# CHAPTER 5

# MODELING WITH NONLINEAR SYSTEMS OF DIFFERENCE EQUATIONS

As we saw in the previous two chapters many situations involving more than one dependent quantity were represented by systems of linear first-order difference equations. Also we know that a system of linear difference equations might be represented by a single matrix difference equation, and therefore, matrix algebra can be used to find an analytical solution and a numerical solution of the matrix equation.

In this chapter, we investigate situations modeled by systems of nonlinear first-order difference equations. If a situation is modeled by a single nonlinear higher-order difference equation, this nonlinear difference equation can be converted to a system of nonlinear first-order difference equations. Because matrix algebra cannot be used for systems of nonlinear equations, we will rely on using MATLAB to iterate the systems and investigate their long-term behavior.

Section 5.1 focuses on the dynamic of interacting species. In Section 5.2 we study infectious disease modeling, particularly the SIR model. In Section 5.3 we introduce modeling with second-order nonlinear difference equations.

# 5.1. MODELING OF INTERACTING SPECIES

In this section we explore the dynamics of interactions between two or more species represented by systems of first-order nonlinear difference equations. The main types of interactions are predator–prey, competition, and mutualism. We will focus on the predator–prey models.

**5.1.1 A Predator–Prey ...**