# 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*.

*p*_{0} = 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 ...