Chapter 9.R, Problem 22E

Populations of birds and insects are modeled by the equations

$\begin{array}{l}\frac{dx}{dt}=0.4x-0.002xy\\ \frac{dy}{dt}=-0.2y+0.000008xy\end{array}$

(a) Which of the variables, x or y, represents the bird population and which represents the insect population? Explain.

(b) Find the equilibrium solutions and explain their significance.

(c) Find an expression for $dy/dx$.

(d) The direction field for the differential equation in part (c) is shown. Use it to sketch the phase trajectory corresponding to initial populations of 100 birds and 40, 000 insects. Then use the phase trajectory to describe how both populations change.

(e) Use part (d) to make rough sketches of the bird and insect populations as functions of time. How are these graphs related to each other?