When we use the word system, we are referring to subject of study and all of the possible quantities associated with that subject. In principle, every object interacts with so many other objects that a system’s complexity is in reality incomprehensible. For the sake of simplification and sanity, we adopt a reductionist viewpoint and consider the system as representing only those quantities that are relevant for addressing specific scientific questions.
A state variable is any individual quantity associated with the subject of study. A state variable might represent a quantity that is directly measurable (whether or not it is measured). It might instead represent a mathematical expression derived from other measurements, in which case the state variable is called a dependent variable. For example, a physical object has a mass, \(m\), as well as a volume, \(V\). Both \(m\) and \(V\) are measurable. Using these state variables, we can also consider the average density, \(\rho = \frac{m}{V}\), which is a dependent variable.
The state of the system is the collection of all state variables, simultaneously measured for a given configuration of the system. In many cases, the system is time-dependent, meaning that the state of the system naturally is changing as time passes. In this case, the time of measurement should be included as one of the state variables. Practically, we only include the state variables that are directly relevant.
Example1.4.1
Consider a population being studied at a certain location. Some of the state variables include the number of individuals in the population, the number of births and the number of deaths during a given time period (e.g., for a year), the total biomass of the population, the physical area required to support the population, and the amount of food used to feed the population during the time period. The state of the population consists of all of these measurements for a given time. The same population will have different states at different times.
One of the primary purposes of mathematical modeling is to establish relationships between different state variables and then use those relationships to make predictions. Plotting graphs of relationships between variables often reveal patterns. This requires tracking the same variables over different states of the system. Each state corresponds to an experimental measurements. Depending on the complexity of the system or the costs of measurement, there may be only a limited number of measurements available.
We usually use scatter plots to look for these relationships. A scatter plot takes two state variables which we measure relative to corresponding units (in order to get numbers), say \(x\) and \(y\). We then plot the points \((x,y)\) coming from different states. If a pattern arises, then we look for an equation that captures the essential features of the relationship. These equations form the essence of our mathematical model. We can then analyze our system equations to see what predictions they allow us to infer.
