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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