This is a preprint of: “Integrability of a class of N-dimensional Lotka-Volterra and Kolmogorov systems”, Jaume Llibre, Rafael Orlando Ramı́rez, Valentı́n Ramı́rez, J. Differential Equations, vol. 269, 2503–2531, 2020. DOI: [10.1016/j.jde.2020.02.001] INTEGRABILITY OF A CLASS OF N –DIMENSIONAL LOTKA–VOLTERRA AND KOLMOGOROV SYSTEMS JAUME LLIBRE1 , RAFAEL RAMÍREZ2 AND VALENTÍN RAMÍREZ3 Abstract. We study the integrability of an N –dimensional differential Kol- mogorov systems of the form N ! X ẋj = xj aj + ajk xk + xj Ψ(x1 , . . . , xN ), j = 1, . . . , N, k=1 where aj , and ajk are constants for j, k = 1, . . . , N and Ψ(x1 , . . ., xN ) is a homogenous polynomial of degree n > 2, with either one additional invariant hyperplane, or with one exponential factor. We also study the integrability of the N –dimensional classical Lotka-Volterra systems (when Ψ(x1 , . . . , xN ) = 0). In particular we consider the integrability of the asymmetric May–Leonard systems. 1. Introduction and main results For the N –dimensional nonlinear differential systems the existence of K < N − 1 independent first integrals means that systems is partially integrable. The existence of N − 1 independent first integrals means that the system is completely integrable, i.e. the intersection of the N − 1 hypersurfaces obtained fixing the N − 1 first integrals provide the trajectories of the differential system. Polynomial differential systems of the form ẋj = xj fj (x1 , . . . , xN ), for j = 1, . . . , N is called the N -dimensional Kolmogorov differential equations, where fj = fj (x1 , . . ., xN ) is a given function for j = 1, . . . , N. We develop a method based in the study of the rank of convenient matrices in order to determine the integrability of a class of N –dimensional Kolmogorov systems of the form N ! X (1) ẋj = xj aj + ajk xk + xj Ψ(x1 , . . . , xN ), j = 1, . . . , N, k=1 where Ψ(x1 , . . . , xN ) is a homogenous polynomial of degree n, with either one ad- ditional invariant hyperplane, or with one exponential factor. 2010 Mathematics Subject Classification. 34C05, 34C07. Key words and phrases. Kolmogorov ordinary differential equations, May–Leonard model, Lotka–Volterra systems, first integral, Jacobi multiplier, center problem, Nambu bracket, com- pletely integrable ordinary differential equations, invariant hyperplanes. 1 2 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ When Ψ(x1 , x2 , x3 ) ≡ 0 systems (1) becomes the classical Lotka–Volterra dif- ferential systems that describe the evolution of N conflicting species in population biology, and appear in many different topics like neural networks, laser physics, plasma physics, etc. (see for instance [3, 6, 11, 13]). For simplicity we shall assume that all the functions which appear below are of class C ∞ although most of the results remain valid under weaker hypotheses. 1.1. Integrability of N - dimensional Lotka-Volterra systems with an ad- ditional invariant plane. We begin by studying the integrability of the N – dimensional classical Lotka–Volterra systems, i.e. systems (1) with Ψ(x1 , . . . , xN ) = 0. We shall study the classical N - dimensional Lotka -Volterra systems. N ! X (2) ẋj = xj aj N +1 + ajk xk := Xj , j = 1, . . . , N, k=1 Since populations can not be negative, the vector x = (x1 , . . . , xN ) is taken to be non-negative, i.e., x is an element of  RN N + = x = (x1 , . . . , xN ) ∈ R : xj ≥ 0, for j = 1, . . . , N . Hence any population might be zero. However if xj = 0 holds at certain point in time, it holds for all time. The Lotka -Volterra systems were introduced independently by Lotka and Volterra in the 1920’s to model the interaction among species, see [11, 13]. Differential system (2) has N invariant hyperplanes gj = 0 with cofactors Kj for j = 1, 2, . . . , N namely N X gj = xj , Kj = aj N +1 + ajk xk , for j = 1, . . . , N. k=1 Clearly that   N X N X (3) divX = aj + 2ajj xj + ajk xk  j=1 k6=j N X ∂ where X = Xj . j=1 ∂xj The aim of this subsection is to study the problem on the integrability of system (2) by considering that (x1 , . . . , xN ) ∈ RN , with the additional invariant hyperplane gN +1 = 0 and with the cofactor KN +1 such that N X N X (4) gN +1 = νN +1 + νj xj , KN +1 = aN +1 N +1 + aN +1 k xk . j=1 k=1 First we introduce some definitions and notions which we will use in this paper. INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 3 Let U be an open subset of RN . We say that a non-locally constant function H : U −→ R is a first integral of the differential system (5) ẋj = Xj (x1 , x2 , . . . , xN ), for j = 1, . . . , N, or its associated vector field ∂ ∂ ∂ X = X1 + X2 + . . . + XN , ∂x1 ∂x2 ∂xN if H = H(x1 (t), . . . , xN (t)) is constant for all values of t for which the solution (x1 (t), . . . , xN (t)) is defined and contained in U . Clearly a C 1 function H is a first integral of system (5) if and only if ∂H ∂H ∂H Ḣ = X (H) = X1 + X2 + . . . + XN ≡ 0 in U. ∂x1 ∂x2 ∂xN If Hr : Ur −→ R for r = 1, . . . , K are K first integrals of system (5), we say that they are independent in ŨK := U1 ∩ U2 . . . ∩ UK if their gradients are independent in all the points of Ũ except perhaps in a zero Lebesgue measure set. First we characterize the integrability of the N –dimensional Lotka-Volterra sys- tems with N + 1 invariant hyperplanes. Theorem 1. The Lotka-Volterra systems (2) with the additional invariant hyper plane gN +1 = 0 with cofactor KN +1 given in (4) has the Darboux first integral   N Y (6) H = log |gN +1 |µN +1 |xj |µj  , j=1 if and only if the rank of the (N + 1) × (N + 1) matrix   a11 ... aN 1 aN +1 1  a12 . . . a aN +1 2   N 2  (7) B1 =  .. ..   . ... .  a1 N +1 ... aN −1,N aN +1 N +1 has (8) rankB1 ≤ N. We say that system (5) is completely integrable in an open set ŨN −1 if it has N − 1 independent first integrals. In this case the orbits of system (5) are contained in the curves n o n o n o H1 = h1 ∩ H2 = h2 . . . ∩ HN −1 = hN −1 where h1 , h2 . . . , hN −1 vary in R. Let J = J(x1 , . . . , xN ) be a non–negative function non–identically zero on an open subset U of RN . Then J is a Jacobi multiplier of the differential system (5) if Z Z J(x1 , . . . , xN )dx1 . . . dxN = J(x1 , . . . , xN )dx1 . . . dxN , Ω ϕt (Ω) for all open subset Ω of U , ϕt is the flow defined by the differential system (5), and ϕt (Ω) is the image of Ω under the flow ϕt . 4 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Theorem 2. The Lotka-Volterra systems with N + 1 invariant hyperplanes has N −1 independent Darboux first integrals H1 , H2 , . . . , HN −1 if and only if rankB1 = 2. Moreover the independent first integrals are Hj = log (|x1 |κ1j |x2 |κ2j |xj |) for j = 3, . . . , N, (9) HN +1 = log (|x1 |κ1 N +1 |x2 |κ2 N +1 |gN +1 |) , where κnj is a convenient constant, for n = 1, 2 and j = 3, . . . , N. Moreover completely integrable N –dimensional Lotka-Volterra systems (2) can be written as   N Y ẋj = |gN +1 | |xj | {H1 , H2 , . . . , HN −1 , xj } , for j = 1, . . . , N, j=1  −1 N Y where J = |gN +1 | |xj | is the Jacobi multiplier. j=1 N Y Theorem 3. Let J = |gN +1 |σN +1 |xj |σj be a non-negative C 1 function non- j=1 identically zero defined on an open subset dense in RN . Assume that the gN +1 = 0 is an invariant hyperplane with cofactor KN +1 . Then J is a Jacobi multiplier of the Lotka-Volterra systems (2) if and only if (10) rank(B1 ) = rank(B2 ) = rank(B3 ), where  N  X  a11 ... aN +1 1 −2a11 − a1j      j6=1  XN     a21 ... aN +1 2 −2a22 − a2j    B2 =  j6=2 .    .. ..   . ... .     XN   a ... aN +1,N +1 − a  1 N +1 j N +1 j=1   a11 ... aN,1 aN +1 1 −a11 + aN +1 1    a21 ... aN 2 aN +1 2 −a22 + aN +1 2      B3 =  .. .. .  . ... .      a1 N +1 ... aN,N +1 aN +1,N +1 aN +1,N +1 Moreover if (11) rank(B1 ) = rank(B2 ) = rank(B3 ) ≤ N, then differential system (2) is completely integrable. INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 5 Corollary 4. The N –dimensional Lotka-Volterra systems (2) with the invariant N X N X hyperplane gN +1 = νN +1 + νj xj = 0, with the cofactor KN +1 = aj j xj has j=1 k=1 the Jacobi multiplier eH J= N . Y |gN +1 | |xj | j=1 N Y µj and the first integral H = log (|xj ||gN +1 |) . j=1 Theorem 5. The Lotka-Volterra system (2) has the Jacobi multiplier 1 J= N , Y |gN +1 | |xj | j=1 and N − 2 Darboux first integral Hj = log (|x1 |κ1j |x2 |κ2j |x3 |κ3j |xj |) for j = 4, . . . , N, (12) HN +1 = log (|x1 |κ1 N +1 |x2 |κ2 N +1 |x3 |κ3 N +1 |gN +1 |) , if and only if (13) rank(B1 ) = rank(B2 ) = rank(B3 ) = 3. Moreover this differential system is completely integrable and the complementary first integral HN −1 can be determined from the equations N ! X (14) {H1 , H2 , . . . , HN −1 , xj } = J xj aj N +1 + ajk xk k=1 where j = 1, . . . , N. The proofs of these results are given in section 3. 1.2. Integrability of N - dimensional Lotka-Volterra systems with one ex- ponential factor. We shall study the Lotka-Volterra system (2) with an exponen- tial factor. Let g and h be relative prime polynomials in the variables x1 , x2 , . . . , xN . Then the function E = eh/g is called an exponential factor of the polynomial vector field ∂ ∂ ∂ X = X1 + X2 + . . . + XN , of degree m if X (E) = LE, where L is ∂x1 ∂x2 ∂xN a polynomial of degree at most m − 1, L is called the cofactor of the exponential factor E. Clearly the exponential factors do not define invariant hypersurfaces of the flow of X . In fact the exponential factor E = eh/g appears when the hypersurface g = 0 is invariant. There are exponential factor of the form E = eg , these appears when the infinity has multiplicity larger then one. For more details see [9, 10]. 6 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Analogously we can to study the problem on the existence of first integrals and Jacobi multipliers for N- dimensional Lotka -Volterra with the invariant hyperplanes gj = xj = 0, for j = 1, . . . , N and with the exponential factor E = eh/g with XN cofactor LN +1 = bN +1 N +1 + bN +1 j xj . Since the results of this subsection can j=1 be proved exactly as the ones of the previous section using the cofactors, we do not provide their proofs. Theorem 6. Assume that the Lotka–Volterra system (2) has the exponential factor E = eh/g . Let T1 be the (N + 1) × (N + 1) matrix   a11 ... aN 1 bN +1 1  a12 ... aN 2 bN +1 2    T1 =  .. .. .  . ... .  