Hamiltonian structure of the Lotka-Volterra equations

Volume 145, number I PHYSICS LETTERS A 26 March 1990

HAMILTONIAN STRUCTURE OF THE LOTKA—VOLTERRA EQUATIONS Y. NUTKU

Department of Mathematics, Bilkent University, 06533 Bilkent, Ankara, Turkey

Received 22 December 1989; accepted for publication 22 January 1990 Communicated by D.D. HoIm

The Lotka—Volterra equations governing predator—prey relations are shown to admit Hamiltonian structure with respect to a generalized Poisson bracket. These equations provide an example of a system for which the naive criterion for the existence of Haniiltonian structure fails. We show further that there is a three-component generalization of the Lotka—Volterra equations which is a bi-Hamiltonian system.

The conditions for a dynamical system where A, B, C and D are constants. Since the vector

~k=xk k=l,2,...,2n, (1) field

to admit Hamiltonian structure are naively given by X= (A—By )x~- +(Cx—D)y (5)

[1] X 3’

Xk —0— (2) is not divergence free, the naive criterion for the ex-istence of Hamiltonian structure fails. On the other Gonzalez-Gascon [2] has noted that this criterion is hand we may consider the following ansatz for the valid only when the variablesx’ are chosen such that symplectic two-form,

the symplectic two-form w is cast into the canonical

form w=f(x,y)dxAdy, (6)

(o=dx’ ~ which is always closed in two dimensions. Hamil-(3) ton’s equations require that

wjX=dH, (7)

according to Darboux s theorem. When the original

variables defining the dynamical system are not of where the Hamiltonian function H is a zero-form. this form, the criterion (2) is too restrictive. In fact The integrability conditions of eqs. (7) are obtained Gonzalez-Gascon has given an example of a Ham- by applying the exterior derivative. Thus we find that iltonian system where this condition is violated. w given by eq. (6) will be symplectic provided

f

resent a familiar dynamical system. We shall show

[(A_By)X.flx+[(CXD)yf]vrz0. (8) that the predator—prey equations of Lotka and

Vol-terra provide another example of a Hamiltonian sys- This first-order equation has the solution tern for which the criterion (2) fails. It is surprising

that the Harniltonian structure of such a well-known

f=

system as the Lotka—Volterra equations has not been XY

noted earlier, plus an arbitrary function of its characteristic, The Lotka—Volterra equations are given by H=A ln y+D ln x— Cx— By, (10) ~=(A—By)x, j’=(Cx—D)y, (4) .

which also plays the role of the Hamiltornan func-0375-9601 /90/S 03.50 © Elsevier Science Publishers B.V. (North-Holland) 27

Volume 145, number 1 PHYSICS LETTERS A 26 March 1990 tion. Eq. (10) is well-known as the Liapunov func- This particular case is a bi-Hamiltonian system. tion for the Lotka—Volterra equations. It can be readily verified that eqs. (14) can be

In the dual representation [3] eqs. (4) can be written as Hamilton’s equations in two distinct ways, written as

~=J~VkH2=J~VkHl, (17)

~iJzkvkH (11)

where the components of x’ are given by x, y, z re-with x1 =x, x2=y, where spectively and

~(

\—Xy

o)

~

—bcxz yz 0 are the structure functions. The Jacobi identities

Jk[m\7Jnp)_O (13) 0 cxy(az+3’) cxz(y+v)

J2= —cxy(az+~) 0 xyz ).

are satisfied automatically because we are in two

(

dimensions. (19)

The Lotka—Volterra equations are a Hamiltonian

system with respect to the generalized Poisson bracket In three dimensions the Jacobi identities (13) re-defined in terms of eq. (12). They do not admit a duce to a single equation which is satisfied by any second Hamiltonian structure as an examination of linear combination of J1 and J2 with constant coef-the above general solution of eq. (8) reveals. There ficients. Thus they are compatible. No new con-exist several generalizations of the Lotka—Volterra served Hamiltonians are generated from the recur-equations

[4,51

Hamiltonian structure and we shall now consider a j, V/H,=0, J2 V/I-I2=0, (20) three-component generalization which is a

bi-Ham-iltonian system. that is, H~ and ‘~‘2are Casimirs of J1 and J2

Grammaticos et al.

[51

The multi-Hamiltonian structure of Lotka—Vol-~=x(cy+z+A), j’_—y(x+az+ i~ terra equations is evidently a rich subject as the above ~=z(bx+y+ ii), (14) examples indicate.

where some ofthe constants appearing in these

equa-tions can be related to those in eqs. (4) by scaling References the dynamical variables and time. It was pointed out

in ref. [5] that subject to the conditions [1] L. Andrey, Phys. Lett. A 111(1985) 45.

abc=—1, v=jth—Aab (15) [2] F. Gonzalez-Gascon, Phys. Lett. A 114(1986) 61. [31P.J. Olver, Graduate texts in mathematics, Vol. 107. eqs. (14) admit two conserved quantities, Applications ofLie groups to differential equations (Springer,

Berlin, 1986).

H1=abln x—b lny+ln z, [4] L. Brenig, Phys. Lett. A 133 (1988) 378.

[5] B. Grammaticos, J. Moulin-Ollagnier, A. Ramani, J.-M. H2 =abx+ y—az + v ln y—iiin z. (16) Strelcyn and S. Wojciechowski, preprint (1989).