Computer Simulation of Population Growth using Populus 5.4

Density-Independent Model

From the main menu, select Model, then select Single Species Dynamics and Density Independent Growth.

Density-Independent growth refers to a situation in which limitations to growth are not related to the density of the population. You will be presented with two models. The discrete (geometric) population growth model can be used for populations with non-overlapping generations. Population growth is calculated using the equation N_t = N₀l^t. The continuous growth model is used for populations with overlapping generations. This model uses the equation N_t = N₀e^rt. Both of these equations are discussed in the chapter on population growth in The Biology Web and in the textbook.

Select "Discrete"  in the Density-Independent Growth input box. The parameters for discrete (geometric) growth below are already entered. 

Drag the base of the graph to the bottom of the screen to enlarge the graph.

Discrete parameters: 

Initial population size (N₀) = 10

Growth rate ( or lambda) = 1.11

Number of generations = 20

Examine a plot of growth for 20 generations. The red dots on the graph are the population size at time t. The dots are connected by a broken line because the population does not grow during the interval between each time period. Growth occurs at the time indicated by the red dots but not during the time between the dots. It would be more accurate to draw a stair-step graph with each point representing a step. The diagram below shows the discontinuous nature of geometric growth.

Select continuous.

Parameters for continuous independent growth have been entered.

Continuous parameters:

Initial population size (N₀) = 10

Growth rate (r) = 0.104

Number of generations = 20

The letter "r" is used for the growth rate in this model because the rate is an instantaneous rate. An instantaneous rate is used for populations that grow continuously. The line is smooth to indicate that growth is continuous; it is always occurring.

Notice that both curves (Discrete and Continuous) are J-shaped. If the X-axis were compressed or if the population grew for more generations, the J-shape would be more obvious.

1.      Describe how the population growth for the discrete model differs from that produced by the continuous model. Describe how they are similar.

 

 

2.    Why does the curve for the discrete model differ from that of the continuous model?

 

 

3.      Choose the discrete model. Change lambda from 1.11 to 2.22 and press Enter. You can also click View to update the graph each time you change one of the parameters. What happens to the population growth rate when lambda is increased?  Be sure you check the numbers on each axis when comparing curves! The digit(s) to the right of the letter E on the Y-axis are the number of zeros. For example 9E07 is 90,000,000. The numbers on the Y-axis are very large, making it look like the population is at zero for the first few generations. However, the population size in these early generations is not zero at all; it is much larger than the population was at that time when lambda was 1.11. It looks like it is zero because the large numbers on the Y-axis. The distance that the dot is located above the zero line might be barely visible but this slight distance represents a large number of individuals.

      The population size for generation 16 through 20 is readable. To answer the question- What happens to the population growth rate when lambda is increased?- look at the population size for generation 16 or 17 on both plots.

 

4.      Choose the continuous model. What happens to the population growth rate when r is increased from 0.1 to 0.8? Be sure that you check the numbers on the Y-axes when comparing the curves. As in the previous question, the distance that the dots are located above the zero line might be barely visible but this slight distance represents a large number of individuals. As with question 3 (above), generations 16-20 are readable so it might be easier to compare the graphs by looking at population size in one of these generations.

 

 

5.  What kind of organisms are best represented by the discrete (geometric) model?

 

6.  What kind of organisms best represented by the continuous model model?

 

Logistic (Density-Dependent) Population Growth Model

The formula for logistic growth is

Click Model on the main menu bar, then choose Single- Species Dynamics: and Density-Dependent Growth. The Density-Dependent Growth: Input box will open. Use the default parameters (listed below) and click View to view a graph of population growth. Drag the top of the box to the top of the screen so that it fills the screen from top to bottom.

Parameters for Logistic Growth

Initial population size(N₀) = 5

Carrying capacity (K) = 500

Growth rate (r) = 0.2

Number of generations = 50

7.   How does the continuous density-dependent growth curve given in this part of the program differ from density-independent growth?  To answer this question, you will need to return to the density-independent growth model and enter the above parameters for continuous growth. Click View after you enter the parameters.

 

 

8.   Return to the continuous density-dependent model. Use the parameters given above but change the growth rate (r) to 1.0. Describe what happens. How does population growth in this population differ from when the growth rate is 0.2?

 

 

9.   Consider the term 1-(N/K) in the equation for logistic growth above. Explain what happens to the value of this term when the population size (N) is equal to the carrying capacity (K). In other words, what is the value of 1-(N/K) if N=K? To answer this question, you will need to plug numbers into the formula and calculate a value.

 

 

10. Replace the term 1-(N/K) in the logistic equation (below) with the value that you calculated in question 9. How does the value calculated in question 9 affect the overall logistic equation (below)? 

 

 

11. Consider the term 1-(N/K) in the equation for logistic growth. Explain what happens to the value of this term when the population size (N) is very small, that is if N is nearly zero. To answer this question, you will need to plug numbers into the formula and calculate a value. Try using a value of 0 for N. Use any value for K.

 

 

12. How does the value calculated in the previous question affect the overall equation?
To answer this question, first consider the overall equation:

 
Next, replace the term 1-(N/K) with the value that you calculated in question 11.