a1 N +1 . . . aN −1,N bN +1 N +1 Then the Lotka-Volterra system (2) has the first integral     h µN +1  N  g Y  H = log  e |xj |µj  , j1 if and only if rank(T1 ) ≤ N. Moreover if rank(T1 ) = 2, then the Lotka-Volterra system (2) is completely integrable. Example 7. Three dimensional Lotka-Volterra system   α4 λ2 − α2 λ4 ) ẋ = x α4 + α2 y + β3 z , λ 2 β4 (15) ẏ = y (β4 + β1 x + β3 z) ,   α2 λ4 − α4 λ2 ż = z λ4 + β1 x + λ2 y . β4 α2 is completely integrable. Indeed in this case we have that  α2 λ4 − α4 λ2  0 β1 β 1  β4 α2     α2 0 λ2 λ 2 β1  T1 =  .  β α4 λ2 − α2 λ4 ) 0 α 2 β3   3  λ2 β4 α4 β4 λ4 0 It is easy to show that rank(T1 ) = 2. Then in view of Theorem 6 we get that differential system (15) is completely integrable. The independent first integrals are  H1 = log |x|β4 |y −α4 |eλ2 β1 x−λ2 α2 y+α2 β3 z)/λ2 , H2 = log |x|−λ4 |z|β4 e(β1 λ2 x−α2 λ2 hy+α2 β3 z)/α2 . Differential system (15) can be written as follows 1 1 1 ẋ = {H1 , H2 , x}, ẏ = {H1 , H2 , y}, ż = {H1 , H2 , z}, J J J 1 where J = , and % is a non-zero constant. %|xyz| INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 7 Theorem 8. Assume that the Lotka–Volterra system (2) has the exponential factor E = eh/g . Let J be a non-negative C 1 function non-identically zero on an open subset dense in RN . Then   h σN +1  N g Y J =e |xj |σj j=1 is a Jacobi multiplier of the Lotka-Volterra systems (2) if and only if rank(T1 ) = rank(T2 ) = rank(T3 ), where  N  X  a11 ... aN,1 bN +1 1 −2a11 − aj1     j6=1   XN     a21 ... aN 2 bN +1 2 −2a22 − aj2    T2 =  j6=2 ,    .. ..   . ... .     XN   a ... aN,N +1 bN +1,N +1 − a  1 N +1 N +1 j j=1 and   a11 ... aN,1 bN +1 1 −a11 + bN +1 1    a21 ... aN 2 bN +1 2 −a22 + bN +1 2      T3 =  .. ..   . ... .      a1 N +1 ... aN,N +1 bN +1,N +1 bN +1,N +1 Moreover if rank(T1 ) = rank(T2 ) = 3, then the Lotka-Volterra system (2) is com- pletely integrable. As we observe the proofs of these results can be  obtained from the proof of the g σN +1 results given in the previous section, by putting e h instead of gN +1 . 1.3. Integrability of a class of N - dimensional Kolmogorov systems with one additional invariant hyperplane. Now we shall study the class of Kol- mogorov differential systems N ! X (16) ẋj = xj aj N +1 + ajk xk + xj Ψ(x1 , . . . , xN ) := Xj , j = 1, . . . , N, k=1 where Ψ(x1 , . . . , xN ) is a homogenous polynomial of degree n > 2, and system (16) has N + 1 invariant hyperplanes. Differential system (16) has N invariant hyperplanes gj = 0 with cofactor Kj for j = 1, . . . , N, namely N X gj = xj , Kj = aj N +1 + ajk xk + Ψ, k=1 8 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ where j = 1, . . . , N. Clearly in view of Euler Theorem for homogenous polynomials we get that ∂(x1 Ψ) ∂(x2 Ψ) ∂(xN Ψ) + + ... + = (N + n)Ψ. ∂x1 ∂x2 ∂xN Then   N X N X (17) divX = aj + 2ajj xj + ajk xk  + (N + n)Ψ. j=1 k6=j Theorem 9. The differential system (16) with the invariant hyperplanes xj = 0 for j = 1, . . . , N, and with the additional invariant hyperplane gN +1 = 0 with cofactor KN +1 given by N X N X gN +1 = νN +1 + νj xj , KN +1 = aN +1 N +1 + aN +1 k xk + Ψ, k=1 k=1 has the Darboux first integral   YN  µj |x j | (18) H = log   j=1 |gN +1 | if and only if the matrix (N + 1) × N if and only if the R(N + 1) × N matrix   a11 − aN +1 1 ... aN 1 − aN +1 1  a12 − aN +1 2 ... aN 2 − aN +1 2    W1 =  .. ..   . ... .  a1 N +1 − aN +1 N +1 ... aN −1,N − aN +1 N +1 is such that (19) rank W1 ≤ N. Theorem 10. Let YN  µj |xj | j=1 |gN +1 | J= N Y |gN +1 |n |xj | j=1 be a non-negative C 1 function non-identically zero on an open subset dense in RN . Then J is a Jacobi multiplier of differential system (16) if and only if rank(W1 ) = rank(W2 ), where   1 ... 1 1 −n + 1  a11 ... aN 1 aN +1 1 −a11 + aN +1 1     a12 ... aN 2 aN +1 2 −a22 + aN +1 2    W2 =  .. .. .  . ... .     a1 N ... aN N aN +1 N −aN N + aN +1 N  a1 N +1 ... aN N +1 aN +1 N +1 aN +1 N +1 INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 9 Theorem 11. Differential system (16) with the additional invariant hyperplane N X gN +1 = νN +1 + νj xj = 0 with cofactor j=1 N X (20) KN +1 = aj j xj + Ψ, k=1 has the Darboux first integral   YN  µj |x j | H = log   j=1 |gN +1 | and the Jacobi multiplier eH (21) J= N Y |gN +1 |n |xj | j=1 if and only if rank W1 = rank W2 ≤ N. Corollary 12. Differential system (16) has a Jacobi multiplier and N − 2 Darboux first integrals if and only if rank W1 = rank W2 = 3. Moreover this differential system is completely integrable with the N − 2 first inte- grals Hj = log (|x1 |κ1j |x2 |κ2j |x3 |κ3j |xj |) for j = 4, . . . , N, HN +1 = log (|x1 |κ1 N +1 |x2 |κ2 N +1 |x3 |κ3 N +1 |gN +1 |) , 1 and with the Jacobi multiplier J = N . Y |gN +1 | n |xj | j=1 The proof of these results are given in section 4 2. Preliminary results The following result of Whittaker [15] plays a main role for detecting a Jacobi multiplier. Theorem 13. Let J be a non–negative C 1 function non–identically zero defined on an open subset of RN . Then J is a Jacobi multiplier of the differential system (5) if and only if the divergence of the vector field JX is zero, i.e. ∂(JX1 ) ∂(JXN ) div(JX ) := + ... + = 0. ∂x1 ∂xN The following result goes back to Jacobi, for a proof see Theorem 2.7 of [5]. 