Chapter 5.5, Problem 4E

Prove that the critical point $\left(8\right)$ of the Volterra-Lotka system is a center; that is, the neighboring trajectories are periodic. [Hint: Observe that $\left(9\right)$ is separable and show that its solution can be expressed as $\left[x_2^A{e}^{-B{x}_2}\right]\cdot \left[x_1^C{e}^{-D{x}_1}\right]=K](26)$.

Prove that the maximum of the function $x^p{e}^{-qx}$ is ${\left(p/qe\right)}^p$, occurring at the unique value $x=p/q$ (see Figure 5.24), so the critical values $\left(8\right)$ maximize the factors on the left in $\left(26\right)$. Argue that if $K$ takes the corresponding maximum value ${\left(A/Be\right)}^A{\left(C/De\right)}^C$, the critical point $\left(8\right)$ is the (unique) solution of $\left(26\right)$, and it cannot be an endpoint of any trajectory for $\left(26\right)$ with a lower value of $K^{†}$.

$†$ In fact, the periodic fluctuations predicted by the Volterra-Lotka model were observed in fish populations by Lotka’s son-in-law, Humberto D’Ancona.

$\begin{array}{rll}{x}_2\left(t\right)& \equiv & \frac{A}{B}\\ x_1\left(t\right)& \equiv & \frac{C}{D}(8)\end{array}$

$\frac{d{x}_2}{d{x}_1}=\frac{-C{x}_2+D{x}_1{x}_2}{A{x}_1-B{x}_1{x}_2}=\frac{x_2}{x_1}\times \frac{-C+D{x}_1}{A-B{x}_2}(9)$