Population Growth—chapter 2

Patterns in Populations

Here's the goat again

Understanding the patterns of population change has become possible by creating mathematical models, or equations, that have the same characteristics as actual populations.

An equation that successfully models population growth must display some of the same patterns that we saw on Goat Island. Making a graph that is a visual expression of a mathematical equation can be very helpful in understanding the meaning of the equation.

Graph various scenarios and then brainstorm additional population patterns.

I. Feedback Loops

Party planning

Another way to understand the reasons for changes in the goat population is to draw a concept map of the island ecosystem. If you do this, you will notice that the map has loops. These are called feedback loops. You probably have heard the term feedback when someone asks you a question such as, “How did you like the party? I would like your feedback.” Just answering a question is not necessarily feedback. Only if the response changes the future situation is it proper to call the response feedback. For example, if the question was, “Do you think there were too many people at the party?” The response is feedback because it can change future actions.

In the language of science, feedback only occurs when the result automatically controls the process of change—the output of a system controls its input. A room thermostat is often used in illustrating the principle.


Use containers of water to model a biological feedback loop.

Photograph of a thermostatThe outer case of an office thermostat. 
Photo Dante Alighieri by via Wikimedia Commons.

The type of feedback loop in a thermostat is called a negative feedback loop. A room thermostat is often a wall-mounted switch which works as follows:

There are two thermometers in the thermostat. One displays the temperature and is located in the cover. The other controls the heating and cooling systems and is simply a coiled bimetallic strip (two different types of metal laminated together). The metals in the strip expand and contract when they are heated or cooled, but each metal has a different rate of expansion, so when heated, the metal on the inside of the coil expands more and the strip tends to uncoil.

The center of the coil is connected to the temperature-adjustment lever, and a mercury switch is mounted to the end of the coil so that when the coil winds or unwinds, it tips the mercury switch one way or the other.

Your concept map of the thermostat may have looked like the one below. The Thermostat represents a negative feedback loop which tends to keep the condition of the system stable.

 In a thermostat, the metal coil expands and contracts with temperature changes, tilting the mercury switch back and forth, alternately turning on and off the furnace.

There are negative feedback loops in your body which maintain stability in your living system. In biology, these negative feedback loops are called homeostasis. A good example is the interaction between your thyroid gland and your pituitary gland. The thyroid gland is under the control of the pituitary gland, a small gland the size of a peanut at the base of the brain (shown in the diagram). When the level of thyroid hormones (T3 & T4) drops too low, the pituitary gland produces Thyroid Stimulating Hormone (TSH) which stimulates the thyroid gland to produce more hormones. Under the influence of TSH, the thyroid will manufacture and secrete T3 and T4 thereby raising their blood levels. The pituitary senses this and responds by decreasing its TSH production. One can imagine the thyroid gland as a furnace and the pituitary gland as the thermostat. Thyroid hormones are like heat. When the heat gets back to the thermostat, it turns the thermostat off. As the room cools (the thyroid hormone levels drop), the thermostat turns back on (TSH increases) and the furnace produces more heat (thyroid hormones).

Question 2.6.
There are many other examples of negative feedback found in nature.
Can you think of any?
Find an example in a book or on internet and diagram it as a negative feedback loop.

Question 2.7.
Can you think of negative feedback in manufactured substances?
Make a diagram of one such feedback loop.

II. Negative and Positive Feedback

 Negative feedback loop.

The kinds of feedback illustrated by all the examples used so far maintain many natural systems at proper working levels. It is called negative feedback because the change to the system is restored each time to the start by canceling out the change. Therefore, the change is added, but then subtracted. This keeps the system within bounds.

Question 2.8.
Can you explain how the amount of light coming into your eye controls how much more light comes in? Diagram this loop.

When the topic of feedback is discussed the idea of positive always being good gets turned on its head. Physicists long ago labeled positive feedback as feedback which added to the change in the system. This creates a situation which escalates until the situation is out of control.

 Ironically, this diagram shows a positive feedback situation.

Question 2.9.
Using the Palestinian-Israeli conflict as an example, can you make a positive feedback loop?

Another example of positive feedback can be when the very condition that makes a place special makes it overused. Ecotourism is an example. These tours usually consist of groups of people who love nature who want to see some of the rare and beautiful wonders of the world. Our beautiful Yosemite National Park has been overrun by so many tourists that traffic jams and air pollution are the norm. So many people wish to view the beauty that they have spoiled the beauty.

Question 2.10.
Can you think of other natural wonders in danger from ecotourism? Make a positive feedback loop illustrating your choice.