 

 

Time Lags

The term 1-(N/K) limits population density, causing growth to level off. A time lag is a delay in the effect of 1-(N/K). As a result of this delay, the population growth will be faster because limitation occurs later. For example, suppose that a population is limited by predators; as population density increases, the rate of predation increases and limits further density increase. There may be a delay in the effect of predation because predators need to reproduce and become mature before they can become effective. Predator reproduction depends on the population size of their prey but it may take time before predators have more offspring. The effect of predation therefore may be more a function of the population size (N) at some previous time rather than at the current time. In the density-dependent (logistic) model, we can add the letter T to represent a time lag. The lagged continuous model in this section adds a time lag.

With a time lag (T), the formula for logistic growth becomes

   

Select Lagged Logistic in the Density-Dependent Growth: Input box. This model calculates population growth using the equation above. Use the parameters for logistic growth that you used in the previous exercise above (N₀ = 5, K = 500, r = 0.2) and set the time lag (T) to 2. View the graph of population growth by clicking the View button.

13. Did the population grow faster or slower than the population without a time lag (T=0)? Why? Your grade for this question will be based on your answer to the question - Why? [Hint: The answer can be found in the discussion of time lags above.]

 

 

14. Increase r to 0.8 and then click View. What happened?  Why? Your grade for this question will be based on your answer to the question - Why? [Hint: As the population density approaches the carrying capacity, think about what will happen to density if the effect of 1-(N/K) is delayed.]

 

 

15. Increase r to 1.0 and leave the other parameters unchanged. What happened?  

 

 

16. The population shown is unstable because of the large fluctuations.  

a) Based on your answers to questions 14 and 15, describe the relationship that r has on population stability when there is a time lag. 

b) Describe the relationship that r has on population stability when there is not a time lag. See your answer to question 8 for help with this one.

 

 

17.      Change r to 0.5 (T = 2). What happened?

 

 

18.   Keep r at 0.5 but increase the time lag by changing T to 4. What happened?

 

 

19.  Summarize the effect of time lags on population growth and stability. You should mention the effect of time lags (see your answer to question 13), the effect of increasing growth rates on time lags (see your answer to questions 14 and 15 above), and the effect of increasing the amount of the time lag (see your answer to questions 17 and 18 above).

Environmental Science (ENV 101) students should stop here. Ecology students (BIO 206) should continue with the section below.

Age-structured Populations

Close all of the open models by clicking the Close button.

Click Model, then select Single-Species Dynamics: and Age-Structured Growth.

In this exercise we will start with a hypothetical population of animals in which all individuals are in the 0 age category; there are no adults. The time units for this hypothetical species will be years.

Select output type S_x/S_x vs t. This is the third choice under Output Type.

Change the number of classes to 7 (7 years).

Change Run Time to 15.

Change Birth Pattern to Birth-Pulse (Discrete). This hypothetical animal breeds once each year.

Census Timing should be Postbreeding.

The L_xM_x schedule at the bottom of the input box should be modified. Enter the values in the table below.

Initially, the population will have only one individual whose age is 0 years (newborn). This is indicated by the value of 1 in the first cell of the S_x(0) column in the table below. The S_x(0) column tells us the proportion of individuals in each age category in the initial population. The value 1.0 for age 0 means that 100% of the individuals are age 0. We will plot the proportion of individuals in each age category (age structure) after each generation.  Use the following survival (l_x) and fecundity (m_x) values. The cells in the table can be changed by double-clicking them and then selecting the number.

Change the value in Age Class to View to 0 and press View. It will be helpful to enlarge the graph by dragging the bottom so that it fills the screen. The graph will display the proportion of individuals that are age 0 for the next 15 years. The graph shows that initially 100% of the population is 0 years old. This is what you entered into the table (see the table above). One year later, none of the individuals in the population are 0 years old because they are all one year old. Notice that at two years, 1/2 of the population is 0 because some of the individuals reproduced. At 3 years it is approximately 0.33, then 0.4 etc. Notice what happens to the proportion of age 0 animals after about 8 years.

Next, use the up arrow to change the value in "Age Class To View" to 1. The proportion of 1-year-olds initially is 0 because the initial population was all 0 years old. After 1 year, it becomes 1 (or 100%), then 0.35 etc. As with the 0 age category, notice that the proportion of 1 year olds stops changing after a few years.

Change "Age Class To View" to 2, then 3. Continue until you have viewed a graph of all of the age classes. For each age class, check to see if the age distribution stops changing.

20. How many years does it take before the age structure of the population stops changing? At this point, the population has reached a stable age distribution.

 

21.  What eventually happens to the age structure of a population that is started with 1 individual in age class 4? This can be done by putting a 0 in the first cell in the S_x0 column (for age x = 0) and putting a 1 in the cell for x = 4 (the 5th cell down in the S_x(0) column). Change Age Class to View to 0 and then click View. Try viewing different age classes. It might be convenient to view all age classes together on the same graph by selecting "View All Age Classes." Does it become stable and stop changing? After a stable age distribution is reached, what proportion of the population is 0 years old? After a stable age distribution is reached, what proportion of the population is 1 year old? 2 years old? 3 years old?

 

22. Try using different values in the S_x(0) column. For example you might starting a population with half of the individuals age 0 and half of the individuals age 1. What eventually happens to the age structure of a population regardless of the initial age composition? Does it become stable and stop changing?

 

* In this part of the exercise (Age Structured Populations), you graphed the proportion of individuals in each age category. You did not look at population size. Population size was increasing exponentially.

The answers to these questions should be submitted using ANGEL.