The most important factor affecting the population growth of a species is its size relative to the resources in its habitat. Most populations grow if their population is small compared to their habitat and decline if their population is too large compared to their habitat. The key phrase here is "compared to their habitat." Population biologists need to focus on population density -- population divided by the area of a habitat. But habitats are not uniform and a population is rarely uniformly distributed throughout its habitat. Some locations may have more food than others and some may be more densely populated than others. Furthermore, most species are mobile and biologists must be concerned with the effects of migration. In this section we discuss the effects of migration on population density.
We represent migration by a vector field. In a two-dimensional habitat, the kind most of us enjoy, migration is described by a vector field of the form --
In a three-dimensional habitat, the kind enjoyed by fish, migration is described by a vector field of the form
|
The applet at the right shows two dimensional migration described by a vector
field of the form
You can change the vector field later but for now look at the vector field that is used when the applet first starts. Notice the north-south, or j component, is zero so migration will always be east to west or west to east. Both axes run from -1 to +1. The red dot is located at (0, 0.8) and the green dot at (0, -0.8). At the red dot the vector field is pointing to the right, so migration will be from west to east. Click on the red dot to see this migration. At the green dot migration will be from east to west. Click on the green dot to see this migration. At points further away from the x-axis the rate of migration is more pronounced and at points closer to the x-axis it is less pronounced. Click on several different points to see this effect. |
Use this applet to experiment with migration described by each of the following vector fields. You can specify the vector field by typing in the two boxes, labeled P(x, y): and Q(x, y):, at the top of the applet.
Describe what you see for each of the following vector fields.
This applet uses a parser written by Darius Bacon to interpret the algebraic expression defining the function P(x). The parser is available at his web site. Please see his file on copying.
In two dimensions the vector field
describing migration describes both the direction and rate of migration and is usually measured in units of individuals per unit of length per unit of time -- for example, people per mile per year. For example, if
then the rate at which people would cross a fence whose length was 3 miles and which ran from north to south would be 1245 people per year. If the fence ran from east to west, no individuals would cross it because they would be migrating parallel to the fence.
In a three dimensional habitat the vector field
describing migration is measured in units of individuals per unit of area per unit of time -- for example, fish per square kilometer per month.
Although we often talk in terms of individuals, population biologists often talk in terms of biomass. This is often a better (and more easily measured) measure of poulation size because, for example, ten robust individuals may be preferable to 20 scrawny ones.
Click here to open a new window with another Java applet. Arrange the two windows so that they overlap and you can move easily back-and-forth between them by clicking on the exposed portion of the inactive window to make it active. Eventually when you are done with the applet, close its window.
When it first starts, this applet looks at migration described by the vector field
with the x-axis and the y-axis both running from -1 to +1. You can change the vector field or the range of the axes later but for now just click on the graph paper. You should see an animation showing how the population density changes as the population migrates as described by this vector field.
At first the population is uniformly distributed, indicated by the color white. Then as the population migrates the population density changes. Higher population densities are indicated by shades of red and lower ones by shades of blue. By the time the animation stops, the highest population density is in the northeast corner and the lowest in the southwest corner. Can you explain why? You may want to try some other vector fields to see what happens. For example, try the vector field
Notice that with this applet absolute value is written abs(x) rather than |x|.
This applet estimates population density by dividing the region shown in the graph into small rectangles like the rectangle shown below.
It estimates the rate at which the population is entering the rectangle on the left by P(x - h, y) 2k and the rate at which the population is leaving the rectangle on the right by P(x + h, y) 2k. It estimates the rate at which the population is entering the region on the bottom by Q(x, y - k) 2h and the rate at which the popluation is leaving on the top by Q(x, y + h) 2h. Thus the rate at which the population is decreasing inside the rectangle is estimated by
P(x + h, y) 2k - P(x - h) 2k + Q(x, y + k) 2h - Q(x, y - k) 2h
and the rate at which the population density in the rectangle is decreasing is estimated by
P(x + h, y) 2k - P(x - h) 2k + Q(x, y + k) 2h - Q(x, y - k) 2h
-------------------------------------------------------------- =
4 h k
P(x + h, y) - P(x - h, y) Q(x, y + k) - Q(x, y - k)
------------------------- + -------------------------
2 h 2 k
We can obtain very good estimates of the rate at which the population density is decreasing at the point (x, y) by using very small values for h and k and we can get an exact answer by taking the limit of these estimates. We use the same terminology, divergence, and the same notation, div F, we used in one dimension.
Verify your answer using the Java applet in your other window.
Verify your answer using the Java applet in your other window.
Verify your answer using the Java applet in your other window.
Verify your answer using the Java applet in your other window.
Verify your answer using the Java applet in your other window.
In three dimensions when migration is described by a vector field of the form
we use the same terminology and notation and get an analogous formula for the divergence of F -- that is, for the rate at which the population density at a point is decreasing.
