Population Growth

Today's exercises are intended to familiarize you with the basic models of population growth, including exponential, logistic, time-delayed logistic, and discrete logistic.

Also, look at the R code for the Ricker's equation covered in class (text file in this directory). Play around with it. (Make sure to turn the history "on" in the plot window after you make the first plot). Can you write code to make a bifurcation diagram?

Density Independent Growth

At the main menu choose "Population Growth". On the population growth menu choose "Density Independent Growth". This will get you into the first part of Populus that we will use this week. Take the time to read the introductory material to re-familiarize yourself with the concepts associated with density independent population growth.

Exercise #1 - Density Independent Growth (Continuous Model)

The program for density independent growth allows you to examine both a discrete and a continuous model. Begin with the continuous model and you will notice that you can set three parameters: initial population size, growth rate, and number of generations.

Begin with the default settings for these parameters (N₀ = 10, r = 0.1, & generations = 10) and run the simulation. Examine the relationship between N and time. What is the pattern. Does the population grow or decline? What is the final population size?

Use the space key to view the graphs of N, log N, dN/dt, and dN/(Ndt) vs time. Examine all four patterns. Which patterns are similar to each other? What is the relationship between dN/(Ndt) vs time and r?

Now, change the value of r to r = .2. What is the final population size? Examine the four graphs again. Is there a change in the patterns from the previous parameter values?

Now change the value of r to r= -.1. Run the simulation and examine the graphs. How have the graphs changed?

Exercise #2 - Density Independent Growth (Discrete Model)

The discrete model presents only the population growth pattern. Vary the values of lambda and examine how the population grows. For what values of lambda does the population grow and for what values does it decline?

Logistic Population Growth

Exercise #3 - Logistic Population Growth (Continuous)

Now exit the density independent growth module, return to the main menu, and choose the logistic growth module. This module has three different programs that you can run: continuous, lagged continuous, and discrete. Choose the continuous model.

The default parameter values are N₀ = 5, K = 500, and r = 0.2. Run the simulation with these parameter values, switch views (using the space bar), and examine the relationships between N and log N vs time, and dN/dt and dN/(Ndt) versus density. How do these relationships differ from the density independent model? At what density (relative to K) is the rate of population growth (dN/dt) maximal? At what density (relative to K) is the per capita rate of population growth maximal? What is the relationship between K and the final population density? What is the relationship between r and the final population density?

Now, modify the value of K and rerun the simulation. What is the relationship between the final population density and the value of K? Are the shapes of the relationships of dN/dt and dN/(Ndt) with density affected by the value of K? Are the shapes of the relationships of N and log N with time affected by K? How long does it take for the population to come within 10% of its final value?

Now modify the value of r, perhaps doubling it, and run the simulation. What is the relationship between the final population density and the value of r? How long does it take for the population to come within 10% of its final value?

Now modify the value of N₀ so that N₀ is greater than K, and run the simulation again. What are the patterns in the four graphs now?

Exercise #3 - Logistic Population Growth (time lagged)

Switch to the time-lagged logistic population growth option, and reset the parameter values to the default values of N₀ = 5, K = 500 and r = .2. Set the time lag, T, to T = 0, and run the simulation. This is the equivalent of a simple logistic population growth. Now do a series of runs in which you set the value of T to T = 1, 2, 3, 4 and 5. What effect does the time lag have on the pattern of population growth?

Return the value of T to T = 1, so that there is a short time lag, and now do a series of runs in which the you set the value of r to larger and larger values. At what value of r do you observe a noticeable overshoot of K? At what value of r do you begin to observe a series of damped oscillations? At what value of r do you begin to observe fluctuations that do not apparently dampen out? Overall, how would you characterize the effect of r on the pattern of growth when there is a short time lag?

Now explore the effect of both T and r together on the pattern of population growth. For example, set the value of T to T = 3, and run through a series of values for r. Or, set the value of r to a higher value, such as r = 1.0 and run through the series of time lags, from T = 0 to T = 5. How would you characterize the interaction between r and T in generating the different patterns of population growth?

Exercise #3 - Logistic Population Growth (discrete)

Switch to the discrete logistic option, and set the parameter values back to their default values (N₀ = 5, K = 500, r = .2). Now, do a simulation with these parameter values. Does the pattern of population growth resemble a normal logistic population growth?

Do a series of runs, slowly increasing the value of r until you notice a change in the pattern of population growth. What is the new pattern of population growth? At what value of r do you notice this change?

Continue increasing the value of r until you notice another qualitative change in the pattern of population growth. What is this new pattern, and at what value of r does it occur?

Continue increasing the value of r until you notice another qualitative change in the pattern of population growth. What is this new pattern, and at what value of r does it occur?

General Questions

The patterns of population growth with time lags and with discrete logistic growth are in many ways very similar. Why do you think this is the case?

What kinds of factors are likely to generate time lags?

Are the conditions that lead to wild patterns of population growth likely to occur in nature, and if so, for what kinds of organisms?

If you saw a pattern of population densities in the field that looked like the patterns seen for discrete logistic growth with high values of r, would you be able to figure out from the field data the rule the population was following?

Thanks to John Addicott