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# 4: Discrete-Time Models I - Modeling

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Discrete-time models are easy to understand, develop and simulate. They are easily implementable for stepwise computer simulations, and they are often suitable for modeling experimental data that are almost always already discrete. Moreover, they can represent abrupt changes in the system’s states, and possibly chaotic dynamics, using fewer variables than their continuous-time counterparts.

• 4.1: Discrete-Time Models with Difference Equations
The discrete-time models of dynamical systems are often called Difference Equations, because you can rewrite any ﬁrst-order discrete-time dynamical system with a state variable x.
• 4.2: Classiﬁcations of Model Equations
There are some technical terminologies I need to introduce before moving on to further discussions:
• 4.3: Simulating Discrete-Time Models with One Variable
Now is the time to do our very ﬁrst exercise of computer simulation of discrete-time models in Python. Let’s begin with this very simple linear difference equation model of a scalar variable x:
• 4.4: Simulating Discrete-Time Models with Multiple Variables
Now we are making a ﬁrst step to complex systems simulation. Let’s increase the number of variables from one to two.
• 4.5: Building Your Own Model Equation
Now that you know how to simulate the dynamics of difference equations, you may want to try building your own model equation and test its behaviors. Then a question arises: How do you build your own model equation? Mathematics is a language that is used to describe the world. Just like that there is no single correct way to describe an idea in English, there is no single correct way to build a mathematical model equation either.
• 4.6: Building Your Own Model Equations with Multiple Variables
We can take one more step to increase the complexity of the model building, by including more than one variable. Following the theme of population growth, let’s consider ecological interactions between two species.