Explorations of Mathematical Models in Biology with MATLAB (eBook)

Mazen Shahin (Autor)

eBook Download: EPUB

2013
John Wiley & Sons (Verlag)
9781118548530 (ISBN)

Infos

 Explore and analyze the solutions of mathematical models from diverse disciplines

As biology increasingly depends on data, algorithms, and models, it has become necessary to use a computing language, such as the user-friendly MATLAB, to focus more on building and analyzing models as opposed to configuring tedious calculations. Explorations of Mathematical Models in Biology with MATLAB provides an introduction to model creation using MATLAB, followed by the translation, analysis, interpretation, and observation of the models.

With an integrated and interdisciplinary approach that embeds mathematical modeling into biological applications, the book illustrates numerous applications of mathematical techniques within biology, ecology, and environmental sciences. Featuring a quantitative, computational, and mathematical approach, the book includes:

• Examples of real-world applications, such as population dynamics, genetics, drug administration, interacting species, and the spread of contagious diseases, to showcase the relevancy and wide applicability of abstract mathematical techniques
• Discussion of various mathematical concepts, such as Markov chains, matrix algebra, eigenvalues, eigenvectors, first-order linear difference equations, and nonlinear first-order difference equations
• Coverage of difference equations to model a wide range of real-life discrete time situations in diverse areas as well as discussions on matrices to model linear problems
• Solutions to selected exercises and additional MATLAB codes

Explorations of Mathematical Models in Biology with MATLAB is an ideal textbook for upper-undergraduate courses in mathematical models in biology, theoretical ecology, bioeconomics, forensic science, applied mathematics, and environmental science. The book is also an excellent reference for biologists, ecologists, mathematicians, biomathematicians, and environmental and resource economists.

MAZEN SHAHIN, PhD, is Professor in the Department of Mathematical Sciences at Delaware State University. He has extensive background and experience in designing interdisciplinary instructional materials that integrate mathematics and other disciplines, such as biology, ecology, and finance. Dr. Shahin's research interests include boundary value problems, dynamical systems, impulsive differential equations, and mathematics education.

Inhalt

PREFACE ix

ACKNOWLEDGMENTS xiii

1 OVERVIEW OF DISCRETE DYNAMICAL MODELING AND MATLAB®
1

1.1 Introduction to Modeling and Difference Equations 1

1.2 The Modeling Process 8

1.3 Getting Started with MATLAB 13

2 MODELING WITH FIRST-ORDER DIFFERENCE EQUATIONS 28

2.1 Modeling with First-Order Linear Homogenous Difference
Equations with Constant Coefficients 28

2.2 Modeling with Nonhomogenous First-Order Linear Difference
Equations 42

2.3 Modeling with Nonlinear Difference Equations: Logistic
Growth Models 58

2.4 Logistic Equations and Chaos 74

3 MODELING WITH MATRICES 85

3.1 Systems of Linear Equations Having Unique Solutions 85

3.2 The Gauss-Jordan Elimination Method with Models 99

3.3 Introduction to Matrices 119

3.4 Determinants and Systems of Linear Equations 147

3.5 Eigenvalues and Eigenvectors 160

3.6 Eigenvalues and Stability of Linear Models 185

4 MODELING WITH SYSTEMS OF LINEAR DIFFERENCE EQUATIONS
195

4.1 Modeling with Markov Chains 195

4.2 Age-Structured Population Models 219

4.3 Modeling with Second-Order Linear Difference Equations
231

5 MODELING WITH NONLINEAR SYSTEMS OF DIFFERENCE EQUATIONS
249

5.1 Modeling of Interacting Species 249

5.2 The SIR Model of Infectious Disease 264

5.3 Modeling with Second-Order Nonlinear Difference Equations
270

REFERENCES 277

INDEX 279

Rezensionen

"Overall, the book is a great resource to use across many
diverse fields." (Mathematical Association of
America, 1 January 2015)

Textauszug

CHAPTER 2

MODELING WITH FIRST-ORDER DIFFERENCE EQUATIONS

2.1. MODELING WITH FIRST-ORDER LINEAR HOMOGENOUS DIFFERENCE EQUATIONS WITH CONSTANT COEFFICIENTS

In this section, we investigate situations that are modeled by first-order linear homogenous difference equations with constant coefficients, that is, equations in the form

(2.1)   

where a is a constant coefficient. We will iterate equation 2.1 with an initial condition to find a numerical solution, and we will also derive an analytical solution of equation 2.1.

2.1.1. Model 2.1: Drugs

Assume that the kidneys remove 20% of a drug from the blood every 4 h. Assume that the initial dose of the drug is 200 mg. Let dn denote the amount of drug in the blood after n 4-h periods, and d0 denote the initial amount of drug in the blood.

i. Find a difference equation that represents this situation.

ii. Find the amount of drug after 12 hours.

iii. Iterate the difference equation obtained in part I with the initial condition to find the ordered pairs (n, dn), n = 1, 2, … , 28. Graph it.

iv. Find an analytical solution of the obtained difference equation. Use this solution to find the amount of drug in the blood after 1 day.

v. When will the amount of the drug reach 1 mg?

Discussion

i. The amount of drug in the blood after (n + 1) 4-h periods, dn+1, equals the amount of drug after n 4-h periods, dn, minus 20% of dn. We obtain

Therefore this situation is modeled by the difference equation

(2.2)   

which is a first-order linear homogenous difference equation with constant coefficients.

ii. There are three 4-h periods in 12 h. So the amount of drug in the blood after 12 h is d3. To find d3, we will iterate the difference equation 2.2 with the initial condition

(2.3)   

To find d1 put n = 0 in equation 2.2 and use equations 2.3,

Similarly,

iii. The following MATLAB code can be used to create the table (n, dn), n = 0, 1, 2, … , 30. Note that this table is 31 × 2 matrix, where the first column is the time n and the second column is the amount of drug dn.

t = 0;

d = 200;

M = [t d];

for k = 1:30

    d = 0.8*d;

    M = [M; k d];

end;

A partial output of the matrix M is

0

200.0000

1.0000

160.0000

2.0000

128.0000

3.0000

102.4000

4.0000

81.9200

5.0000

65.5360

....…….

