Single Species Population Growth – Populus Simulations
Biol 301 L
Populus is a modeling program developed at the University of Minnesota. Ecologists often use models (Mathematics) to understand the dynamics of population change. In this exercise you will investigate models that describe density independent, and density dependent population growth. By manipulating parameters of each model, you should gain an understanding of how these parameters influence changes in population dynamics over time.
The models used here include the following parameters:
N_{0}_{ }= Population size (number of individuals) at time 0. This parameter ranges from 1 to infinity.
r = intrinsic rate of growth. This parameter ranges from 0.1 to 5.
K = carrying capacity. (Density dependent population growth only). This parameter ranges from 1 to infinity, and represents the maximum number of individuals the environment can support.
T = time lag (lagged logistic population growth only). This parameter ranges from 010, and represents the influence of population size in previous generations on the current generation’s population growth rate.
Single Species Models

Begin the Populus program by double clicking the icon on the desktop.

Once the program has loaded, click the ‘model’ menu at the top lefthand corner of the screen.

Select ‘single species dynamics’ followed by ‘density independent growth’ from the sub menu.

A new window will appear that displays model type, plot type, and model parameters. Select ‘continuous’, plot N vs. t, and leave the parameters as the default values. Click view to model the population growth.
Describe the growth curve of this population simulation in the space provided below.

Select the ‘B’ tab under population parameters. Increase growth rate (r) to 0.2. Be sure to check the box next to population parameters to display the new population. Select the ‘C’ tab under population parameters. Increase the growth rate (r) to 0.3. Be sure to check the box next to population parameters to display the new population. You should see three populations displayed on the graph.
Describe the growth curves of the new populations with respect to the original population you modeled above.
Does the population stop growing? Explain.
Which model discussed in class describes this growth curve? Write the formula.
When you finish answering the questions, select the close tab on the graph, and the model menu.
Single Species Models (cont)

Once again, click the ‘model’ menu.

Select ‘single species dynamics’ followed by ‘density dependent growth’ from the sub menu.

Select ‘continuous logistic’ under model type. Again, plot N vs. t, and leave the parameters set as the default under the ‘A’ tab.

Select the ‘B’ tab and set N_{o} = K. Be sure to select the box next to population parameters to display the new population. Select the ‘C’ tab, and set N_{o} > K. Check the box next to population parameters, and select the view button from the input menu. You should see three populations.
Describe the effect of initial population size on continuous logistic population growth.
What model described in class explains these growth curves? Write the formula.
Explain why this model behaves this way.
How would an increase in growth rate (r) influence the models behavior? Before you investigate further, provide your prediction here.
Use Populus to test your prediction. Close the current graph, and model input menus, and start a new density dependent, continuous logistic growth model. Again, manipulate the ‘A’, ‘B’, and ‘C’ population parameters. Use the default values for N_{o} and K in all three tabs. Increase growth rate (r) in tab ‘B’, and increase it even further in tab’C’. Provide your parameters in the space below.

Plot N vs. t, and describe the effect of increasing growth rate on logistic population growth. Was your prediction accurate?
Now, do the same thing, only this time vary the value for K keeping N_{o} and r constant. Provide your parameters below, and explain your findings.

Describe the effect of increasing K on logistic population growth.
Continuous vs. Discrete vs. Timelagged logistic population growth.
A continuous logistic growth model assumes that reproductive generations overlap, and population growth and density dependent feedback are instantaneous. However, many populations in the wild do not behave this way.
In species that reproduce only during specific seasons, population growth (dN/dt) occurs in discrete time intervals. This may result in unpredictable (chaotic) dynamics from generation to generation for some combinations of r and K. Discrete logistic models are useful for understanding populations that behave this way.
Also, in many species, the rate of population increase many be dependent on population size in previous generations. That is, there is a time lag from the introduction of new individuals in the population to reproductive maturity and the birth of new individuals in the next generation. The “grandmother effect” in whitetailed deer is a documented phenomenon. If conditions are good when your grandmother is gestating your mother, then your mother tends to produce offspring that survive and breed well. This effect often results in a 2 –generation time lag in density dependent population growth. Therefore, a timelagged logistic model is more useful and maybe more accurate for understanding populations that behave this way.
Use the following parameters to investigate continuous logistic population growth, as well as discrete population growth. Record your observations in the space provided.
Continuous population growth:
Trial

N_{o}

r

K

run time

Observations

A

5

0.5

500

50


B

5

1.5

500

50


C

5

2.65

500

50


D

5

3.6

500

50








Discrete population growth:
Trial

N_{o}

r

K

run time

Observations

A

5

0.5

500

50


B

5

1.5

500

50


C

5

2.65

500

50


D

5

3.6

500

50


Run each set of trials for continuous and discrete logistic population growth. Compare and contrast the results from these two models.
For each of the models explored above, give two species whose population growth may be described by the models.
Are there other species whose population growth may not be explained by any of these models? Give an example and explain why these models do not apply. (hint: Think about some of the assumptions of these models). 