1550078481-Ordinary_Differential_Equations__Roberts_

134 Ordinary Differential Equations

and the rate of population growth will approach zero. Using the techniques
of separation of variables and partial fraction decomposition, it can be shown
that the explicit solution of the initial value problem

(5)

is

(6) P( t ) = ePo + (k -kPo ePa)e-k(t-ta)

where Po is the size of the population at time t 0. Notice that as t -+ oo,

P(t) -+ k/ E. So, regardless of the initial population size, the population

ultimately approaches the limiting value of k/ e. Consequently, the constants

k and E are called the vital coefficients of a population and the constant

K = k/e is called the carrying capacity of the population. The graph of

equation (6) has an elongated S-shape and is called the logistic curve. See

Figure 3 .10.

p

8e+09

6e+09

4e+09

2e+09

0

1700 1800 1900 2000 2100 2200

Figure 3.10 Graph of the "Logistic Curve"

for the Earth's Human Population

Experts have estimated that the earth's human population has a vital coef-

ficient k = .029. Given that the population of the world in 1960 was 3 billion

people and the growth rate, (dP/dt)/P, was 1.8% per year, we can deter-

mine the vital coefficient E in the following manner. Dividing the differential

equation of (5) by P, we find that the growth rate expressed as a percentage

satisfies

dP/dt p = k - EP.