Concept
A dynamical flow associated with an observation vector ${\bf y}(t)$ may have functions, $I({\bf y})$ that are time independent, being $dI/dt=0$. The number of invariants and the length of the observation vector have an effect on overall dynamics.
Lotka-Volterra (LV) System
The LV dynamics explains the behaviour between population of the prey $v$ and population of predators $u$, a case of predator-prey model. We will use a special case of the LV dynamics, remember the dot notation, meaning time derivatives, for predators,
$$ \dot{u} = u (v-2) $$
and for prays,
$$ \dot{v} = v (1-u) $$
Observation vector will consist of $y=(u,v)$.
If we divide these equations, hoping that we can collect $u$ and $v$ in separate terms,
$$
\begin{eqnarray}
\frac{\dot{u}}{\dot{v}} & = & \frac{u (v-2)}{v(1-u)} \\
\dot{u} v (1-u) & = & \dot{v} u (v-2) \\
\dot{u} v (1-u) - \dot{v} u (v-2) & = & 0 \\
\dot{u} (1-u) - \dot{v} u/v (v -2) & = & 0\\
\dot{u} (1-u)/u - \dot{v}(v-2)/v & = & 0
\end{eqnarray}
$$
If we integrate both sides over time $dt$,
$$
\begin{eqnarray}
\int \frac{1-u}{u} \frac{du}{dt} dt - \int \frac{v-2}{v} \frac{dv}{dt} dt & = &0 \\
\int \frac{1-u}{u}du - \int \frac{v-2}{v} dv & = &0 \\
\end{eqnarray}
$$
Solution of these indefinite integrals yields to a an invariant of the LV dynamics
$$ d I({\bf y})/dt = ln u - u + 2 ln v - v $$
We have shown one invariant of the system. This is important to determine the structure of the system, such as volume preserving dynamics, i.e., Hamiltonian Dynamics.
Further Reading
- Geometric Numerical Integration, Ernst Hairer, Christian Lubich, Gerhard Wanner, Springer (2002)
- Arnold, V. I. and A. Avez (1968). Ergodic Problems of Classical Mechanics. New York, Benjamin.
