In the predator-prey systemL represents a lady bug population and P represents a pest that...

In the predator-prey system

L represents a lady bug population and P represents a pest that lady bugs like to eat. Enter the system in the PPLANE6 Setup window, set the display window so that 0 ≤ L ≤ 2 and 0 ≤ P ≤ 3, then select Arrows for the direction field.

a) Use the Keyboard input window to start a solution trajectory with initial condition (0.5, 1.0). Note that the lady bug-pest population is periodic. As the lady bug population grows, their food supply dwindles and the lady bug population begins to decay. Of course, this gives the pest population time to flourish, and the resulting increase in the food supply means that the lady bug population begins to grow. Pretty soon, the populations come full cycle to their initial starting position.

b) Suppose that the pest population is harmful to a farmer’s crop and he decides to use a poison spray to reduce the pest population. Of course, this will also kill the lady bugs. Question: Is this a wise idea? Adjust the system as follows:

Note that this model assumes that the growth rate of each population is reduced by a fixed percentage of the population. Enter this system in the PPLANE6 Setup window, but keep the original display window settings. Create and set a parameter H = 0.2. Start a solution trajectory with initial condition (0.5, 1.0).

c) Repeat part b) with H = 0.4, 0.6, and 0.8. Is this an effective way to control the pests? Why? Describe what happens to each population for each value of the parameter H.