Exponential Growth and Decay: Population, Radioactive Substances, and Temperature, Study notes of Differential Equations

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Differential Equations

7.4 Exponential Growth and Decay

Exponential Growth and Decay

Now let’s take another 1000 bacteria of the same type and put them with the first population. Each half of the new population was growing at a rate of 300 bacteria per hour. We would expect the total population of 2000 to increase at a rate of 600 bacteria per hour initially (provided there’s enough room and nutrition). So if we double the size, we double the growth rate. In general, it seems reasonable that the growth rate should be proportional to the size.

Exponential Growth and Decay

The same assumption applies in other situations as well. In nuclear physics, the mass of a radioactive substance decays at a rate proportional to the mass. In chemistry, the rate of a unimolecular first-order reaction is proportional to the concentration of the substance. In finance, the value of a savings account with continuously compounded interest increases at a rate proportional to that value.

Exponential Growth and Decay

 =  k dt

ln | y | = kt + C |y | = e kt+C = e C e kt y = Ae kt where A (=  e C or 0) is an arbitrary constant. To see the significance of the constant A , we observe that y (0) = Ae k  0 = A Therefore A is the initial value of the function.

Exponential Growth and Decay

Because Equation 1 occurs so frequently in nature, we summarize the following.

Population Growth

What is the significance of the proportionality constant k? In the context of population growth, we can write or The quantity is the growth rate divided by the population size; it is called the relative growth rate.

Population Growth

According to (3), instead of saying “the growth rate is proportional to population size” we could say “the relative growth rate is constant.” Then (2) says that a population with constant relative growth rate must grow exponentially. Notice that the relative growth rate k appears as the coefficient of t in the exponential function y 0 e kt .

Example 1 – Modeling World Population with the Law of Natural Growth Assuming that the growth rate is proportional to population size, use the data in Table 1 to model the population of the world in the 20th century. What is the relative growth rate? How well does the model fit the data? Table 1

Example 1 – Solution

We measure the time t in years and let t = 0 in the year

1900. We measure the population P ( t ) in millions of people. Then the initial condition is P (0) = 1650. We are assuming that the growth rate is proportional to population size, so the initial-value problem is = kP P (0) = 1650 From (2) we know that the solution is P ( t ) = 1650 e kt

Example 1 – Solution

Thus the relative growth rate is about 0.88% per year and the model becomes P ( t ) = 1650 e 0.0087846 t cont’d

Example 1 – Solution

Table 2 and Figure 1 allow us to compare the predictions of this model with the actual data. cont’d Figure 1 A possible model for world population growth Table 2

Example 1 – Solution

The estimate for the relative growth rate is now 1.16% per year and the model is P ( t ) = 1650 e 0.0115751 t Figure 2 illustrates the second model. This exponential model is more accurate after 1970 but less accurate before

cont’d Figure 2 Another model for world population growth

Radioactive Decay