Mathematical Modeling

Unit III, Focus on Mathematical Models. In Unit I, Witchcraft in Salem Village, we were trying to understand the past, in Unit II, Earnings and Discrimination, the present, and in this unit, Unit III, Population and Resources, we are interested in predicting the future. We are interested in questions, such as: 'What will population levels be like 20 years from now?' 'Will resources keep up?' 'Can we make a better future by our actions today?' Such are the questions, and we'll try to answer them by using mathematical models.

Examples and Objectives of Mathematical Models. What do you think of when you hear the words, "Mathematical Modeling?" What are some examples of mathematical models? Click below for responses of students:

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General weather forecasting, global warming, flight simulation, hurricane forecasting, nuclear winter, nuclear arms race, ... might come to mind as examples of large mathematical models with a large potential impact on us all. Mathematical models are also used to describe traffic flow, stock market options, predator-prey relations, and techniques of search.

What are the objectives of mathematical modeling? Forecasting the future, preventing an unwanted future, and understanding various 'natural' and unnatural phenomena are some possibilities expressed in very general terms. These might all be put into the category of problem solving by using mathematics to mirror an aspect of the world.

Example of Model

Objective

Overarching Objective

Weather Prediction

Explanation

and

Understanding

Flight Simulation Training
Nuclear Arms Race Strategy Development
Traffic Flow Regulation
Predator-Prey Management

It's important not to confuse the mirror with what it's mirroring. As the linguist S.I. Hayakawa put it: 'The symbol is NOT the thing symbolized; the word is NOT the thing; the map is NOT the territory it stands for.' In the same vein, the model is NOT the real-world.

The Example of Malthus' Modeling. An example of a simple mathematical model, but one that has had long-term effects, is in the Malthus lecture: Malthus assumed that population would grow exponentially while subsistence would grow at best linearly. From these assumptions, Malthus derived mathematical consequences and proposed policies to try to prevent, or at least soften, the consequences.

Below is a schematic for a general mathematical modeling framework and, following the schematic, what Malthus' model and his proposal look like in that framework.

Real World: Lots of stuff going on. The French Revolution a decade ago - Ottoman Empire in decline - East India Company entrenched in South Asia - Population growth generally considered good by European intellectuals - Rapid population growth in the resource-rich United States.

Observation (construction): Through his lens of experience, goals, and intellect, Malthus observes or constructs the idea of hard times, of misery and vice. He'd like to do something about these problems. Lots of stuff is ignored.

The Model and Its Formulation:

Focus/Variables Malthus has to decide on what his focus should be and what aspects of the real world he should ignore. He picks just two variables to work with, ignoring all else.
Assumptions He makes assumptions about the rates of change of his two variables.
Derivations He draws conclusions purely from the mathematics.

Predictions/Comparison to Real World: Now Malthus interprets his mathematical conclusions in terms of the real world and compares the real world to the model. Well, ideally, he would do that. But, he doesn't live in an information-rich age, and he's dealing with lengthy time spans, so he can't make such comparisons very easily.

Revise Model: If he did this, he didn't tell us about it.


Policy Changes: On the basis of his model, Malthus makes some recommendations concerning agricultural, labor, and manufacturing policies, personal restraint, and public assistance policies.

Looking at Malthus' model more closely, we see the following:

 

Population

Birthrate - Deathrate = fixed percent of population per unit time

Food

Agricultural Growthrate = fixed absolute amount per unit time

 

Population and Food Supply are both determined by rates of growth - The rates of growth are unaffected by anything, except for the modeler's (Malthus') assumptions - a sort of 'invisible hand' (using the words of Adam Smith, but in a context other than Smith intended).

What to Focus on: A Critical Choice in Modeling. An early mathematical model was formed for the psychology of perception.Things to include or ignore: Whether the perceiving was going on inside or outside, if inside, what size the room is, the temperature, noise level, type of stimulus, distance from the stimulus, length of the stimulus if it is a card, color of the stimulus card, shortest perceptible difference in the length of the stimulus card, ... . By focusing on just two variables, (Magnitude of the stimulus and the least perceptible difference in the magitudes of the stimuli) the concept of 'just noticeable difference' was constructed and the Weber-Fechner Law formulated.

Start Simply. If you were going to model population growth, what factors or variables would you want to include? (We'll come back to this question near the end of class.) Obviously, a lot of potentially valuable factors related to population growth have been left out. However, the Malthus model illustrates a principle for beginning a mathematical model: Start simply - then gradually add complexity, so long as complexity also adds insight.

To see be more precise about the process of modeling, including the revision part, let's look at the model formulation, guided by Lab 1 and the modeling software Stella.

Decide on the variables. In Lab 1, we choose at first two variables, Population and Births (per year), with Population represented with the Stella idea of a stock and Births per year represented with the Stella idea of a flow. Time is also a variable, but it isn't explicitly controlled in any way. Stella assumes that all models involve time, and we choose Stella as our model-creating tool. The initial Population is assumed to be 1,000, and Births per year is assumed to be 200. The interrelationship between the two variables is:

Population at a particular point in time

is

Population in the previous year + Births

Symbolically, in Stella language this looks like:

Population (t) = Population (t-dt) + Births per year * dt
INIT Population = 1000
INFLOWS
Births per year = 200

So, we have a very simple model that we can compare to reality. How do they compare. Well, if we run this Stella-implemented model, we see the linear growth of population over time. What we actually have is a model, supposedly for population growth, that matches Malthus' model for food growth.

We know that we have at least omitted a very obvious variable: Deaths! We could make our model a little more complex by adding a Death variable, represented in Stella as a flow away from Population.

