases, not described as integrable by Theorem 5.2. Such an example is given in Fig.5(b) and corresponds to (k 1 , k 2 , k 3 , k 4 ) = (−2, −2, −2, 2). It turns our that the variables x 2 and x 4 tend asymptotically to zero as e −2t , while x 1 and x 3 asymptotically converge to the periodic orbit shown in Fig.6(b). Furthermore, we carefully examine the largest Lyapunov exponent λ in Fig.7 for constantly increasing energies and we find that λ ∝ 1/t, even when E = 72, which strongly indicates that the system is integrable in this case too.

Conclusions
We presented a new class of Hamiltonian parametric Lotka-Volterra systems with non-zero linear terms and we proved that, for particular choices of parameters, Liouville integrability and superintegrability is established. Different choices of parameters when n = 4, not described by the theory, were studied numerically, showing that both chaotic and new integrable cases appear. Concerning these new cases with integrable behavior, we aim to investigate them in the future to find additional integrals.
In the present work we restricted our analysis to the even-dimensional case; however, a similar approach can be considered for odd dimensions. Finally, we believe that a similar approach can be considered for integrable Lotka-Volterra systems with different community matrices, or integrable deformations of them such as the systems presented in [6,7], by inserting parametric linear terms in the corresponding vector fields.