Population Ecology - Example Exercises

Shown below are two homework problems. Typically, there are about 10-12 such homework assignments during the semester.

Even the uninitiated can see that this is a classic example of logistic population growth.

Homework 4

Do problems 2, 3, and 5 at the end of Chapter 2 in Gotelli.

Show analytically that in the case of limited growth described by the logistic equation, the maximum growth rate occurs when N^ = K/2 and that the time to reach N^ is equal to:

t^* = ln ( K / N_o -1)
______________
             r

Using the linear regression approach that I mentioned in class and the following data from Gause (1934), estimate the parameters of a logistic model of population growth. Plot the data along with your estimated curve. Explain why it seems to fit well at some points but not at others. How did you choose K? Did you use all the data? Do you understand how the fit has resulted in the “oscillations around K” toward the end?

1.  Use POPULUS to consider Lotka-Volterra competition between two organisms limited by the same resource. You might begin by running the model with the default parameters just to get a feel for changing the conditions.

a. In class I gave you an example of two species that coexist at equilibrium in the long run. Start with these conditions:

Species 1: N₀ = 2, r₁ = 0.039, K₁ = 20, α₁₂ = 0.50

Species 2: N₀ = 2, r₂ = 0.030, K₂ = 30, α₂₁ = β = 0.91

Run POPULUS and study the dynamics.  What are the population sizes of the two species at equilibrium? How long does it take the populations to reach the equilibrium? Is this affected by the values for r?

b. Let's compare the standard conditions above with a situation where species 2 always wins the competition and species 1 always goes to extinction.   Based on the graphical isoclines from above, now change the parameters for species 2 so that species 2 always wins.  Explain the change in analytical terms and interpret these conditions ecologically.

c.  Now consider the case where the elements of competition influence the outcome based on the stochastic element of the initial populations of the two species.   Calculate parameter conditions for this case by altering the isocline for either species. Show output that demonstrates that either species may win depending on the initial population mixture.   What ecological conditions might change for one or the other species that would influence the population parameters and the relative competitive ability of a species

2.  Use POPULUS to simulate the conditions described in Gotelli's Problem 1 (page 136). What value would be a reasonable r value for scorpions? Does the value of r influence the outcome?

3.  Use POPULUS to simulate the conditions of Gotelli's Problem 2 (page 136).