HYPOTHETICAL CASES
Your case is about assault and battery. Shawn Chan an Asian-American citizen was coming out
of the Pub, a local bar which Shawn Chan regularly attended, when he was attacked by two
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 d
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 Genetic
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 Logisti
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, a
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 carryin
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
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 worthwhil
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
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
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
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
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 g
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
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
185
70"
Which of the following statements must be true? In the gure abOVe 1' m- y
1 3.0: A. 150
II. a+ca180 B. 140
III. a+d=90 C- 130
A. I 9' 120
B. II 0
m ."é'4!«.7."_If-§ét2";7 "
3*
2.
30°
x0
y
i]?
PercenT Word Problems
Ratio and proportion method
Here are several aids ThaT will help you solve word problems:
1. Make sure you undersTand The quesTion ThaT is asked
2. SorT ouT The informaTion T
CHAPTER / TRIANGLES 135
_ PRACTICE SAT QUESTIONS
1. What is the value of x? 4. Which of the following statements is correct?
B
.11 . I). -. ._
61°
130°
«1215?. .1.- imli ne-qr.
B. 70 A 59 C
A. AB<BC
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
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The Edinburgh Building, Cambridge CB2 2RU, United Kingdom
Published in the
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
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