**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.

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