Question: The following equations describe the rates of growth of an insect and bird population in a particular region, where x is the insect population in millions at time t and y is the bird population in thousands:
(dx/dt) = 3x - 0.02xy
(dy/dt) = -10y + 0.001xy
(a) Describe in words the growth of each population in the absence of the other, and describe in words their interaction.
(b) Find the two points (x, y) at which the populations are in equilibrium.
(c) When the populations are at the nonzero equilibrium, 10 thousand additional birds are suddenly introduced. Let A be the point in the phase plane representing these populations. Find a differential equation in terms of just x and y (i.e., eliminate t), and find an equation for the particular solution passing through the point A.
(d) Show that the following points lie on the trajectory in the phase plane that passes through point A:
(i) B (9646.91, 150)
(ii) C (10,000, 140.43)
(iii) D (10,361.60, 150)
(e) Sketch this trajectory in the phase plane, with x on the horizontal axis, y on the vertical. Show the equilibrium point.
(f) In what order are the points A, B, C, D traversed? [Hint: Find dy/dt, dx/dt at each point.]
(g) On another graph, sketch x and y versus time, t. Use the same initial value as in part (c). You do not need to indicate actual numerical values on the t-axis.
(h) Find dy/dx at points A and C. Find dx/dy at points B and D.