Section 2.1

Population ModelsIn Chapter 2, we will spend a great deal of time on a more in-depth discussion of the uses

of differential equations for modeling physical phenomena. In particular, in this section we willdiscuss the uses of differential equation in modeling populations. In homework and in previousclasses, we have already seen a very basic population model, which assumes that a population willgrow proportionally to its size, i.e. exponential growth; the exponential model for population growthis given by

Exponential Model: P ′ = kP for some number k.

Below we will discuss an example and see the model’s limitations.

Example. At time t = 0, a herd consists of 100 unicorns. After 1 year, the herd size has increasedto 110. Assuming that the population grows at a rate proportional to its size, write and solve adifferential equation to model the population size.

Letting P = P (t) represent the population size at time t, we are told that P ′ is a multiple ofP , so we will model the population’s growth with the exponential model

P ′ = kP.

This equation is separable, thus easy to solve: rewriting as

dP

P= k dt,

we integrate both sides, ∫dP

P=

∫k dt,

to getlnP = kt+ C

so thatP = ekt+C .

We can solve for C using the initial condition P (0) = 100:

100 = P (0) = e0+C ,

so C = ln 100. We haveP = 100ekt,

and we can find k using the fact that P (1) = 110:

110 = P (1) = 100ek

means that

k = ln(1110

).

So the exponential growth equation that models the population is

P = 100e(ln1110

)t.

Let’s graph the curve P :

1

Section 2.1

Exponential model of Unicorn population

Notice that this model estimates that, in 50 years, the unicorn population will be over 11, 000,and that the model suggests that the population will approach infinity. This seems unlikely, evenfor unicorns.

The exponential model is rather naive in its assumptions; the model assembles a tiny amountof information and proceeds to make grand inferences about the population. In particular, themodel does not specifically account for the fact that birth rates are rarely constant, and it isunclear as to how the model accounts for death rates. Thus the exponential population model isquite unrealistic and limited in its practicality, and we would like to develop a model that betterestimates population.

Overall, the pros and cons of the exponential model can be summarized as follows:Pros

• Easy to use/solve

• Requires minimum data to use

• Decent approximation over short time periods

Cons

• Over time, model always predicts infinite growth.

The Logistic EquationA population’s birth and death rates are generally the most important factors in determining

the population’s size. The logistic model that we will study in this section specifically accounts forthese two mechanisms of population growth (or decline), and assumes that population change isproportional to the difference between the birth rate and death rate. While this may still seemssimplistic, it turns out that the logistic model will provide a surprisingly accurate prediction forpopulation size.

Given a population P = P (t) viewed as a function of time t, let β be the population’s birthrate and δ be its death rate. It turns out that birth rates tend to be linearly decreasing over time,so we will assume that β is of the form

β = β0 − β1P,

2

Section 2.1

where β0 and β1 are constants. Death rates, on the other hand, tend to be constant, so we willassume that δ is a constant. The logistic model is

dP

dt= (β − δ)P = (β0 − β1P − δ)P,

which is often rewritten as

Logistic Model:dP

dt= kP (M − P ),

where

k = β1 and M =β0 − δ

β1.

The information with which we are provided will determine which form of the logistic modelwe will use, and which parameters we will actually need to calculate. For example, the problembelow allows us to bypass a calculation of the birth and death rates and simply determine valuesfor k and M .

Example. A herd of 100 unicorns is initially growing at a rate of 10 unicorns per year. In 5 years,the herd has 144 unicorns and is growing at a rate of 9 unicorns per year. Assuming that thepopulation size follows the logistic model, find the function P (t) that models the population’s size.

The logistic model states thatP ′ = kP (M − P );

we are given enough information to be able to determine k and M . Since

P (0) = 100 and P ′(0) = 10,

we know that10 = 100k(M − 100).

Similarly, sinceP (5) = 144 and P ′(5) = 9,

we know that9 = 144k(M − 144).

Since M and k are both constants, let’s solve both equations for M : using

10 = 100k(M − 100), we see that M = 100 +1

10k.

