13
1.2. Nonlinear Models
However, when the population is small (i.e., P is much smaller than K ), the
factor (1 P/K ) in the per-capita growth rate should be close to 1. Therefore,
for small values of P, our model is approximately
Pt+1 (1 + r )Pt .
In oth
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The Edinburgh Building, Cambridge CB2 2RU, United Kingdom
Published in the United States of America by Cambridge University Press,
14
Dynamic Modeling with Difference Equations
next_p = p+.7*p*(1p/10)
Population P
12
10
8
6
4
2
0
0
5
10
15
Time
Figure 1.2. Population values from a nonlinear model.
If we measure population size in units such as thousands, or millions of
individuals, t
4
Dynamic Modeling with Difference Equations
It may seem odd to call Pt+1 = (1 + f d)Pt a difference equation, when
the difference !P does not appear. However, the equations
Pt+1 = (1 + f d)Pt
and
!P = ( f d)P
are mathematically equivalent, so either one
MATHEMATICAL MODELS IN BIOLOGY
AN INTRODUCTION
ELIZABETH S. ALLMAN
Department of Mathematics and Statistics,
University of Southern Maine
JOHN A. RHODES
Department of Mathematics,
Bates College
1.3. Analyzing Nonlinear Models
be expressed as the ratio
23
pt+1
for small values of pt . But
pt
pt+1
Pt+1 P
F(Pt ) P
F(Pt ) F(P )
=
=
=
,
pt
Pt P
Pt P
Pt P
where Pt+1 = F(Pt ) is the equation defining the model. (Note that we used
P = F(P ) for the
vi
6.
7.
8.
A.
B.
Contents
5.4. Tree Construction: Maximum Parsimony
5.5. Other Methods
5.6. Applications and Further Reading
Genetics
6.1. Mendelian Genetics
6.2. Probability Distributions in Genetics
6.3. Linkage
6.4. Gene Frequency in Populations
Infec
Contents
page vii
xi
Preface
Note on MATLAB
1. Dynamic Modeling with Difference Equations
1.1. The Malthusian Model
1.2. Nonlinear Models
1.3. Analyzing Nonlinear Models
1.4. Variations on the Logistic Model
1.5. Comments on Discrete and Continuous Models
6
Dynamic Modeling with Difference Equations
1.1.2. In the early stages of the development of a frog embryo, cell division
occurs at a fairly regular rate. Suppose you observe that all cells
divide, and hence the number of cells doubles, roughly every hal
1.3. Analyzing Nonlinear Models
25
a population below the carrying capacity of the environment may in a single
time step grow so much that it exceeds the carrying capacity. Once it exceeds
the carrying capacity, the population dies off rapidly enough that
30
Dynamic Modeling with Difference Equations
replace P + r P(1 P) by these approximations in Pt+1 = Pt +
r Pt (1 Pt ). Use this to determine the stability of the equilibria. Your
answer should agree with the preceding two problems.
1.3.11. Many biologica
Note on MATLAB
Many of the exercises and projects refer to the computer package MATLAB.
Learning enough of the basic MATLAB commands to use it as a high-powered
calculator is both simple and worthwhile. When the text requires more advanced commands for ex
viii
Preface
Our writing style is intentionally informal. We have not tried to offer definitive coverage of any topic, but rather draw students into an interesting field.
In particular, we often only introduce certain models and leave their analysis
to ex
33
1.4. Variations on the Logistic Model
p. Your model should be something like
Pt+1 = floor(Pt + r Pt (1 Pt /K ),
where K is first a constant and then is made to vary randomly.
1.4. Variations on the Logistic Model
In presenting the discrete logistic mod
20
Dynamic Modeling with Difference Equations
intuitively? (Note that r will be very small, because we are using a
small time interval.) The logistic growth model is sometimes also
referred to as the autocatalytic model.
1.3. Analyzing Nonlinear Models
Un
1.2. Nonlinear Models
17
At this point, you can learn a lot more from exploring the logistic model
with a calculator or computer than you can by reading this text. The exercises
will guide you in this. In fact, you will find that the logistic model has so
1.4. Variations on the Logistic Model
37
Problems
1.4.1. For a discrete population model, the relative growth rate is defined as
Pt+1
.
Pt
a. Complete: For a particular value of Pt , if the relative growth rate
over the next time
is larger than 1, then th
1.1. The Malthusian Model
9
evidence. Can you think of factors that might be responsible for
any deviation from a geometric model?
b. Using the data only from years 1920 and 1925 to estimate a growth
rate for a geometric model, see how well the models res
xii
Note on MATLAB
The MATLAB files made available with the text are:
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
!
aidsdata.m contains data from the Centers for Disease Control and
Prevention on acquired immune deficiency syndrome (AIDS) cases in
the United States
cob
Note on MATLAB
!
!
!
xiii
seqgen.m generates DNA sequences with specified length and base
distribution
sir.m displays iterations of an SIR epidemic model, including time
and phase plane plots
twopop.m displays iterations of a two-population model, includi
10
Dynamic Modeling with Difference Equations
Table 1.4. Changing Time Steps in a Model
t
Nt
0
A
t
Pt
0
A
1
2A
1
2
2A
2
4A
3
4
4A
3
8A
5
6
8A
t by 1 represents 0.1 years (so, Q 10t = Nt ). You should begin by
producing tables similar to those in part (a).
1
Dynamic Modeling with Difference Equations
Whether we investigate the growth and interactions of an entire population,
the evolution of DNA sequences, the inheritance of traits, or the spread of
disease, biological systems are marked by change and adapt
1.1. The Malthusian Model
3
Table 1.1. Population Growth
According to a Simple Model
Day
Population
0
1
2
3
4
.
.
500
(1.07)500 = 535
(1.07)2 500 = 572.45
(1.07)3 500 612.52
(1.07)4 500 655.40
.
.
is the difference or change in population between two cons
16
Dynamic Modeling with Difference Equations
next_p = p+.7*p*(1-p/10)
20
Population at time t+1
18
16
14
12
10
8
6
4
2
0
0
2
4
6
8
10
12
14
16
18
20
Population at time t
Figure 1.4. Cobweb plot of a nonlinear model.
to the parabola to find the point (P1
Preface
ix
genetics applications. Chapter 5, which has an algorithmic flavor different
from the rest of the text, depends in part on the distance formulas derived
in Chapter 4. Chapter 8s treatment of infectious disease models naturally
depends on Chapter
1.3. Analyzing Nonlinear Models
31
modeled well by the discrete logistic difference equation
!P = r P(1 P/K ).
Of course, the dynamics of the population will depend on the value of r ,
but by choosing appropriate units, we may assume K = 1.
Investigate th
2
Dynamic Modeling with Difference Equations
To begin to address these questions, we start with the simplest mathematical
model of a changing population.
1.1. The Malthusian Model
Suppose we grow a population of some organism, say flies, in the laboratory
19
1.2. Nonlinear Models
a.
b.
Pt+1
Pt+1
Pt
P0
c.
Pt
P0
d.
Pt+1
Pt+1
P0
Pt
P0
Pt
Figure 1.5. Cobweb graphs for problem 1.2.9.
occurs in amount K , is converted to chemical 2, which occurs in
amount Nt at time t. Explain why !N = r (K N ). What values
of r