1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 463

3. Without using any computer software, decide how the results of exercise 2
   will be affected in the following cases:
   a. Only the prey population is harvested with harvesting coefficient

h1 < r = 2. That is, for const ants as given in exercise 2, what is the ef-

fect of adding t he term -h 1 x to the first equation of system (1)?

b. Only the predator population is harvested with harvesting coefficient h 2.

That is, for constants as given in exercise 2, what is the effect of adding the

term -h 2 y to the second equation of system (1)?

c. Both prey and predator populations are harvested with harvesting coef-

ficients h1 and h2, respectively.

4. a. For h 1 = h 2 = 0 find and classify all critical points of the internal prey
   and internal predator competition with harvesting model, system (2), in the
   first quadrant.
   b. For h 1 =f=. 0 and h2 =f=. 0 find and classify all critical points of system (2)
   in the first quadrant.
   c. How does harvesting affect the prey and predator populations? Is the
   answer the same as for t he Volterra-Lotka prey-predator model?


5. Use computer software to solve the internal prey and internal predator com-

petition with harvesting model, system (2), for the initial conditions x(O) = 3,

y(O) = 2 on the interval [O, 5] for r = 3, C = 2, H = s = P = Q = 1, and the

following five values for the harvesting coefficients:

a. h 1 = h2 = 0 b. h1 = 1, h2 = 0 c. h1 = 0, h2 = 1 d. h1 = h2 = .5

e. h1 = h2 = .25

For each case,

(i) Display a graph of x(t) and y(t) on the interval [O, 5].

(ii) Display a phase-plane graph of y versus x.

(iii) What happ ens to x(t) and y(t) as t increases?

6. Use SOLVESYS or other computer software to solve the three species
   model, system (3), for t he initial conditions Y1(0) = 5, y2(0) = 4, y3(0) = 3
   on the interval [O, 5] for the following values of the constants.

a. a =d=g=l, b=c = e =. 25 and f=h=i=.l

b. a = d = g = 1, b = f = h = i = .1 and c = e = .25

In each case, display a single graph showing Y1 ( t), Y2 ( t), and y3 ( t) on the
interval [O, 5]. What happens to y 1 (t), Y2(t), and y3(t) as t increases?

7. a. Find the critical points of the three species model, system (4).
   b. Use computer software to solve system ( 4) for the init ial conditions
   y 1 (0) = 1, Y2(0) = 1, y3(0) = 1 on the interval [O, 6] for the following values
   of t he constants.