39 Introduction
To introduce this next chapter we begin with the formal definition for a mathematical model [9].
Definition.
Mathematical models is math construct designed to study a particular real-world system or phenomenon.
As the definition suggests, we utilize mathematical modeling, in the real world, every day. There are different uses for applying math models but in this section we will only cover a few specific applications. We begin by understanding what math modeling can do for us. In the following subsections we will see how we can calculate trends and form dynamic formulas to help illustrate the trends we see. We can also utilize the formulas we create to help predict future values. These abilities can help businesses predict future sales, or help nature conservancies track and predict the changes in animal population in the coming years. With an ability to roughly approximate future values for different scenarios we can better prepare or adjust for the coming years or make appropriate alterations to attempt to improve future trends.
Before delving into different types of models we need to cover some of the basics of what makes a model appropriate and useful. Firstly, we look at the Fidelity of a model [9].
Definition. Fidelity
The preciseness of a model’s representation of reality.
This means we want to judge if a model really gives a good representation for a real world scenario. It needs to include the appropriate factors that may effect change over time. This leads us into the second property we want to look at, flexibility [9].
Definition. Flexibility
The ability to change and control conditions affecting the model as required data are gathered.
The real world has a multitude of interconnections, we have learned to illustrate these connection by the Butterfly effect. A small change on one side of the world could potentially lead to a drastic result on the other side. With this in mind we know that most models will not be perfect, take for example the predator prey model. We know that there are other predators to a mouse than just an owl and there are some predators that prey upon owls. All these small changes will effect how we structure our model as we begin to collect data. Essentially we will see if a model is flexible enough to adapt to the ever changing data being collected or not, this is why we have the flexibility property to consider.
When we begin to construct a model we need to follow a set of steps. Beginning with identifying the problem. In the next subsection we will be given examples of different scenarios that require us to construct a model. Knowing what we are attempting to utilize this model for is how we identify the problem. The next step is to make assumptions, this could mean assumptions are being made about taking external factors into account or not. In this step we consider different factors and their relevance to our problem to create the most concise model to fit the problem, which may aid in the flexibility of the model. Once you have distinguished which factors to include we move to classify the dependent and independent variables, and occasional a variable will be neither. Along with distinguishing the variables we need to determine their interrelationships before we create the model we will then solve for and later verify [9]. In the following section we will be given various scenarios and asked to construct and solve models, and later utilize them to predict future values or illustrate trends graphically.