Contents: (Click on a link to go to that part of the page)

• General Information on the Computer Assignments
• Assignment 1: The Population Growth Computer Assignment
• Assignment 2: The Lotka Volterra Competition Model Assignment

General Information on the Computer Assignments: You will complete two assignments in which you explore ecological models by making graphs of computer-generated data.  The computer programs you will use are available on the web (links are given below.) For the assigned exercises described below, you will be required to:

• make the assigned graphs ON GRAPH PAPER.  The computer will give you a large number of points.  DO NOT plot every point!  Generally, you will need to plot about every 10th point; choose points to show the shape of the curve (for example, be sure to include largest and smallest values and a selection in between.)
• clearly label each graph with ALL THE NUMBERS you plugged into the model to obtain the graph.

In some cases, you will also be required to answer questions about the models.  These will be brief (fill-in or multiple choice.)

For both assignments, you are encouraged to work in groups.  The maximum group size is six. If you do work in a group, hand in JUST ONE group project with names of group members on it.  Each group member must have primary responsibility for at least one graph; on each graph, put the name of the person responsible for it FIRST, then put the names of the rest of the group members. If there are assigned questions, each group member must be responsible for, and put his/her name by, one question. If you donít work in a group, youíre responsible for doing the whole thing yourself.  Whether or not you work in a group, you are responsible for knowing all the information (not just what you did!)  If you work in a group, you should get together as a group and work on it (don't just assign different parts to different people and plan to meet up again) because sometimes later questions depend on answers to earlier questions so you will need to know what other group members are finding out.  Note that all members of a group will receive the same grade; it is the responsibility of all group members to check what others are doing and make sure you all agree on the answer. EACH GROUP MEMBER SHOULD KEEP A COPY OF THE ASSIGNMENT.

Assignment 1: The Population Growth Computer Model Assignment

The goal of this assignment is to explore the population growth models you have learned about in lecture, exponential and logistic growth, to make sure you understand the effects of the different parameters of the model.  To complete the exercise,  run computer models of population growth.

CLICK HERE to access the population growth computer model.

EXERCISE I: Compare the logistic and exponential growth models.

Required graphs: (1) N vs t plotted for both models on the same graph, (2) dN/dt vs N plotted for both models on the same graph, (3) dN/Ndt versus N plotted for both models on the same graph.  For all graphs, use an initial population size of 2 and an r value of 0.3.  Run the model for enough generations to show the shape of each graph for logistic growth; note that within his time the population size for exponential growth is likely to get very large so you will probably not be able to fit all the points for exponential growth on the graph -- that's fine, your goal is to make graphs that clearly show the shapes for both models.

AS YOU STUDY THESE GRAPHS, BE SURE YOU CAN EXPLAIN THE FOLLOWING: How does the pattern of population growth differ between exponential and logistic growth?  Why?  At what population size is the population growth rate maximum for logistic growth?  Does it have the same maximum for exponential growth?  Explain the difference.  At what population size is per capita growth maximum for logistic growth?  At what population size is it zero?  What causes the difference in the pattern of per capita growth between logistic and exponential growth?

EXERCISE II: Time lags. There are three possible results if there is a time lag in population response: (1) Damped oscillations.  The population size will cycle, but the amplitude of the cycling will decrease and eventually the population will stop cycling. (2) Stable oscillations.  The population size will cycle, and the amplitude will stay the same so the population cycles indefinitely.  (3) Unstable oscillations that increase in magnitude until the population goes extinct.

MAKE THE FOLLOWING TWO REQUIRED GRAPHS:
 A.  Using the values K=150,  r_m=0.4,  and initial population size=2 make plots of population size versus time for situations that vary in the length of the time lag. Find a time lag that results in damped oscillations, a time lag that results in stable oscillations, and a time lag the results in unstable oscillations (or at least one unstable oscillation followed by extinction.) Make a graph of each situation.

 B. Using the values K=150, initial population size=2, and time lag t = 2, make plots of population size versus time for situations that vary in r_m.  Choose r_m values less than or equal to 1 to be realistic.  Find a value of r_m that result in damped oscillations,  a value of r_m that results in stable oscillations, and a value of r_m that results in unstable oscillations (or at least one unstable oscillation followed by extinction.) Make a graph of each situation.