10 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Theorem 14. Consider the differential system (5) and assume that it has a Jacobi multiplier J and N − 2 independent first integrals H1 , H2 , . . . , HN −2 . Then the system admits an additional first integral functionally independent of the previous ones given by Z ˜  J HN −1 = X̃2 dx1 − X̃1 dx2 , ∆˜ where ˜ denotes quantities expressed in the variables (x1 , x2 , h1 , . . . , hN −2 ) with Hj = hj for j = 1, . . . , N − 2 and ∂H1 ∂H1 ∂H1 ... ∂x3 ∂x4 ∂xN ∂H2 ∂H2 ∂H2 ∆= ... ∂x3 ∂x4 ∂xN .. .. .. . . ... . ∂HN −2 ∂HN −2 ∂HN −2 ... ∂x3 ∂x4 ∂xN Then system (5) is completely integrable. Theorem 15. Suppose that a polynomial vector field X defined in RN admits M invariant algebraic hypersurfaces gj = 0 with cofactor Kj for j = 1, . . . , M, i.e. the following equations hold ∂gj ∂gj ∂gj X gj = X1 + X2 + . . . + XN = Kj gj , ∂ x1 ∂ x2 ∂ xN for j = 1, . . . , M. If there exist µj ∈ R not all zero such that M X µj Kj + divX = 0, j=1 M Y if and only if the function |gj |µj is a Jacobi multiplier. j=1 For a proof of Theorem 15 see [4]. The following result is proved in [?]. Theorem 16. A differential system (5) is completely integrable with C 2 first inte- grals Hj for j = 1, . . . , N − 1 if and only if it can be written as 1n o ẋj = H1 , H2 , . . . , HN −1 , xj = Xj , for j = 1, 2, 3, J n o where J = J(x1 , x2 , . . . , xN ) is a Jacobi multiplier and H1 , H2 , . . . , HN −1 , ∗ is the Nambu bracket (see for instance [8, 12]), i.e. INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 11 ∂H1 ∂H1 ∂H1 ... ∂x1 ∂x2 ∂xN ∂H2 ∂H2 ∂H2 ... n o ∂x1 ∂x2 ∂xN H1 , H2 , . . . , HN −1 , ∗ := .. .. .. . . . ... . ∂HN −1 ∂HN −1 ∂HN −1 ... ∂x1 ∂x2 ∂xN ∂ ∂ ∂ ... ∂x1 ∂x2 ∂xN Theorem 16 extends results of [14] and [1]. From Theorem 16 it is immediate to prove the following result (see [7]). Corollary 17. Assume that differential systems (5) has N − 2 independent first integrals H1 , H2 , . . . , HN −2 and a Jacobi multiplier J, then another independent first integral HN −1 can be obtained as a solution of the partial differential equation o {H1 , H2 , . . . , HN −2 , HN −1 , xj = JXj , for j = 1, . . . , N. Theorem 18. Suppose that an N –dimensional polynomial vector field X admits p invariant algebraic hypersurface gj = 0 with cofactors Kj for j = 1, . . . , p and q exponential factors Ej = ehj /gj with cofactors Lj for j = 1, . . . , q. Then there exist complex numbers µj for j = 1, . . . , p + q not all zero such that p X q X µj Kj + µp+j Lj = 0, j=1 j=1  ! p q gj Y Y µj+p if and only function H = log  |gj |µj e hj  , is a first integral of the j=1 j=1 vector field X . For the proof of Theorem 18 see [4]. 3. Proof of Theorems 1, 2 3, 5 and Corollary 4 Proof of Theorem 1. First we suppose that H given in (31) is a first integral of N X +1 system (2), then by Theorem 18 we have µj Kj = 0, with not all the µj are j=1 zero. Hence µ1 a1 1 + µ2 a2 1 + . . . + µN +1 aN +1 1 = 0, µ1 a1 2 + µ2 a2 2 + . . . + µN +1 aN +1 2 = 0, (22) .. .. .. . . ... . , µ1 a1 N +1 + µ2 a2 N +1 + . . . + µN +1 aN +1 N +1 = 0. 12 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Consequently B1 µ = 0 where µ = (µ1 , µ2 , . . . , µN +1 )T , where B1 is the matrix given in (7). Since µ is a non-zero vector we obtain condition (8). Now we suppose that (8) holds and we shall prove that there exists the Darboux first integral (31). Indeed, if (??) holds then (for example) aN +1 1 = γ1 a11 + γ2 a21 + . . . + γN aN 1 , aN +1 2 = γ1 a12 + γ2 a22 + . . . + γN aN 2 , .. .. .. . . ... ., aN +1 N +1 = γ1 a1 N +1 + γ2 a2 N +1 + . . . + γN aN N +1 , where γ1 , γ2 , . . . , γN are convenient constants. Hence we get that the cofactor of the invariant plane gN +1 = 0 becomes N X N X KN +1 = aN +1 N +1 + aN +1 k xk = γj Kj . k=1 j=1 Thus from the equation   ∂gN +1 ∂gN +1 γ1 γN X1 + . . . + XN = KN +1 gN +1 = X1 + . . . + XN gN +1 , ∂x1 ∂xN x1 xN we deduce that     ∂gN +1 gN +1 γ1 ∂gN +1 gN +1 γN − X1 + . . . + − XN = 0, ∂x x1 ∂xN xN Hence  N   N  Y Y γj γj  ∂ log |xj |   ∂ log |xj |   ∂g   ∂g   N +1 j=1   N +1 j=1   − gN +1  X1 +. . .+ − gN +1  XN = 0,  ∂x1 ∂x1   ∂xN ∂xN           |g  ∂ H̃ ∂ H̃ ˙  N +1 |  which is equivalent X1 + . . . + XN = H̃ = 0, where H̃ = log  N . ∂x1 ∂xN Y   |x |γj  j j=1 µj Thus H̃ is a first integral. Hence taking γj = − after some computations we  µN +1 µN +1 get H =  H̃ µj  . In short the theorem is proved.  γj =− µN +1 Proof of Theorem 2. By assumption the Lotka –Volterra system has N +1 invariant hyperplanes. Then by Theorem 1 the system (2) has a Darboux first integral of the form (31) because rank B1 = 2 ≤ N. Thus (22) holds. Since the rankB1 = 2, by the Rouche– Frobenius Theorem the set of solutions of system (22) form a subspace of dimension N − 1. Then without loss of generality INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 13 we can assume that (23) an = −κ1n a1 − κ2n a2 , for n = 3, . . . , N + 1, where an = (a1n , a2n , . . . , aN +1,n ) for n = 1, . . . , N + 1, and κ1n and κ2n are convenient constants. Substituting (23) in system (22) we get a1 (µ1 − (κ13 µ3 + . . . + κ1N +1 µN +1 )) + a2 (µ2 − (κ23 µ3 + . . . + κ2N +1 µN +1 )) = 0, where µj for j = 3, . . . , N + 1 are arbitrary constants. Hence since a1 and a2 are independent vectors we obtain µ1 = κ13 µ3 + . . . + κ1N +1 µN +1 , µ2 = κ23 µ3 + . . . + κ2N +1 µN +1 . Consequently the Darboux first integral H becomes   H = log |x1 |µ1 |x2 |µ2 . . . |xN |µN |gN +1 |µN +1   = log |x1 |κ11 µ3 +...+κ1N +1 µN +1 |x2 |κ23 µ3 +...