Lotka-Volterra 3d

Lotka-Volterra Model in 3D

The Lotka–Volterra model in 3D is an example of a Hamiltonian system with degenerate Poisson structure.

The equations read

\[\begin{aligned} \dot{q}_{1} &= q_{1} ( - a_{2} q_{2} + a_{3} q_{3} - b_{2} + b_{3} ) , \\ \dot{q}_{2} &= q_{2} ( \hphantom{-} a_{1} q_{1} - a_{3} q_{3} + b_{1} - b_{3} ) , \\ \dot{q}_{3} &= q_{3} ( - a_{1} q_{1} + a_{2} q_{2} - b_{1} + b_{2} ) , \\ \end{aligned}\]

which can be written in Poisson-form as

\[\dot{q} = P(q) \nabla H(q) ,\]

with Poisson matrix

\[P(q) = \begin{pmatrix} 0 & - q_{1} q_{2} & \hphantom{-} q_{1} q_{3} \\ \hphantom{-} q_{1} q_{2} & 0 & - q_{2} q_{3} \\ - q_{1} q_{3} & \hphantom{-} q_{2} q_{3} & 0 \\ \end{pmatrix} ,\]

and Hamiltonian

\[H(q) = a_{1} q_{1} + a_{2} q_{2} + a_{3} q_{3} + b_{1} \ln q_{1} + b_{2} \ln q_{2} + b_{3} \ln q_{3} .\]

References:

• A. M. Perelomov. Selected topics on classical integrable systems, Troisième cycle de la physique, expanded version of lectures delivered in May 1995.

• Yuri B. Suris. Integrable discretizations for lattice systems: local equations of motion and their Hamiltonian properties, Rev. Math. Phys. 11, pp. 727–822, 1999.