Population Computer Models

 

In this section we describe in simple terms what mathematical and computer models are. These models are the basis for the program or computer model that you'll be using in the last sections of the lesson. Models are also basic tools used by Population Ecologists in the study and analysis of populations.

Mathematical models. Scientists have always tried to come up with models that will behave, as close as possible, to the actual problem that they are trying to understand and analyze. When these models can accurately describe a system's behavior, in addition to helping provide an explanation of how the system works, they can also be used to predict how the system will work under different conditions. Before there were computers the models developed by scientists were entirely mathematical. Some models use pretty simple equations, while others can be very complex.

To better understand what is meant by a mathematical model, let's do a simple example. Let's assume that the problem that we are trying to analyze and understand is that of the growth of a squirrel population. Assume that there is a special cabin in the woods that you visit once a year and that squirrels live around the cabin. Let's also assume that since you've always been fascinated with the squirrels, every year you visit this nice cabin you count the total number of squirrels and keep good notes in your yearly visits diary. After several years, you go over the data that you collected and you pose yourself a problem by asking, "How quickly does the squirrel population grow? How many squirrels will I find this coming summer when I go back to the cabin?" In order to try to answer these questions you first look at the data that you've collected over the last five years:

Year	Number of Squirrels
1995	5
1996	10
1997	20
1998	40
1999	80

After observing the data and looking for a pattern, you suddenly realize that the squirrel population is doubling every year, i.e. the population of any given year is twice that of the previous year. The thing to do at this point is to write this mathematically, that is, with a formula or equation. In order to do this we need to use variables to represent the different quantities. The following table shows the variables that we will use:

Variable	What it represents	Examples and explanations
n	The year	n can represent any year, for example n can have a value of 1998. This can be written as n = 1998, or it can have a value of 1995 or n = 1995 If n is equal to 1998, then n - 1 represents the previous year or 1997
p	The squirrel population	p can have a value of 40. This can be written as p = 40 squirrels, or a value of 5 or p = 5 squirrels
p(n)	The squirrel population on a given year	This special notation (or way of writing) is used to indicate the population on year n. Remember that n can represent any given year and that is why they are called variables. To indicate the population on a specific year such as 1998, one can write p(1998) = 40 squirrels

Once the concepts introduced with the previous table are well understood, it is very simple to write the equation that describes the squirrel population growth:

p(n) = p(n-1) x 2

To conclude this example, this mathematical model (or formula) does indeed describe how quickly the population grows; it doubles every year. It can also be used to predict the expected population next year, year 2000.

Computer models. The mathematical model used in the previous section is a very simple one; however, some models can be very complex. In those cases a computer model can be very helpful in implementing the mathematical model. A computer model is essentially a computer program that contains the mathematical model and uses it to calculate solutions to given problems. In the early days of computers, the computer models or programs would read in a set of numbers, do their calculations, and print out the results. The computers of today are much more powerful and can do much more. A computer model can display the results graphically as it does the calculations, and it can also take the input through a nice user interface (by pointing and clicking with the mouse). These are advantages, but a more important advantage is that they can handle more complex mathematical models that better reflect the reality of the system being modeled.

Computer models can be developed for many different systems. A few examples are:

• Car traffic patterns on a given freeway
• Weather prediction
• Operation of a fast food restaurant
• Population growth and balance

Since we are dealing with a population growth model in this lesson, from here on that is the system that we will be describing; however, remember that many of the concepts apply to other systems.

There are three general ways to develop computer models:

1. By taking known nature rules and using them to develop the model. For example, if it is known that a certain species has 5 to 7 offspring every year and that it eats 50 pounds of food every month, one would use those numbers in the model.
2. By analyzing and sampling nature, doing tables and charts, coming up with mathematical equations that can produce the charts as accurately as possible, and then using those equations in the model.
3. A combination of both.

Why use computer models? There are many advantages to using computer models. Here are some of the most important ones. If you can think of others, let me know and I might include them here.

1. To gain a good understanding of the system being modeled -- In order to come up with an accurate model, one needs to understand how the real physical system behaves. Many times one needs to do many observations, collect data, graph data, come up with a first attempt to describe the model (that is, write an initial formula or mathematical model), do more observations, compare the initial model against the observations, improve the model, etc. etc. One needs to get very involved into understanding the real system's behavior in order to develop a good model.
2. To discover special behaviors of the system -- In general terms, models don't reflect the physical system 100%. One could say that there is always room for improvement. Here is a typical example. There are many cases where you have your real system and a model of it, and they are not identical. You do tons of observations and measurements on the real system followed by comparisons against the computer model to find out that they correlate (match or agree) most of the time. However, there might be a few places or conditions under which they disagree. These might be very interesting, surprising, and challenging situations. To be able to explain the differences one might have to do more and different kinds of observations, lots of analysis, lots of study, lots of exchange of information with other scientists, etc., to be able to come up with a reasonable explanation. Once there is a sensible explanation, one that several people agree with, the model can be refined or improved to reflect this new condition. The model will now reflect the realities of the physical system more accurately. The important aspect here is that scientists might think that they have a reasonable understanding of the physical system. After going through the whole exercise to refine the model, what they are really doing is discovering new mechanisms or behaviors of the system as well as coming up with explanations for these.
3. To predict behavior under different conditions -- A common use of models is to use them to attempt to predict what the system will do if some of the conditions change. For example, when dealing with a population model of several species, one may want to try to figure out what would happen if, say:
   
   
   • one of the species grows faster or slower than normal, or
   • the reproduction rate changes for one of the species, or
   • one of the primary food supplies decreases, or
   • a new predator comes to the area, etc.

If the model can handle these changes and accurately predict what would happen under any of these new conditions, it can be very helpful to scientists.

4. Last but not least, to learn about the physical system -- A computer model can also be very useful for people to interact with it and gain a good understanding of the behavior of the physical system. This is essentially the objective of this lesson, for you to interact and control a population growth and balance computer model to learn and appreciate how species interact and depend on each other. The next section describes the model that will be used in the rest of this lesson.

 

Population Fundamentals

Table of Contents

Description of our Population Model

Last Updated: Monday, 13-Nov-2000 05:22:04 GMT

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