+κ2N +1 µN +1 |x3 |µ3 |xN |µN |gN +1 |µN +1  µ3  µN = log |x1 |κ13 |x2 |κ23 |x3 | + . . . + log |x1 |κ1N |x2 |κ2N |xN | N X +1 µN +1 + log (|x1 |κ1 N +1 |x2 |κ2 N +1 |gN +1 |) := µj Hj . j=3 µ In view of arbitrariness of µ3 , . . . , µN +1 we have that Hj j for j = 3, . . . , N + 1 are independent first integrals (9). Thus, under the assumption rankB = 2 the N-dimensional Lotka-Volterra systems is completely Darboux integrable. The  reciprocity it is easy  to obtain. Indeed, from Theorem 18 and 1 if Hk = N Y k k log |gN +1 |µN +1 |xj |µj  for k = 1, . . . , N − 1 are independent first integrals, j=1 then the following relations hold N X −1 (k) (k) (k) µN +1 KN +1 + µN KN + µj Kj = 0, for k = 1, . . . , N − 1. j=1 (k) We consider that the matrix Γ = ( µj ) is a non-degenerated matrix. j,k=1,...,N −1 Hence after some computations we prove that Kα = %N KN + %N +1 KN +1 , for α = 1, . . . , N − 1. Consequently if we denote by an = (a1n , a2n , . . . , aN +1,n ) for n = 1, . . . , N + 1, we get an = κN aN + κN +1 aN +1 , for n = 1, . . . , N − 1. for convenient constants κN and κN +1 . Hence we easily obtain that the matrix B1 is such that rankB1 = 2. In short the corollary is proved.  Example 19. Three dimensional Lotka-Volterra systems with the invariant plane not passing through the origin g4 = ν1 x + ν2 y + ν3 z + ν4 = 0 with cofactor K4 = 14 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ α1 x + β2 y + λ3 z, is     α1 ν4 ν2 (β1 − α1 ) ẋ = x + α1 x + β2 − y + α3 z , ν1 ν1     β2 ν 4 ν3 (λ2 − β2 ) (24) ẏ = y + β1 x + β2 y + λ 3 − z , ν2 ν2     λ3 ν4 ν1 (α3 − λ3 ) ẋ = z + α1 − x + λ2 y + λ3 z ν3 ν3 The matrix B1 under the conditions σ1 , becomes   ν1 (α3 − λ3 )  α1 β1 α1 − α1   ν3     ν (β − α1 )   β2 − 2 1 β2 λ2 β2   ν1   B1 =  . ν3 (λ2 − β2 )     α3 λ3 − λ3 λ3   ν2     α1 ν4 β2 ν4 λ 3 ν4  0 ν1 ν2 ν3 By considering that ν4  2 det (B1 ) = − ν1 β2 (λ3 − α3 ) + ν2 λ3 (α1 − β1 ) + ν3 α1 (β2 − λ2 ) . ν1 ν2 ν3 we get that if det B1 = 0, then it is easy to prove that rank ( B1 ) = 2, hence in view of Theorem 2 differential system (24) has two independent Darboux first integrals H1 and H2 . In particular if α3 β2 + λ3 (β1 − β2 ) (25) ν1 = ν2 = ν3 = −ν4 = −1, α1 = , λ3 − λ2 + β2 we get that the first integrals are  H1 = log |x|−λ3 (λ3 −λ2 +β2 |z|α3 β2 +λ3 (β1 −β2 ) |1 − x − y − z|(α3 +β2 −β1 )λ3 −α3 λ2 ,  H2 = log |x|−β2 (λ3 −λ2 +β2 ) |y|α3 β2 +λ3 (β1 −β2 ) |1 − x − y − z|(β2 −β1 )(λ3 −λ2 +β2 ) , After some computations we obtain that differential system (24) under the con- ditions (25) can be written as 1 1 1 {H1 , H2 , x}, ẏ = {H1 , H2 , y}, ż = {H1 , H2 , z}, ẋ = J J J 1 where J = , is the Jacobi multiplier, and % is nonzero con- %|xyz(1 − x − y − z)| stant. Proof of Theorem 3. From Theorem 13 we get that system (2) has a Jacobi multi- plier if and only if XN ∂ log J Xj + divX = 0. j=1 ∂xj INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 15 N Y N X +1 Consequently, if J = |gN +1 |σN +1 |xj |σj then we get that σj Kj + divX = 0. j=1 j=1 In view of (3) we get that this equation holds if and only if N X σ1 a1 1 + σ2 a2 1 + . . . + σN +1 aN +1 1 = −2a11 − a1j , j6=1 N X σ1 a1 2 + σ2 a2 2 + . . . + σN +1 aN +1 2 = −2a22 − a2j , (26) j6=2 .. .. .. . . ... . N X σ1 a1 N +1 + σ2 a2 N +1 + . . . + σN +1 aN +1 N +1 = − aj . j=1 After the change σj = −1 + µj for j = 1, . . . , N + 1 we obtain that µ1 a1 1 + µ2 a2 1 + . . . + µN +1 aN +1 1 = −a11 + aN +1 1 , µ1 a1 2 + µ2 a2 2 + . . . + µN +1 aN +1 2 = −a22 + aN +1 2 , (27) .. .. .. . . ... . , µ1 a1 N + µ2 a2 N + . . . + µN +1 aN N +1 = −aN N + aN +1 N , µ1 a1 N +1 + µ2 a2 N +1 + . . . + µN +1 aN +1 N +1 = aN +1 N +1 . Consequently in view of the Rouché–Frobenius Theorem we get that system (27) has solution if and only if (10) holds. On the other hand if (11) holds then in view of Theorem 1 we obtain that there exists the first integral (31), consequently from Theorem 14 we get that the Lotka-Volterra system is completely integrable. In short the theorem is proved.  Proof of Corollary 4. The proof of this corollary is obtained from the proof of The- orem 3. Indeed, from (26) we get that if aN +1 j = aj j , and aN +1 N +1 = 0 then   N Y J= |gN +1 |σN +1 |xj |σj  j=1 σ1 =−1+µ1 ,...,σN +1 =−1+µN +1 N Y |gN +1 |µN +1 |xj |µj j=1 eH = N := N , Y Y |gN +1 | |xj | |gN +1 | |xj | j=1 j=1 16 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ where µ1 , . . . , µN +1 are solutions of (27), i.e., µ1 a1 1 + µ2 a2 1 + . . . + µN +1 aN +1 1 = 0, µ1 a1 2 + µ2 a2 2 + . . . + µN +1 aN +1 2 = 0, .. .. .. . . ... . , µ1 a1 N + µ2 a2 N + . . . + µN +1 aN N +1 = 0, µ1 a1 N +1 + µ2 a2 N +1 + . . . + µN +1 aN +1 N +1 = 0.   N Y Hence in view of Theorem 1 we obtain that H = log |gN +1 |µN +1 |xj |µj  is the j=1 Darboux first integral see [?]. In short the corollary is proved.  