.……....

25.0000

0.7556

26.0000

0.6045

27.0000

0.4836

28.0000

0.3869

29.0000

0.3095

30.0000

0.2476

To graph dn vs. n we need to graph the second column of M vs. the first column. You may enter the code to have the graph in black points and label the axes.

plot(M(: , 1), M(: , 2), ‘k.’);

xlabel (‘Time n in four-hour periods’);

ylabel (‘Amount of drug d(n)’)

The graph is shown in Figure 2.1A. If you want to have the graph in black points and connect these points with black line graph, you may replace the above plot command with

plot(M(: , 1), M(: , 2), ‘k.’, M(: , 1), M(: , 2), ‘k’);

The graph is shown in Figure 2.1B.

iv. We will derive a closed-form of the solution (analytical solution) of the difference equation 2.2. We obtain

Following this pattern, we can conclude that

(2.4)   

Equation 2.4 is the analytical solution of the difference equation 2.2. Because there are six 4-h periods in one day, the amount of the drug in the blood after 1 day, d6, can be calculated from equation 4.4 by setting n = 6 and d0 = 200. We obtain d6 for the nearest hundredth

FIGURE 2.1. Graphs of the amount of drug in the blood after n time periods, dn, vs. time, n, in 4-h periods. The graphs are in disconnected black points, and the axes are labeled.

Thus the amount of the drug in the blood after 1 day is 52.43 mg.

v. We want to know the value of n when dn = 1. Setting dn = 1 in equation 4.4, we get

To solve this exponential equation, compute

Hence n = 23.7439. This means that the amount of drug in the blood reaches 1 mg after approximately 24 4-h periods—that is, after 96 h, or 4 days.

Analytical Solution

The analytical solution of the first-order linear homogenous difference equation 2.1,

can be derived in a similar way to model 2.1. Setting n = 0 in equation 2.1, we obtain

Put n = 1 in equation 2.1 to obtain

Setting n = 2 in equation 2.1, we get

From this pattern, we conclude that

(2.5)   

Solution of a First-Order Linear Homogenous Difference Equation with Constant Coefficients

An analytical solution to the difference equation 2.1

where a is a constant, is a formula that expresses yn in terms of n, and the constants y 0 and a. That is, yn as a function of n. The analytical solution of equation 2.1 is equation 2.5:

where y0 is the initial condition.

Analytically we can check that equation 2.5 is a solution of equation 2.1 by showing that yn in equation 2.5 satisfies equation 2.1. We have

Therefore, LHS = RHS.

2.1.2. Model 2.2: Population Dynamics, First Pass

A population of owls is growing at 4% per year and there are 1000 owls now. For simplicity we will ignore the interaction of owls with other species. Let Pn be the population of owls n years from now.

i. Model the owls’ population dynamic by a difference equation, which is a difference equation describing the change of the population year after year.

ii. Find the ordered pairs (n, Pn), n = 0, 1, 2, … , 60. Graph Pn vs. n.

iii. Find an analytical solution of the difference equation obtained in a part ii. Use this solution to determine the population of owls after 10, 40, and 55 years.

iv. When does the population of owls double and triple?

v. What is the long-term behavior of the owls’ population?

Discussion

i. The population of owls can be modeled by the difference equation

(2.6)   

FIGURE 2.2. Graph of owls’ population, Pn, vs. the time, n, in years. The initial population is P 0 = 1,000.

ii. The following MATLAB code may be used to find the ordered pairs (n, Pn), n = 0, 1, 2, … , 60 and graph Pn vs. n, which is shown in Figure 2.2.

t = 0;

P = 1000;

M = [t P];

for k = 1:60

  P = 1.04*P;

  M = [M; k P];

end;

plot(M(: , 1), M(: , 2), ‘k.’, M(: , 1), M(: , 2), ‘k’);

xlabel (‘Time n in years’);

ylabel (‘Population P(n)’);

iii. Using equation 2.5, the analytical solution of the difference equation 2.6 is

(2.7)   

Substitute n = 10, n = 40 and n = 55 in equation 2.7 to find P10, P40 and P55, respectively.

iv. We will use equation 2.7 to determine the value of n which satisfies Pn = 2,000. We have,

To solve for n we take log of both sides of this equation:

That, is the owl’s population will double in approximately 18 years. Similarly, the population will triple in 28 years. These...

Produkt-Details

• Erscheint lt. Verlag: 24.12.2013
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
• Themenwelt: Naturwissenschaften ► Biologie
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Angewandte Physik
• Themenwelt: Technik
• Schlagworte: Ökologie / Methoden, Statistik • Biologie • Biowissenschaften • Computational Biology • Life Sciences • mathematical biology, biology, medicine, biotechnology, biomathematics, theoretical biology, mathematical models, applied mathematics, environmental science, linear and nonlinear difference equations, matrix algebra, population dynamics, growth model, maximum sustainable yield, harvest strategies, Markov chains, genetics, eigenvalues, eigenvectors, Matlab • Mathematical Modeling • Mathematics • Mathematik • Mathematik in der Biologie • Mathematische Modellierung • MATLAB • Methods & Statistics in Ecology • Modell (Math.) • Ökologie / Methoden, Statistik
• ISBN-13: 9781118548530 / 9781118548530