Prove that the critical point (8) of the Volterra–Lotka

Chapter 5, Problem 4E

(choose chapter or problem)

Prove that the critical point (8) of the Volterra–Lotka system is a center; that is, the neighboring trajectories are periodic. Hint: Observe that (9) is separable and show that its solutions can be expressed as (26) Prove that the maximum of the function xpe-qx is (p/qe)p, occurring at the unique value x = p/q (see Figure 5.24), so the critical values (8) maximize the factors on the left in (26). Argue that if K takes the corresponding maximum value (A/Be)A(C/De)C, the critical point (8) is the (unique) solution of (26), and it cannot be an endpoint of any trajectory for (26) with a lower value of K. Figure 5.24 Graph of xe-x