Population at a particular point in time

is

Population in the previous year + Births - Deaths

If Deaths per year is assumed to be 100, we get symbolically,

Population (t) = Population (t-dt) + Births per year * dt - Deaths per year * dt
INIT Population = 1000
INFLOWS
Births per year = 200
OUTFLOWS
Deaths per year = 100

The Stella Diagram is

However, the graph of Population over time is still a linear one, just increasing at a constant absolute rate that is less than before. Can we do better?

Add Complexity. A key point to Malthusian growth is the idea of proportional or percentage growth, but the models thus far are assumed to have constant absolute growth rates. To introduce percentage growth, think about the increase in population when that increase is proportional to the population level itself. In other words, the increase is dependent upon two things: The proportion and the Population.

Rather than having Births determined by an 'invisible hand,' we'll have it determined by a fixed proportion and by the Population itself. To do this in Stella, introduce a converter, called Per Capita Births per year, which will be constant. Then draw connections from Per Capita Births per year and from Population to the flow Births per year, so that the diagram looks like:

Do the same sort of thing with Deaths, making it dependent upon a constant Per Capita Deaths per year and upon Population. Now the diagram looks like:

Now if we put in some reasonable numbers for Per Capita Births per year and Per Capita Deaths per year and run the model, we get a graph that is an exponential one. How does this compare to Malthus? How does it compare to our knowledge of past population growth?

Well, the model pretty well captures Malthus' ideas about population growth, but it turns out not to fit well with the data on population growth either on a world-wide basis or looking at smaller segments, say by country or region of the world. This isn't surprising, since we haven't taken any real-world features into account, other than the propensity of populations to grow!

We'll put in one last factor to try to make the model more realistic, leaving it to your laboratory work to take into account food, agricultural innovation, fertility rates, education, social security systems, etc. This last factor is the idea of a carrying capacity. It seems reasonable that our earth and solar system have some limit to the population it can support. If so, it has a carrying capacity, or maximum number of individuals that can survive on the planet.

The previous model is extended by adding a converter called Carrying Capacity, which is a (pretty big) constant. It is connected to Births per year, so that the diagram is:

The equations for the model are:


Population (t) = Population (t-dt) + Births per year * dt - Deaths per year * dt
INIT Population = 1000
INFLOWS
Births per year = Per Capita Births per year * Population * (Carrying Capacity - Population)
OUTFLOWS
Deaths per year = Per Capita Deaths per year * Population

[Note on the term 'Per Capita Births per year' in this model that includes Carrying Capacity.]

Thus, the number of births per year is assumed in this model to be jointly proportional to the population and how close the population is to carrying capacity. When population is graphed against time it is an elongated, roughly S-Shaped curve shown below: (Population in billions, time in tens of years)

Does the Model Explain? Is the graph above consistent with Malthus? Other population data? What is the significance of the flat growth for the latter portions of time? The graph does bear some resemblance to the graph distributed in the Malthus lecture. (You can obtain this graph at the United Nations Population Information Network.)

Although the graph above is not consistent with Malthus' assumption about population growth having a constant doubling time, it perhaps captures some of the spirit in the following way: There is early on an increasing rate of growth as the graphs 'curves upward.' The fact that the graph increases at smaller and smaller rates of growth as it approaches a carrying capacity is possibly what Malthus had in mind by the term 'misery.' This slow approach to a carrying capacity is perhaps the result of war, pestilence, and starvation as more and more people contend for the resources that are now at their upper bound.

What is clear is that even if the graph were a good depiction of actual world population growth, it doesn't explain much. The dynamics of population growth remain a mystery. None of the dynamic interaction of the factors related to population growth are either assumed in or deduced from the present model.

So, let us turn to brainstorming other factors and variables to add to the model.

Brainstorming The Choice of Factors to Include when Modeling Growth of Populations. If you were going to model population growth to explain the dynamic interaction of variables involved with growth, what factors or variables would you want to include? Click below for student responses:

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A Note on the Idea of Parameter.The notion of parameter is inherent to mathematical modeling. Roughly speaking, the parameters of a model are the constants involved in the model.

For example, we initially set Births per year equal to a constant, and we could have set that constant equal to anything we wanted. For that reason, Births per year was a parameter in the initial model. Once we changed the model by adding Per Capita Births per year and set Births per year equal to the product of Per Capita Births per year and Population, Births per year was no longer a parameter, but Per Capita Births per year became a new parameter, which we could set equal to something like 0.03. Introducing Carrying Capacity into the model introduced yet another parameter into the model.

Values for the parameters of a model are usually decided upon by collecting data or experimenting. However, values may be set in any way the modeler wants and the resulting model 'run' to see what the consequences are. The ability to experiment in this way is a very useful property of a mathematical model.

The Values of Mathematical Modeling.

1. One is forced to choose what to focus on. You must prioritize factors.
2. The modeling process helps make thoughts more precise.
3. A model helps one go beyond the surface of a phenomenon to an understanding of mechanisms and relationships.
4. One can play out different scenarios, modifying assumptions, initial values, and values of parameters, to see the resulting effects.

Problems Associated with Mathematical Modeling.

1. The model doesn't address what you want to accomplish.
2. The model is very sensitive to initial conditions or to the values of parameters.
3. The model creates a mathematical solution to a problem that doesn't lend itself to a mathematical solution.
4. The model is too simple to mirror adequately.
5. The model is too complex to aid understanding.
6. The results are too technical to communicate.
7. The results aren't in a form that can be implemented.
8. Resources aren't adequate to implement a suggested solution.