Let’s return now to Goat Island in the light of our new knowledge about positive and negative feedback loops. The diagram below shows the island ecosystem before the introduction of dogs.

Feedback loop on goat island

Question 2.11.
Does the diagram show a positive or negative feedback loop?

Here are some clues.

Looking at the bottom line, we can see that the amount of food depends on the environmental conditions (which determine how quickly the food grows) and consumption (which determine how rapidly it is depleted). Looking at the top line, we see that the population of goats depends on the birthrate and death rate.

The connection between the upper and lower parts of the diagram is crucial. Looking at the vertical arrows, we can see that the population of goats affects the consumption of plants; and that the number of plants affect the death rate of the goats. The number of goats affects the food supply, which affects the number of goats, which affects food supply, and so on.

The second diagram shows what happens to the system when the dogs are introduced. The bottom and middle layers are the same as before, but a top layer has been added. The death rate of the goats is now affected not only by the available food supply, but by the population of predators as well. The population of goats, in turn, affects the death rate of dogs. When the population of goats falls, the dogs have less to eat. There are now two feedback loops in the system; one linking the goats and plants, and one linking the goats and dogs.

A feedback loop with the goats and the dogs

The ecosystem map is a simplified picture of reality that can show us things. As an example, consider this question:

Question 2.12.
What would happen if a blight sweeps the island, and most of the plants that the goats depend on for food die?
How would that affect the goats and the dogs?
Try making a feedback loop for this situation.

Positive feedback loops often can result in the total breakdown of the system, and with living organisms, possible extinction or huge cycles of feast and famine. This is often the result of populations exceeding the carrying capacity of their environments due to the exponential growth in numbers.

III. What is Exponential Growth?

When populations grow rapidly because there are few limiting factors the growth is exponential. This happened on Goat Island at the beginning when there were no dogs and there was plenty of food and water for the goats. Exponential growth can have tremendous impacts.

An activity to visualize large numbers.

Investigate exponential growth in bacterial populations

A number of things might keep the cells from growing to such a great mass. Did you guess that these are:

  • Limited food supply
  • Amount of space for growth
  • Death rate of the cells
  • Buildup of waste products
  • Attack by or competition with other cells
  • Weight of the mass of cells upon itself

There may be still other possibilities, too. The growing body of cells after ten hours or so may begin to change the temperature of the air around it. As a result, the temperature change may slow the growth rate.

We are talking here of things in the surroundings that may hold back the growth of the cells. The surroundings can only support a certain number of the cells. This is called the carrying capacity.

When that number is reached the population growth slows. Finally, it levels off. The same ideas would hold for humans and their population growth rate. Has this happened yet with the human population? How can you find out?

How much time do you think it will take to fill a bucket if you double the number of drops you add every 30 seconds?

IV. Exponential Growth, Road Kill,
     and Sexual Reproduction

An armadillo

If you have traveled in the southern part of the United States and in Mexico, you may have noticed a particular animal which is often run over by automobiles. Why are these animals often the victims of drivers, and how do they manage to survive despite the large numbers being killed on the highways?

So far you have only applied the idea of exponential growth to nonliving objects or to organisms which reproduce by simple cell division. What about organisms which reproduce sexually like the goats and the dogs? Are they also subject to periods of exponential growth? Let us find out by meeting Mamadillo and Papadillo and some of their descendents.

Question 2.15.
What might prevent the numbers of armadillos from being as large as you calculated? Refer back to the activity on the bacteria to get some ideas. Do the limits on bacterial populations also apply to armadillos, goats, or dogs? Discuss your ideas with your classmates and prepare to present them to the class.

Question 2.16.
What are some limiting factors caused by humans that are affecting plants and animals around the world?


2-6. Adding Armadillos

Trace the lives on three generations of an armadillo family.

V. How Do We Know Population Size?

We often talk about how big a population is and how fast it is growing, how many are surviving in a certain area, and whether or not a population is endangered. How can ecologists find out what is happening in a population in the wild? After all, they cannot go out into the field and see every individual organism. They can collect pelts and count them as they did with the lynx and the hare, or they can use what field biologist call “mark-recapture” You will perform an activity which shows how this works.

If weather and your terrain around the school permit, you can do a mark-recapture activity with real organisms other than grasshoppers. There are many suggestions on the internet.

Obviously, we normally do not go out and mark and recapture humans for study. (Can you think of a way that a mark-recapture activity could be done in a high school to estimate the total number of students?) Instead, we use a method called a survey. The Nielson ratings used for television are an example of this type of counting, as are telephone surveys where a computer randomly dials numbers to be called.

Go outside in the autumn to estimate the population size or grasshoppers (or other animals). 

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