Modeling Population Growth: Exponential Models[1]

The concept of population growth patterns is a key component of understanding evolution by natural selection and the population dynamics in ecosystems. Organisms have the potential to achieve exponential growth under ideal conditions, yet sustained exponential growth is not found in nature. This observation is a cornerstone in the theory of evolution. Factors that limit growth can lead to evolutionary change in a population and can have profound effects on the interactions between organisms.

The plant model used is the bell pepper plant (Capsicum annuum). Bell peppers are annual plants (Crockett, 1972) and the fruits are readily available in supermarkets. Seeds are the plants’ potential offspring and are found in fruits. Annual plants produce seeds that typically germinate the next year (or later), and the parent plant only survives one year. Some plants, including the bell pepper, can self-fertilize. Using an annual plant allows for simplifying assumptions to be made about the potential for population growth.

In this model of plant growth, the assumptions are:

  1. The peppers distributed to the class represent the fruits from one pepper plant.
  2. The number of seeds in the peppers is the offspring of one pepper plant.
  3. The number of seeds produced by every pepper plant in a population will be equal to the total number of seeds in the peppers counted by the class.
  4. Conditions are ideal. There are unlimited space and unlimited nutrients to support growth. There is no plant predation or disease. Climate is stable and favorable for growth.
  5. All the plants die at the end of the summer.
  6. All seeds produced by the pepper plants will grow a plant the following year.
  7. There is only one generation of pepper plants per year.
  8. The number of plants at the beginning of the population model is one.

These assumptions allow you to calculate and graph a simple population growth equation by eliminating survivorship of plants in the next generation. The death rate is assumed to be 100%. You will develop your own equation to use in graphing the population potential over a number of years.

Procedure:

  1. Cut open the pepper and count the number of seeds inside. Record the total number of seeds in your notebook and on the board so that a class total can be obtained. Assume that the number of bell peppers in the class represents the number of peppers produced by one bell pepper plant in a growing season.
  2. Record the total number of seeds counted for the class.
  3. To complete the Data Table, assume that:

(a) Only one bell pepper plant existed the first year and that it produced the number of seeds represented by the class total.

(b) Each seed always grows into a new bell pepper plant the next year.

(c) Each new plant always produces the class total number of seeds.

(d) All pepper plants die at the end of the year.

  1. Find the number of bell pepper plants that grow in the second year and record this number in Data Table 2. (Check with other groups and the teacher to make sure that this step is performed correctly before continuing.)
  2. Calculate the number of bell pepper plants that will grow in the second, third, and fourth years. Describe the method you used to calculate the number of plants for each generation.
  3. Write the method as a mathematical equation.
  4. Graph your data!Watch the scale of your graph, since the y-axis will have to cover a wide range!

Data Table

Total Number of Seeds in Your Bell Pepper
Total Number of Pepper Seeds of All Bell Peppers in the Class. (This is how many seeds each plant will produce.)
Number of Pepper Plants at Year 0 / 1
Number of Pepper Plants Year 1
Number of Pepper Plants Year 2
Number of Pepper Plants Year 3
Number of Pepper Plants Year 4

Analysis

  1. What is the mathematical equation or rule governing how to calculate the pepper population for future years?
  2. Using your graph and/or equation, predict what the population of the bell pepper plant will be in the sixth year if the assumptions are valid. Would this growth of pepper plants be likely to occur?
  3. What will the growth curve look like if only 10 seeds produce plants each generation? Is this growth rate sustainable?
  4. What factors might influence which individual plants survive? If there is a genetic trait that favors survival, how might the characteristics found in the plant population change?
  5. How many seeds, out of the potential number of seeds, can survive to produce population growth that is not exponential?
  6. Using the data provided, predict what the human population will be in the year 2020 if population follows the predicted curve. Predict the population for the year 2050.
  7. Is the current growth rate of the human population sustainable? What factors might limit or promote human population growth?
  8. Identify and discuss the ethical and social justice issues associated with the growth of human populations.

[1] Adapted from the American Biology Teacher article: “Modeling Exponential Population Growth.”