Goal: Investigate the qualitative behavior of a nonlinear system of differential equations. Tools needed: pplane8
Description: A farmer has ladybugs and aphids in her fields. The helpful ladybug (predator) eat the destructive aphid (prey) who decour her crops.
Let
x(t) = aphid population (in millions) at time t,
y(t) = ladybug population (in millions) at time t.
The farmer knows that the growth rates of the aphid and ladybug populations are given respectively by
dx/dt = x(1 − y),
dy/dt = y(x − 1).
Assume there are initially 800, 000 aphids and 400, 000 ladybugs in all that
follows below.
1. Use pplane8 to plot the trajectory through (0.8, 0.4). As t increases, describe what happens to each population. Is the aphid population ever smaller than 300, 000? Are the aphids ever eradicated? Does the ladybug population ever exceed 2 million?
2. If she were to use a pesticide, the growth rates would then become
dx/dt = x(1 − y) − sx,
dy/dt = y(x − 1) − sy. (*)
where s ≥ 0 is a measure of the "strength" of the pesticide – the larger the s, the stronger the pesticide. Currently there are only two commercially available strengths:
s = 0.5 and s = 0.75. Plot the trajectories for the new system of equations (∗) with these values of s. Will the aphids ever be totally eliminated?
3. If she knows her crops will survive if the aphid population never exceeds 2.6 million, which strength (if any) would you recommend she use: s = 0.0 (no pesticide), s = 0.5, s = 0.75?
4. By special permission, she could get a pesticide with the maximum strength of s = 1.5. Plot this trajectory. What happens to the ladybugs and aphids if she uses this pesticide?