On the other hand, using

9 = 144k(M − 144), we see that M = 144 +1

16k,

which means that

100 +1

10k= 144 +

1

16k.

3

Section 2.1

Multiplying both sides through by k, we have

44k =6

160so that k =

3

3520.

Now we can determine M , say using the first relationship: since

M = 100 +1

10k,

we see that

M = 100 +1

10 · 33520

=652

3.

So our logistic equation is given by

P ′ =3

3520P

(652

3− P

).

Since this equation is separable,dP

P (6523 − P )=

3

3520dt,

we can solve it by integrating both sides:∫dP

P (6523 − P )=

∫3

3520dt.

The right hand side is straightforward:∫3

3520dt =

3t

3520+ C.

The left hand side is harder–we’ll need to rewrite the integrand using partial fractions. Thefraction

1

P

(6523 − P

)may be rewritten as

1

P (6523 − P )=

A

P+

B6523 − P

.

Adding the two fractions on the right gives us

1

P (6523 − P )=

A(6523 − P ) +BP

P (6523 − P )

so that the numerators yield the equality

A(652

3− P ) +BP = 1

or652

3A+ (−A+B)P = 1 + 0P.

4

Section 2.1

We may separate this into the two equations

652

3A = 1 and −A+B = 0.

Thus

A =3

652,

and the second equation tells us thatA = B

so that

B =3

652.

So the original integral may be rewritten as∫dP

P (6523 − P )=

∫3

652P+

3

652(6523 − P )dP

=3

652ln |P | − 3

652ln |652

3− P |

=3

652(ln |P | − ln |652

3− P |)

=3

652ln | P

6523 − P

|.

Now we have the equation

3

652ln | P

6523 − P

| = 3t

3520+ C,

so that

ln | P6523 − P

| = 652

3· 3t

3520+ C =

163t

880+ C;

exponentiating both sides, we have

P6523 − P

= e163t880

+C = e163t880 eC .

This is equivalent to

P =652e

163t880 eC

3− Pe

163t880 eC ,

or

P (1 + e163t880 eC) =

652e163t880 eC

3.

Finally, we have the solution

P (t) =652e

163t880 eC

3(1 + e163t880 eC)

.

5

Section 2.1

We can determine C by using the initial condition P (0) = 100 and an earlier equation,

P6523 − P

= e163t880 eC .

We have100

6523 − 100

= e0eC ,

so that

eC =75

88.

Thus our population function is given by

P (t) =652e

163t880 · 75

88

3(1 + e163t880 · 75

88),

or

P (t) =12225e

163t880

66(1 + 7588e

163t880 )

.

The curve modeling the first 50 years of the population is graphed below:

Logistic model of Unicorn population

Notice that the logistic model predicts that the herd approaches 218 unicorns over time–thiscertainly seems more reasonable than the unlimited growth predicted by the exponential model.There is a sense in which 218 is a sort of ”magic number” here; the environment is specificallywell-suited to support this herd size. This number (M in our calculations above) is known as thelimiting population or carrying capacity.

The pros and cons of the logistic model are summarized below:Pros

• Significantly more accurate than the exponential model.

Cons

6

Section 2.1

• Harder to solve than the equation produced by exponential model.

• Requires more data than does the exponential model.

While the logistic model is still a fairly simplistic population model, it appears to be significantlymore realistic than is the exponential model. In fact, the balance of simplicity and accuracy is goodenough that the logistic model, or variations thereof, tends to be the population model of choicefor many applications.

As a quick illustration, let’s look at the chart from page 84 in your book. The chart uses U.S.population data from the 19th century to predict the population of the United States using boththe exponential and the logarithmic model. In addition, the chart provides the actual populationdata for comparison.

U.S. Population vs. Predicted Population

Both models provide a decent approximation to the actual population throughout the 19thcentury; however, near the beginning of the 20th century the exponential model data starts todiffer dramatically from the actual population data. By 1950, when the prediction supplied bythe logistic model is still close to the actual population, the exponential model is off by well over100%. Clearly the logistic model is a significantly better choice for population predictions than isthe exponential model.

7