Name
Problem Set - Population Dynamics
Perhaps the most successful conjunction of math and biology has been in the study of ecology and
evolution.
This problem set focuses on one aspect of population biology: population size. We will try to get an
understanding, via mathematics, of how populations change over time.
THE EXPONENTIAL EQUATION
Let's think of a population of individuals that reproduces asexually via fission. It could be yeast or bacteria,
it doesn't really matter what it is, just what it does. We start with 1 individual at time (t) = 0.
1. If after every hour, each individual divides into two individuals, list how many individuals there
will be at each time interval:
time, t (h)
0
1
2
3
4
5
6
#indiv (N)
But tables like this would get very unwieldy after even one day of growth. So, let's try to simplify the
relationship with a function that tells us how many individuals there are at any interval.
2. Complete the following function by giving 2 an exponent (HINT: it will be a variable):
N₁= 2-
Throughout this problem set, N will refer to population size and N, is population at time t. We're not going
to worry about what the units of time are, however.
Now we will try to make this a more general equation that can be applied to populations with different
growth rates. It is possible to represent 2 as any combination of numbers, e.g., : 2=1+1,=√
=√√2²
148
So
74
we can begin to make the function in question 2 more general by substituting a new combination of values
for 2. Because it is an exponential relationship, we'll try using e and to make it equal to 2, we will raise it to
the power of 0.693. For this population:
N₁ = 2¹ = e0.693t
We're still not at the general equation, however, because we haven't really changed the equation, just its
form. It's still just 2¹. Now we said that this population doubles every hour. But not every population does.
So, which part of this equation is responsible for making it double every hour? 0.693. e is a constant, so if we want the population's rate of change to be something different from doubling every hour, we should
change the exponent. To make the exponent a variable, let's replace 0.693 (which applies just to this
population) with r:
rt
N₁ = e't
So now depending on the population, we can have r = whatever is appropriate: 0.693 in these yeast or 0.214
in some flies or 0.014 in elephants, for example. But there's one more thing! Not all populations begin at
the same size. If you have a sexually reproducing population, you definitely need more than one individual,
and populations can start with many more than one. So, where should we add the value for the population at
time 0? Remember, that in this example No = 1, therefore, our final, general equation becomes:
N₁ = Noert
This equation demonstrates the size of a population as a function of time and is called the exponential
equation. (NOTE: This is the "integrated form" of the equation. We will also look at the differential form
later.) No is a constant it is a constant for any one population, but it will differ among populations. It is the
variable time in whatever units are appropriate. We'll just consider it unitless. r is perhaps the most
important component of the equation, and it determines the rate at which a population grows over time. ris
called the intrinsic rate of natural increase, and it is a constant for each population that represents
essentially how quickly a population can grow when there are no limitations.
POPULATION DEMOGRAPHICS
- integrated form
Population size will change based on the number of (3.)
minus the number of
(4.)
(Ignore immigration and emigration.) r is essentially the first answer
minus the second answer per each individual under conditions that are optimal for population
growth.
5. Give an example of a biological factor that would result in a difference in r between two
populations. All else being equal, there would be a difference in r between two populations, if one
population had a higher/larger
than the other.
Now this gives us the growth of a population over time. But what we are really interested in is taking any
given point in time and predicting how quickly the population will grow at that point in time. So let's make
a differential equation relative to time. No
6. Differentiate the exponential equation with respect to time. The result is below, just show how to get
there. (HINT: before differentiating, start by taking the natural log of both sides. If you get to the end and it
N
doesn't look like the final answer, then it might help to remember that: ert
7. First, let's plot the integrated
form of the exponential equation
on the axes below. Plot three
different functions, each with
N(0) = 1 but with three different
r's: 0.1, 0.2 and 0.5. You do not
need to show your work, but of
course you must label the
functions:
1000
dN(t)
dt
means. It
Now what does this tell us about population size and how it changes? Keep in mind what
means the change in population size over a given time. Most of the time we think about dN (t) as dt⇒0.
That is, at any instant, how quickly the population is growing.
dt
900
800
700
dN,
dt
600
€500
400
300
200
100
0
0
=rN₁ - differentiated form
10
=
20
30
40
t
50 60
70
80 8. OK, so what do these three curves tell you about r? As r increases, how do the curves change?
dN₁
dt
9. Another approach to problem 6 above is to ask what happens to
the differential equation that you derived in question 6.)
as r increases? (Think about
10. What does the exponential equation (in its integrated/non-differential form) predict about
population size after a long time? Solve for the limit as t→∞. Show your work.
11. Does your answer to 10 make biological sense? Explain. THE LOGISTIC EQUATION
So let's agree that the exponential equation is unrealistic in at least one way: there are no populations that
are infinitely large. Real populations can generally be grouped into two major groups: (1) those that are
unstable and fluctuate wildly and (2) those that are relatively stable. The ones that fluctuate wildly tend to
have booms and busts. They increase rapidly and then quickly die back. Locusts that swarm and destroy
crops are a good example.
The populations that are more stable are the ones we're going to discuss now. The idea is that in a given
habitat, each population has a carrying capacity. This is sort of a ceiling of stable population size. That is,
there is a certain amount of room and food and water and other resources in a habitat and these combine to
determine the number of individuals that can live there sustainably. That population size that is the
maximum sustainable population size is the carrying capacity, symbolized as K.
dN₁ t
=
dt
So, how do make the model better fit reality and incorporate K into our equation? We want the population
size not to change when it is equal to K. In other words, when N = K, 0. Because we are talking
about changes, we should start with the differential form of the equation.
of K by adding to the differential version of the exponential equation:
K-N₁
So, we can incorporate the effect
dN₁
=rN₁
dt
K
Set N = K and see what you get for
This equation is called the logistic equation and when the
integrated form is graphed it produces the logistic growth curve. Note that this form here is the
differentiated form.
dN₁
t
dt
The version above is the more common way to see this equation, but if you want to see how many
individuals there are in a population at any time, you would need the integrated form:
Ket
- integrated form
N₁
Note that we use No here. It's a constant
just like in the exponential equation.
A) r= 0.1,
B) r = 0.1,
C) r = 0.2,
12. On the graph to the right, plot the
integrated form of the logistic
No= 1,
No= 250,
No=1,
e +
equation (using whatever means you
wish) for the three given populations.
Label them, of course (3 pts.):
K = 500
K = 500
K = 500
K-No
No
- differentiated form
N(t)
550
500
450
400
350
300
250
200
150
100
50
0
0 10 20 30 40 50 60 70 80 90 100
t