Proof of Theorem 5. If rank(B1 ) = 3 ≤ N, hence (28) an = −κ1n a1 − κ2n a2 − κ3n a3 , for n = 4, . . . , N + 1. where aj = (a1j , . . . , aN +1j ) for j = 1, . . . , N + 1, and κ1n , κ2n and κ3n are con- stants. From Theorem 1 it follows that there exists the first integral (31). Consequently inserting (28) into (22) we have that N X +1 N X +1 N X +1 a1 (µ1 − κ1j µj ) + a2 (µ2 − κ2j µj ) + a3 (µ3 − κ3j µj ) = 0. j=4 j=4 j=4 On the other hand from (13) we obtain that rank(B1 ) = rank(B2 ) = rank(B3 ) = 3, thus in view of Theorem 3 we obtain that there exists the Jacobi multiplier N Y J = |gN +1 |σN +1 |xj |σj . j=1 σj =−1+µj Then from (27) we have N X +1 N X +1 N X +1 0 = a1 (µ1 − κ1j µj ) + a2 (µ2 − κ2j µj ) + a3 (µ3 − κ3j µj ) = −aN +1 + b, j=4 j=4 j=4 where b = (a11 , a22 , . . . , aN N , 0) . Hence by considering that a1 , a2 and a3 are independent vectors we get that N X +1 N X +1 N X +1 µ1 = κ1j µj , µ2 = κ2j µj , µ3 = κ3j µj , j=4 j=4 j=4 and aN +1 = b, thus aN +1 j = ajj , for j = 1, . . . , N, aN +1 N +1 = 0, where µ4 , . . . , µN +1 are arbitrary constants. Consequently the Darboux first integral H INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 17 becomes H = log (|x1 |µ1 |x2 |µ2 . . . |xN |µN |gN +1 |µN +1 ) = log (|x1 |κ11 µ3 +...+κ1 N +1 µN +1 |x2 |κ23 µ3 +...+κ2N +1 µN +1 |x3 |κ33 µ3 +...+κ3 N +1 µN +1 |x4 |µ4 ) + . . . + log (|xN |µN |gN +1 |µN +1 )  µ4  µ5 = log |x1 |κ13 |x2 |κ23 |x3 |κ33 |x4 | + log |x1 |κ15 |x2 |κ25 |x3 |κ35 |x5 |  µN + . . . + log |x1 |κ1N |x2 |κ2N |x3 |κ3N |xN |  µN +1 N X +1 + log |x1 |κ1 N +1 |x2 |κ2 N +1 |x3 |κ3 N +1 |gN +1 | := µj Hj , j=4 which in view of arbitrariness µ4 , . . . , µN +1 we get the existence of first integral (12). The proof of (14) follows from Corollary 17. The reciprocity it is easy to obtain. Thus the theorem is proved.  4. Proofs of Theorems 9, 10 11 and Corollary 12   N  Y µj |x j | Proof of Theorem 9. Let H = log   be a first integral of (16). j=1 |gN +1 | Then by Theorem 18 we have N X N X N X µj Kj − µj KN +1 = µj (Kj − KN +1 ) ⇐⇒ j=1 j=1 j=1 N N ! X X µj aj N +1 − aN +1 N +1 + (ajk − aN +1 k ) xk = 0. j=1 k=1 Hence N X N X N X µj = 0, µj (aj N +1 − aN +1 N +1 ) = 0, µj (aj k − aN +1 k ) = 0, j=1 j=1 j=1 N X N X for k = 1, . . . , N, which is equivalent to µj = 0 and µj (aj k − aN +1 k ) = 0, j=1 j=1 for k = 1, . . . , N + 1, or equivalently (a11 − aN +1 1 )µ1 + . . . + (aN 1 − aN +1 1 )µN = 0, (a12 − aN +1 2 )µ1 + . . . + (aN 2 − aN +1 2 )µN = 0, .. .. .. . . ., (a1 N +1 − aN +1 N +1 )µ1 + . . . + (aN −1 N − aN +1 N +1 )µN = 0, This system can be written in matrix form W1 µ = 0, with µ = (µ1 , . . . , µN )T 6= 0 and 0 = (0, . . . , 0) ∈ RN +1 . So since W1 is a (N + 1) × N matrix. From the Rouche–Frobenius Theorem we get that this linear system has a nontrivial solution if and only if rank (W1 ) ≤ N. 18 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Now we suppose that (19) holds and we shall prove that then there exits the Darboux first integral (18). Indeed, if (19) holds then N X a1 − aN +1 = λj (aj − aN +1 ) , j=2 N X −a1 + λj aj j=2 where an = (a1 n , . . . , aN +1 n ). Hence aN +1 =   . Consequently we XN  λj − 1 j= N X −K1 + λj Kj   N N j=2 ġN +1 X ẋ1 X ẋj get that KN +1 =   , i.e. λj − 1 = − + λj , XN gN +1 j= x1 j=1 xj  λj − 1 j=  N  X  λj − 1 N  d  Y  After some computations we obtain that |gN +1 | j=1 |x | |x | −λj   dt  1 j = j=2  N  −λj |x1 | Y |xj | 0. Thus the function H̃ = is a first integral. By choosing |gN +1 | j=2 |gN +1 | λj for j = 2, . . . , N, properly we get the first integral (18).    N Y Remark 20. If we suppose that H = log |gN +1 |µN +1 |xj |µj  is a first integral j=1 of (16) then   N X +1 N X +1 N X µj Kj = µj aj N +1 + ajk xk + Ψ = 0. j=1 j=1 j=1 Hence N X +1 µj = 0, j=1 (29) N X +1 µj ajk = 0, for k = 1, . . . , N + 1. j=1 Clearly this system can be written in matrix form W2 µ = 0, INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 19 where µ = (µ1 , . . . , µN +1 )T , and 0 = (0, . . . , 0) ∈ RN +2 , and   1 ... 1 1  a11 ... aN 1 aN +1 1     a12 ... aN 2 aN +1 2  W2 =  .  .. ..   . ... .  a1 N +1 ... aN −1,N aN +1 N +1 From the last equations we obtain the system B1 µ = 0, where B1 is the matrix (7) and µ = (µ1 , . . . , µN +1 )T , is a non-zero vector, 0 = (0, . . . , 0) ∈ RN +1 . Hence in view of Theorem 1 we get that there exist the fist integral given by formula (31) if and only if rankB1 ≤ N.. Consequently (30) rankB1 = rankW2 ≤ N. N X On the other hand from the firs equation of (29) we get that µN +1 = − µj . j=1 Inserting this relation into the first integral we obtain     N Y N  Y µj |x j | H = log |gN +1 |µN +1 |xj |µj  N = log  . X |gN +1 | j=1 j=1 µN +1 =− µj j=1 Proof of Theorem 10. From Jacobi’s Theorem we get that (16) has a Jacobi mul- tiplier if and only if N X ∂ log J Xj + divX = 0. j=1 ∂xj N Y Consequently, by considering that J = |gN +1 |σN +1 |xj |σj , where σj = µj − 1 for j=1 N X j = 1, . . . , N and σN +1 = −n − µj , from Theorem 15 we obtain that j=1 N X +1 (31) σj Kj + divX = 0. j=1 20 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ In view of (17) and by considering the coefficients of the polynomial (31) we get that (31) holds if and only if N X +1 σj = −n − N, j=1 N X σ1 a1 1 + σ2 a2 1 + . . . + σN +1 aN +1 1 = −2a11 − a1j , j6=1 N X σ1 a1 2 + σ2 a2 2 + . . . + σN +1 aN +1 2 = −2a22 − a2j , j6=2 .. .. .. . . ... . , N X σ1 a1 N + σ2 a2 N + . . . + σN +1 aN N +1 = −2aN N − aj , j6=N N X σ1 a1 N +1 + σ2 a2 N +1 + . . . + σN +1 aN +1 N +1 = −2aN +1 N +1 − aN +1 j , j6=N +1 After the change σj = µj − 1 for j = 1, . . . , N + 1 we obtain that N X +1 µj = −n + 1, j=1 µ1 a1 1 + µ2 a2 1 + . . . + µN +1 aN +1 1 = −a11 + aN +1 1 , (32) µ1 a1 2 + µ2 a2 2 + . . . + µN +1 aN +1 2 = −a22 + aN +1 2 , .. .. .. . . ... . , µ1 a1 N + µ2 a2 N + . . . + µN +1 aN N +1 = −aN N + aN +1 N , µ1 a1 N +1 + µ2 a2 N +1 + . . . + µN +1 aN +1 N +1 = aN +1 N +1 Consequently in view of Rouché–Frobenius Theorem we get that system (32) has solution if and only if (10) holds. In short the theorem is proved.  . Proof of Theorem 11. Clearly, by Theorem 9 and (30) it follows that rank B1 = rank W1 ≤ N , then there exists the first integral H (see Theorem 1). Hence (22) holds. On the other hand from (32) follows that aN +1 N +1 = 0, aN +1 j = ajj for j = 1, . . . , N N X and µj = 1 − n. Since the cofactor of gN +1 = 0 is (20). From the relation j=1 rank W1 = rank W2 it follows that there exist a Jacobi multiplier which in view of Theorem 10 becomes (21). In short the theorem is proved.  INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 21 Proof of Corollary 12. It is analogously to the proof of Corollary 5 and by consid- XN ering that σj = −1 + µj and µj = 1 − n.  j=1 Remark 21. It is possible to study the Kolmogorov systems (16) with rankW1 = 2 and proved that this system is completely integrable and we can study Kolmogorov systems with an additional exponential factor, analogously the study the Lotka- Volterra systems (2) with and additional exponential factor. Example 22. A three dimensional cubic Kolmogorov differential system with com- plementary invariant plane g4 = x + y + z = 0 with cofactor s1 x + s2 y + s3 z + Ψ2 (x, y, z) can be written as ẋ = x (s1 x + (s2 + s1 − β1 )y + (s1 + s3 − λ1 )z) + xΨ2 (x, y, z), (33) ẏ = y (β1 x + s2 y + (s3 + s2 − λ2 )z) + yΨ2 (x, y, z), ż = z (λ1 x + λ2 y + s3 z) + zΨ2 (x, y, z), where Ψ2 (x, y, z) = κ1 x2 +κ2 y 2 +κ3 z 2 +κ4 xy+κ5 xz+κ6 yz is such that rank W1 = 3. Hence system (33) has a first integral given by  s −λ  λ1 −s1  s −β ! |x| 2 2 |y| |z| 1 1 H = log . |g4 | |g4 | |g4 | λ1 s2 − β1 s3 If s1 = , then rank W1 = rank W2 = 3. Therefore system (33) is com- λ2 − s3 pletely integrable with the first integral H and the Jacobi multiplier J = |y|σ1 |z|σ2 |g4 |σ3 , where (λ1 + λ2 − s3 )s2 + (λ1 + s3 − β1 )s3 − λ2 (2s3 + λ1 ) σ1 = , (λ2 − s2 )(λ2 − s3 ) (β1 + 2s2 + s3 − λ2 )λ2 − (2s3 + λ1 )s2 σ2 = , (λ2 − s2 )(λ2 − s3 ) (s3 + 2s2 + λ1 − β1 − 3λ2 )(λ2 − s3 ) σ3 = . (λ2 − s2 )(λ2 − s3 ) 5. Integrability of the asymmetric May-Leonard model. 5.1. Asymmetric May-Leonard model with four invariant planes. A par- ticular 3–dimensional Lotka–Volterra system is ẋ = x (1 − x − a1 y − b1 z) = X1 , (34) ẏ = y (1 − y − a2 z − b2 x) = X2 , ż = z (1 − z − a3 x − b3 y) = X3 . This system is known as the asymmetric May–Leonard model (see for instance [3]). This model describes the competitions between three species and depending on six non–negative parameters aj and bj for j = 1, 2, 3. The state space is the set  R3+ = (x, y, z) ∈ R3 : x ≥ 0 y ≥ 0and z ≥ 0, . 22 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ We shall study the integrability of system (34) under the condition that it has an additional invariant plane. Proposition 23. Differential system (34) under the conditions a3 (a2 − 1) − a2 b3 + 1 a3 b1 − b1 b3 + b3 − 1 (35) b2 = , a1 = , 1 − b3 a3 − 1 i.e. the differential system   a3 b1 − b1 b3 + b3 − 1 ẋ = x 1 − x − y − b1 z = X̃1 , a3 − 1   a3 (a2 − 1) − a2 b3 + 1 ẏ = y 1 − y − a2 z − x = X̃2 , 1 − b3 ż = z (1 − z − a3 x − b3 y) = X̃3 . has the additional invariant plane g := g(x, y, z) = (1 − a3 )(a2 − 1)x + (1 − b1 )(b3 − 1)y + (a2 − 1)(b1 − 1)z = 0, with cofactor K = 1 − x − y − z. Moreover this differential system is completely integrable with the first integral H1 and the Jacobi multiplier J given by  α1 α α  β β x y 2 z 3 y 2 z 3 −3 (36) H1 = log , J= |g| , g g g g g where α1 = (a3 − 1)(b3 − 1)(a2 − 1), α2 = (1 − a3 )(b3 − 1)(b1 − 1), 1 − a2 − b1 a3 (b1 + b3 − 1) − b1 b3 α3 = (1 − a2 )(b1 − 1)(a3 − b3 ), β2 = , β3 = , a2 − 1 (1 − b3 )(a3 − 1) Proof. After some computations we can check that X (g) = (1 − x − y − z)g, thus g = 0 is an invariant plane with cofactor K = 1 − x − y − z. The existence of the first integral and of the Jacobi multiplier given in (36) follows from Theorem 3 by considering that in this case (see condition (11)) rank( (B1 )|σ ) = rank( (B2 )|σ ) = 3, where by σ we denote conditions (35), and     −1 −b2 −a3 −1 −1 −b2 −a3 −1 0  −a1 −1 −b3 −1   −a1 −1 −b3 −1 0  B1 =  −b1 −a2 , B2 =  . −1 −1   −b1 −a2 −1 −1 0  1 1 1 1 1 1 1 1 1  5.2. Asymmetric May-Leonard model with five invariant planes. Proposition 24. Differential system (34) under the conditions (37) a1 = 2 − b2 , a3 = 2 − b1 , a2 = 2 − b3 INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 23 i.e. the differential system ẋ = x (1 − x − (2 − b2 )y − b1 z) , (38) ẏ = y (1 − y − (2 − b3 )z − b2 x) , ż = z (1 − z − (2 − b1 )x − b3 y) , has the five invariant planes namely g1 = x, K1 = 1 − x − (2 − b2 )y − b1 z, g2 = y, K2 = 1 − y − (2 − b3 )z − b2 x, g3 = z, K3 = 1 − z − (2 − b1 )x − b3 y, g4 = 1 − x − y − z, K4 = x + y + z, g5 = x + y + z, K5 = 1 − x − y − z. Proof. It is follows from the relations (39) d d (x + y + z) = (1−x−y−z)(x+y+z), (1 − x − y − z) = (x+y+z)(1−x−y−z). dt dt  Remark 25. From (39) it follows that 1 x(t) + y(t) + z(t) = 1 + . 