Editing Lotka–Volterra equations (section)

====Second fixed point (oscillations)====
Evaluating ''J'' at the second fixed point leads to
<math display="block">J\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right) = \begin{bmatrix}
 0 & -\frac{\beta \gamma}{\delta} \\
 \frac{\alpha \delta}{\beta} & 0
\end{bmatrix}.</math>

The eigenvalues of this matrix are
<math display="block">\lambda_1 = i \sqrt{\alpha \gamma}, \quad \lambda_2 = -i \sqrt{\alpha \gamma}.</math>

As the eigenvalues are both purely imaginary and conjugate to each other, this fixed point must either be a center for closed orbits in the local vicinity or an attractive or repulsive spiral. In conservative systems, there must be closed orbits in the local vicinity of fixed points that exist at the minima and maxima of the conserved quantity. The conserved quantity is derived above to be <math>V = \delta x - \gamma \ln(x) + \beta y - \alpha \ln(y)</math> on orbits. Thus orbits about the fixed point are closed and [[Dynamical system#Conjugation results|elliptic]], so the solutions are periodic, oscillating on a small ellipse around the fixed point, with a frequency <math>\omega = \sqrt{\lambda_1 \lambda_2} = \sqrt{\alpha \gamma}</math> and period <math>T = 2{\pi}/(\sqrt{\lambda_1 \lambda_2})</math>.

As illustrated in the circulating oscillations in the figure above, the level curves are closed [[orbit (dynamics)|orbit]]s surrounding the fixed point: the levels of the predator and prey populations cycle and oscillate without [[Damping ratio|damping]] around the fixed point with frequency <math>\omega = \sqrt{\alpha \gamma}</math>.

The value of the [[constant of motion]] {{math|''V''}}, or, equivalently, {{math|1=''K'' = exp(−''V'')}}, <math>K = y^\alpha e^{-\beta y} x^\gamma e^{-\delta x}</math>, can be found for the closed orbits near the fixed point.

Increasing {{math|''K''}} moves a closed orbit  closer to the fixed point. The largest value of the constant {{math|''K''}} is obtained by solving the optimization problem
<math display="block">y^\alpha e^{-\beta y} x^\gamma e^{-\delta x} = \frac{y^\alpha x^\gamma}{e^{\delta x+\beta y}} \longrightarrow \max_{x,y>0}.</math>
The maximal value of ''K'' is thus attained at the stationary (fixed) point <math>\left(\frac{\gamma}{\delta}, \frac{\alpha}{\beta}\right)</math> and amounts to 
<math display="block">K^* = \left(\frac{\alpha}{\beta e}\right)^\alpha \left(\frac{\gamma}{\delta e}\right)^\gamma,</math>
where {{math|''e''}} is [[e (mathematical constant)|Euler's number]].