Homework Solutions (1



Solutions to Problem Set 3

1.

a. [pic]

h=hunting parameter

b. [pic]

m=moving parameter

c. There are several things you could do, but the simplest is to increase your α. This says that the growth rate for the rabbit population increases.

2.

a. [pic]

Solve for x first. You can either use integrating factor, separation, or undetermined coefficients. I choose to use integrating factor.

[pic]

Plug in your solution for x into the first equation to solve for y. The only method you can use to solve for x is separation.

[pic]

[pic]

b. (-1,0)

c. [pic]

[pic]

[pic]

[pic]

d. Print out graph from HPGSolver.

3. [pic]

a. Guess: [pic]

[pic]

b.

[pic]

[pic]

[pic]

c) Use HPGSolver to view direction field.

d) [pic]

2.1 (2)

i. Equilibrium points (0,0) and (10,0). For the latter equilibrium point prey alone exists; there are no predators. There is also an equilibrium point at (100, -4.5), but this one has no validity – population can’t be negative.

ii. The equilibrium points are (0,0), (0,15), and (3/5,30). For the latter equilibrium point, both species coexist. For (0,15), if the prey are extinct the predators survive.

2.1 (8)

All solutions go to the equilibrium solution point A. So all x(t) graphs go to about 1.2 and all y(t) graphs go to about .75. Use HPG Solver to view x(t) and y(t) graphs for each point A, B, C and D.

2.2 (8) [pic]

a. [pic]

b. [pic]

c. Print graph from HPGSolver.

d. [pic]

e. Solutions are periodic. They form ellipses instead of circles because y and v do not have the same period.

2.2 (14)

a. Equilibrium solutions (2,1)

b. Print phase portrait and direction field from HPGSolver.

c. A typical solution (solution with initial condition with both populations positive) goes to coexistence (the equilibrium point (1,2)).

b.

[pic]

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