Classical Mechanics (Kibble & Berkshire) · Problem 8.10

The energy principle — T.W.B. Kibble, F.H. Berkshire

Physics — Mechanics · predator-prey-phase-paths

Problem

Predator--prey Consider the symmetrical predator--prey equations

\dot{x} = x - xy, \qquad \dot{y} = xy - y,

where

x(t)

and

y(t)

are positive functions. Show that the phase paths satisfy the equation

\left(xe^{-x}\right)\left(ye^{-y}\right) = A,

where

A

is a constant whose value determines the particular phase path. By considering the shape of the surface

z = \left(xe^{-x}\right)\left(ye^{-y}\right),

deduce that each phase path is a simple closed curve that encircles the equilibrium point at

(1, 1)

. Hence every solution of the equations is periodic! [This prediction can be confirmed by solving the original equations numerically.]