1 − ce−t Hence we get that lim (x(t) + y(t) + z(t)) = 1, lim (x(t) + y(t) + z(t)) = 0. t−→+∞ t−→−∞ Thus the α-limits are in the plane x + y + z = 0 and the ω- limits are in the plane x + y + z = 1. Proposition 26. Differential system (38) having the invariant plane ν4 −x−y−z = 0 with cofactor x + y + z + s4 , where ν4 and σ4 are constants such that ν4 s4 = 0, is completely integrable if and only if (s24 + (b1 + b2 + b3 − 3)2 ) (1 − s4 ) = 0. (i) If b1 + b2 + b3 = 3 and s4 = 0, then system (38) has two independent first integrals (40)     |x| |z| H1 = log , H2 = log . |y(1 − x − y − z)1−b2 | |y(1 − x − y − z)b3 −1 | (ii) If s4 = 1, then system (38) has the following first integral and the Jacobi multiplier  H = log |x|b3 −1 |y|b1 −1 |z|b2 −1 |x + y + z|3−b1 −b2 −b3  b3 −1  b1 −1  b2 −1 ! |x| |y| |z| = log , (41) |x + y + z| |x + y + z| |x + y + z| 1 J= , |xyz(x + y + z − 1)| 24 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ Proof. First we observe that the matrix B1 and B2 in this case are   −1 −b2 −2 + b1 −1  −2 + b2 −1 −b3 −1  B1 |(37) =   −b1 , −2 + b3 −1 −1  1 1 1 s4   −1 −b2 −2 + b1 −1 0  −2 + b2 −1 −b3 −1 0  B2 |(37) =   −b1 . −2 + b3 −1 −1 0  1 1 1 1 1 After some computations we obtain that (42)   −1 −b2 −2 + b1 −1   −2 + b2 −1 −b3 −1  2, if s4 = 0 and b1 + b2 + b3 = 3  rank   = −b1 −2 + b3 −1 −1  3, if s4 = 1. 1 1 1 s4 Hence from Theorem 2 we obtain the proof of the first statement. The indepen- dent first integrals are given by formula (40). Differential system (38) under the conditions b1 + b2 + b3 = 3, and s4 = 0, can be written as ẋ = {H1 , H2 , x}/J, ẏ = {H1 , H2 , y}/J, ż = {H1 , H2 , z}/J, where J = 1/|xyz(x + y + z − 1)|.   Now we prove statement (ii). From (42) we get that if s4 = 1, then rank B1 |(37) = 3. On the other hand   −1 −b2 −2 + b1 −1 0    −2 + b2 −1 −b3 −1 0  rank B2 |(37) = rank  −b1  = 3. −2 + b3 −1 −1 0  1 1 1 1 1 Thus in view of Theorem 5 we get that there exits a Darboux first integral and a Jacobi multiplier which in this case are given in (41). In short the proposition is proved.  Remark 27. Differential system (34) under the conditions a1 = a2 = a3 = a, b1 = b2 = b3 = b, becomes the so called symmetric May-Leonard model ẋ = x (1 − x − ay − bz) , (43) ẏ = y (1 − y − az − bx) , ż = z (1 − z − ax − by) . The integrability of system (43) was study in [2] . Condition (35) in this case becomes a2 + b2 − ab − a − b + 1 = 0 with a 6= 1 and b 6= 1, which have not solution in R. Condition (37) becomes a + b = 2 then if s4 = 0 and b = a = 1, then the first |x| |z| integrals (40) can be written as H1 = log and H2 = log . If a + b = 2 and |y| |y| INTEGRABILITY OF A CLASS OF N – DIFFERENTIAL SYSTEMS 25 xyz s4 = 1 then first integral (41) is H = log and the Jacobi multiplier is x+y+z 1 J= . |xyz(1 − x − y − z)| Acknowledgements This work is supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grants MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. References [1] F. Crespo, F. Molero and S. Ferrer, Poisson and Integrable Systems through the Nambu Bracket and its Jacobi Multiplier, J. of Geometric Mechanics 2 (2016), 169–178. [2] G. Blé, V. Castellanos, J. Llibre and I. Quilatán, Integrability amd global dynamics of the May-Leonard model, Nonlinear Analysis-, Real World Applications, 14 (2013), 280–293. [3] Chiai-Wei Chi, Sze-Bi Hsu and Lin-Ing Wu On the asymmetric May-Leonard model of three competing species, SIAM J. Appl. Math. 58 (1998) 221–226. [4] F. Dumortier, J. Llibre and J.C. Artés, Qualitative theory of planar differential systems, Universitext, Springer, 2006. [5] A. Goriely, Integrability and nonintegrability of dynamical systems, Advances Series in Nonlinear Dynamics 19, World Scientific Publishing Co., Inc., River Edge, NJ, 2001. [6] J. Hofbauer and K. Sigmund, Evolutionary games and population dynamics, Cambridge University Press, Cambridge, 1988. [7] J. Llibre, R. Ramı́rez and V. Ramı́rez, Complete integrability of vector fields in RN , preprint (2019). [8] J. Llibre and R. Ramı́rez, Inverse problems in ordinary differential equations and applica- tions, Progress in Math. 313, Birkhäuser, 2016. [9] J. llibre and X. Zhang, Darboux theory of integrability in CN taking into account the multiplicity, J. Differential Equations 246 (2009), 541–551. [10] J. llibre and X. Zhang, Darboux theory of integrability in RN taking into account the multiplicity at infinity, Bull. Sci. math., 133 (2009), 765–778. [11] A.J. Lotka Analytical note on certain rhythmic relation in organic systems, Proc. Natl. Acad. Sci. USA 6 (1920), 410–415. [12] Y. Nambu, Generalized Hamiltonian dynamics, Phys. Rev. D 7 (1973), 2405–2412. [13] V. Volterra, Lecons sur la Théorie Mathématique de la Lutte pour la vie, Gathier-Villars, Parı́s, 1931. [14] R. Tudoran, A normal form of completely integrable systems, J. of Geometry and Physics 62 (2012), 1167–1174. [15] E.T. Whittaker, A treatise on the Analytic Dynamics of Particles and Rigid Bodies, Dover, New York, (1944). 1 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bel- laterra, Barcelona, Catalonia, Spain. Email address: jllibre@mat.uab.cat 2 Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Vir- gili, Avinguda dels Paı̈sos Catalans 26, 43007 Tarragona, Catalonia, Spain. Email address: rafaelorlando.ramirez@urv.cat 26 J. LLIBRE, R. RAMÍREZ AND V. RAMÍREZ 3 Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain. Email address: valentin.ramirez@e-campus.uab.cat