Explorations of Mathematical Models in Biology with MATLAB

Mazen Shahin

• Publisher: Wiley
• Year: 2013
• ISBN:
  9781118548530
• Edition: 1
• Topic: Biological Sciences
• Subtopic: Biophysics
• Index: Biological Sciences

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.

CHAPTER 1

OVERVIEW OF DISCRETE DYNAMICAL MODELING AND MATLAB^®

1.1. INTRODUCTION TO MODELING AND DIFFERENCE EQUATIONS

In this section we introduce dynamical systems, discuss discrete dynamical systems vs. continuous dynamical systems and informally define a mathematical model.

1.1.1. Model 1.1: Population Dynamics, A Discrete Dynamical System

Consider the population of a city with a constant growth rate per year. The population is counted at the end of each year. For simplicity, assume that there is no immigration to or emigration from the city.

i. Model the population dynamic and predict the long-term behavior of the system.

ii. In 2010, the city’s population was 100,000. The natural annual growth rate of the population is 1% per year. Predict the city’s population in 2020. Estimate the population over the next 30 years and graph it. What is the long-term behavior of the population?

Discussion

i. We will measure the population at discrete time intervals in one-year units. Let

p_n = population size at the end of the time period (year), n.

p₀ = the initial population size, 0.

r = the constant growth rate per period (year).

The relationship between the current population, p_n, and the next population, p_n+1, is

(1.1)

Therefore, the population dynamics can be modeled by equation 1.1.

Equation 1.1 is a difference equation (or recurrence equation). The system 1.1 and the initial value, p₀, represent the population dynamics. Because the population changes over time, this system is a dynamical system. Because this dynamical system changes over discrete time intervals, the system is called a discrete dynamical system. We say that the population dynamics is modeled by the discrete dynamical system (or the difference equation 1.1).

To find p_k, use p₀ in equation 1.1 to find p₁, then use p₁ to find p₂ and so on until p_k. This process is called iteration of the difference equation 1.1, and the sequence 1.2,

(1.2)

for any value of k (positive integer) is called a solution or numerical solution of the given difference equation 1.1.

From equation 1.1, if the current value of p_n is known, the next value, p_n+1, can be calculated. For example, if we have p₅ we can calculate p₆. However, if we have p₀, equation 1.1 does not allow us to calculate, for example, p₆ in one step. Therefore, we are in need of a closed form to calculate p_n in one step if we know the values of p₀ and n. It can be easily proven that

(1.3)

Equation 1.3 is called the analytical solution of the difference equation 1.1.

Equation 1.3 is an exponential function and will grow or decay exponentially, depending on the value of r. If r > 0, then (1 + r) > 1, and therefore population size, p_n, grows unbounded when n is very large. If r < 0, then (1 + r) < 1, and consequently, the population size approaches zero when n is very large.

ii. Let’s apply the model just discussed to the given information, where r = 0.01 and p₀ = 100,000. The city’s population is modeled by the system

(1.4)

To find the population in 2020 (10 years from 2010), we use equation 1.3 with n = 10 and p₀ = 100,000. We are looking for p₁₀. We have

One way to find the...