1550078481-Ordinary_Differential_Equations__Roberts_

136 Ordinary Differential Equations

the population of the United States on the interval [1800, 1900]. Com-

pare your results with the actual results given in Table 3.1. Graph the

solution. (HINT: In the initial value problem (5) let to = 0 correspond

to the year 1800. So that t = 10 corresponds to 1810 , t = 20 corresponds

to 1820, and so forth. Thus, the interval of integration becomes [O, 100]

instead of [1800, 1900]. Also let the population be expressed in millions.

That is, let P(t) = p(t) x 106. Substituting for P in the differential

equation of (5), we find d(p x 106 ) / dt = k(p x 106 ) - E(p x 106 )^2. Divid-

ing by 106 , we see that the population expressed in millions satisfies the

differential equation dp/dt = kp - Ep^2 (10^6 ) and the initial condition is

p(O) = 5.31. Thus, in this example, you have translated the independent

variable, t , and scaled the dependent variable, P.)

4. If a population which is known to grow according to the logistic law
   model begins to be harvested at a constant rate, H , at time to, then for

t > t 0 the population satisfies the following initial value problem

dP = kP - EP^2 - H.

dt ) P(to) =Po

where k, E and Hare known positive constants and Po is the population

size at time t 0. Suppose when the population reaches the size Po = 200

a catfish farmer decides to h arvest fish from his pond at a constant rate

H. And suppose for his species of catfish the vital coefficients of the

population are k = .2 and E = .0004. Numerically compute and graph

the population of the catfish on the interval [O, 25] for

a. H = 10 b. H = 25 c. H = 50.

What do you think will happen to the catfish population as time in-

creases in each case? That is, what is limt_, 00 P(t) in each case?

5. Suppose the catfish farmer in exercise 4 waits to start harvesting fish
   until the population size reaches 1000 instead of 200. Compute and
   graph the catfish population on the interval [O, 25] for

a. H = 10 b. H = 25 c. H = 50.

What do you think will happen to the catfish population in each case

now?

Several other population growth models have been proposed by

various researchers. In the exercises which follow we present some

of these models.

6. The following population growth model, which is sometimes used in
   actuarial predictions, was proposed by Gompertz:
   dP

dt = P(a - blnP); P(O) =Po.