MA354
January 24th, 2011
Project 1: Modeling of Population Growth Using Excel
Objective
The objective of today’s laboratory is to use the spreadsheet program Excel to model population
growth under a variety of scenarios. The purpose is to familiarize you with the diversity of
functions available within Excel, and understand the differences between linear, exponential and
logistic models. Read pages 10-12 in your book.
Overview
Changes in Population Size
Population growth models are used to predict changes in population size over time. A model for
population size over time may be expressed as a difference equation that indicates the change in
the population over a time interval t.
Difference equation for population growth:
P
= change in P
= change in population size over an understood time interval
= Pn+1 – Pn 1
The change in population size over an interval can be any kind of function; each choice of
function results in a different growth model. We will explore different models (constant,
proportional and logistic) with these exercises.
Growth and Growth Rate
The growth of a population over a time period is simply the change in size that has
occurred over that time period. Thus growth is equivalent to P:
growth  P
The growth rate of a population is the amount of growth that has occurred relative to the
population size. Let the growth rate of a population be denoted by G (P ) . Then:
growth rate  G ( P) 
1
Here, (n+1)-(n) =1, indicating a single discrete time interval t.
P
P
Exercise 1: Linear Growth Model – Growth is Independent of Population Size
Suppose that the change in population size over a time interval is constant. In this case, the
change in population size doesn’t depend upon anything, as it is fixed. We describe this model
with the difference equation P = k where k is the constant growth over a fixed, understood time
interval.
P = k *P
1. Suppose that the initial size of a population is 60. (P0 = 60). Use Excel to model the
population growth for 20 time intervals for an arbitrary value of k. We say that k is a free
parameter in the model.
2. On a single graph, plot the population size verses time over 20 time intervals for the following
values of k:
Sketch your graph here:
(a) k=3
(b) k=0
(c) k=-1
3. A scientist presents you with the following data for the number of leaves counted per week on
an exotic palm tree. Assuming that the growth can be approximated using a linear model,
estimate the growth constant k.
# Leaves per Week
W1
60
W1
65
W2
68
W3
72
W4
72
W5
75
W6
82
W7
85
W8
89
W9
89
W10
90
k ≈ _____________
Do you think that the palm growth is described well by a linear model? Why or why not?
What do you estimate is the degree of accuracy you have in estimating the parameter k?
The scientist would like a more precise measure of k. He asks if more data points would
help. He wants to know if you would like more frequent data points, or data points over a longer
interval. How would you answer?
Exercise 2: Proportional Growth Model – Exponential Model
In exponential growth, growth depends upon the population size.
P = k *P
1. Use Excel to model the population growth described by exponential growth. Copy your
Excel worksheet for linear growth and modify the equations to reflect exponential growth.
Sketch your results below for 20 time intervals, given the initial population size is P0 = 60,
k=0.3.
2. Distinguishing the growth verses the growth rate: Write down an expression for the
growth as a function of the population size. Write down an expression for the growth rate as
a function of the population size.
growth =
growth rate =
3. Compare with the linear model: Write down an expression for the growth and the growth
rate for the linear model.
growth =
growth rate =
Exercise 3: Biologically Motivated Growth Model
Although there are some examples in biology of linear and exponential growth, growth and
growth rate are both usually size dependent. This is true over a range of scales, from bacteria
populations to tumors to whole organisms. On the one hand, the amount of growth over a fixed
interval is often proportional to size. On the other hand, the growth rate is often inversely
proportional to size: the growth rate is fastest when the population is small and slowest when the
population is large. (Due, for example, to resource management and availability.)
In this exercise, we will build a model for a scenario where the growth rate depends upon the
population size in the following ways:

When the population is small, the growth rate is a maximal growth rate k. In particular, in
the limit of a vanishingly small population size, the growth rate is k:
P = kP when P = 0.

When the population is a maximal size M (the carrying capacity), the growth rate is zero:
P = 0P when P = M.
In this simple scenario, the growth rate depends upon
the population size. In other words, the growth rate is a
function of P. The two assumptions above provide two
points for this function, at the limits when P is minimal
and maximal. The two points are shown on the plot
below.
1. Let GB (P) be the function which describes the growth rate as a function of the population size.
What is the simplest continuous function GB (P) you can think of that passes through these two
points? Sketch your function on the plot above and write the equation in terms of k, P and M:
GB (P) =
2. For a given function G(P) for the growth rate, we have P =G(P)*P. Write down the
difference equation for growth of the population:
P = GB (P) * P =
3. Use Excel to model the population growth described by P = G1(P) * P. Sketch the
population size verses time for 20 time intervals, given the initial population size is P0 = 60,
k=0.3 and M = 300.
Exercise 4: Logistic Growth
The function GB(P) in the previous exericise was chosen as
the simplest function you thought of that yielded a maximal
growth rate of k at small pop sizes and a minimal growth
rate of 0 at the maximal population size M. Obviously, this
function can vary in different ways between points as a
monotonically decreasing function. For example, we can
imagine that the growth rate changes slowly at first or
quickly at first:
Each different function represents a different growth model that may or may not be a better
description of growth in a real-world scenario. Model construction is largely concerned with
finding the function which is most descriptive for a particular problem.
In your book, the difference equation for logistic population growth is written as:
P = k (M – P)*P
(page 407, Eqn 11.9)
1. Use Excel to model the population growth described by logistic growth. Sketch the
population size verses time for 50 time intervals, given the initial population size is P0 = 60,
k=0.3 and M = 500. Notice that the population size initially changes fast (exponential growth)
and then changes slowly (logarithmic growth).
2. For the logistic growth equation, solve for GL(P) given that P = GL (P)*P.
GL(P) =
3. How does the function GL (P) used to describe logistic growth compare with your function GB?
4. The logistic function can be written in equivalent ways:
P = k1 (M – P)*P = k2 (1 – P/M)*P
Convince yourself these two equations are the same. What is the relationship between k1 and k2?
Exercise 5: Relationships Between Models
1. Note that logistic growth is exponential when the population size is small. In particular, the
growth is exponential when the population size is farthest from the carrying capacity. Derive
the equation for exponential growth from the logistic equation by considering the limit as M
approaches infinity.
2. Each of the models (linear, exponential, logistic) has been described with a free parameter k.
Compare the k in these three models. Do they represent the same thing?
3. Only the logistic model explicitly mentions M. What would M be for the linear and
exponential models?