BE SURE YOU CAN EXPLAIN:
How does increasing the time lag affect the size of population cycles?  Why? How does it affect the stability of the cycles? Why? How does increasing the rm values affect the size of population cycles? Why? How does it affect the stability of the cycles? Why?

EXERCISE III: Non-linear response of per capita growth rate to population size.
MAKE THE FOLLOWING THREE REQUIRED GRAPHS:
 A. On the same axes, plot population size versus time for three values of q:  a value less than 1, a value equal to 1, and a value greater than 1.  Choose r_m=0.4, K=150, and initial population size of 2; there should be NO time lag (t equals 0).

 B. On the same axes, plot population growth rate versus population size for three values of q:  a value less than 1, a value equal to 1, and a value greater than 1.  Choose r_m=0.4, K=150, and initial population size of 2; there should be NO time lag (t equals 0).

 C. On the same axes, plot per capita population growth rate versus population size for three values of q:  a value less than 1, a value equal to 1, and a value greater than 1.  Choose r_m=0.4, K=150, and initial population size of 2; there should be NO time lag (t equals 0).

MAKE SURE YOU CAN EXPLAIN:   How does the value of theta affect the response of per capita population growth to population density?  How does the value of theta affect the population size at which population growth is maximum?  In what situations might we expect to see values of theta larger than 1?  Smaller than 1?

EXERCISE IV:

1. You are managing a population of largemouth bass; your goal is to allow a level of fishing on this bass population that will maintain population growth (dN/dt) at a maximum level.  The carrying capacity of the area for your population is 130 bass.  The maximum possible value of r is found to be 0.2 per year.

A. Assume that per capita bass population growth responds linearly to population size.
REQUIRED GRAPH: Make the plot that will most clearly show the population size at which population growth rate is maximum. Put a star (*) on the graph to denote the population size at which population growth rate is maximum.  Next to the graph state how many bass should be allowed to be taken from a bass population at its carrying capacity (you can hand write this on the graph)

B. Now suppose you get more data on bass per capita population growth and find that it responds non-linearly to population size; there is very little effect of population size on per capita growth for small populations, but for large populations there is a strong decrease in per capita population growth.  REQUIRED GRAPH: Select the appropriate model for the situation described.  Make the plot that will most clearly show the population size at which population growth rate is maximum. Put a star (*) on the graph to denote the population size at which population growth rate is maximum.  Next to the graph state how many bass should be allowed to be taken from a bass population at its carrying capacity (you can hand write this on the graph)

2.  A species of vole (small rodent) occurs in populations in the far north and also in southern populations.  Northern populations undergo population cycles with a peak in vole density every four years.  Southern populations remain at a constant population size of 70 per hectare.  It is estimated that the resources in the north could also sustain a constant population of 70 per hectare if the populations didnít cycle (but they do).  Both populations have the same potential for reproduction and the same life expectency, so their maximum possible per capita growth rate will
 (a) Determine the appropriate model and numbers to model a cycling northern vole population in a one hectare plot.  REQUIRED GRAPH: Make a plot of population size over time for this population.
 (b) Determine the appropriate model and numbers to model the southern vole population in a one hectare plot.  REQUIRED GRAPH: Make a plot of population size over time for this population.

The Lotka-Volterra Competition Model Assignment:

General instructions: Answer the following questions with the help of the Lotka-Volterra Competition Model web page.

CLICK HERE to access to Lotka-Volterra Competition Model web page!

You will be required to identify values for the parameters of the Lotka-Volterra competition models and to draw graphs.  The graphs you will need to draw include:

a. An isocline/trajectory plot.  Make a graph of species 1 (N₁) versus species 2 (N₂).  On it, plot the zero-growth isoclines for each competing species on a plot of species 1 (N₁) versus species 2 (N₂).  CLEARLY LABEL EACH ISOCLINE as the N₁ or N₂ isocline.  Then draw an additional line to show the trajectory of N₁ and N₂ over time.  A trajectory line refers to the combinations of numbers of species 1 and species 2 through which the populations pass up to the point at which an equilibrium, either competitive exclusion or coexistence, is reached.

b. graphs of population size for each species through time.  Plot numbers of both species on the same set of axes (so you will have two lines on this kind of graph, one for species 1 versus time and one for species 2 versus time.)  CLEARLY LABEL each line as representing species 1 or species 2.

For lines representing trajectories and population sizes versus time, DO NOT plot every point given to you by the computer.  Plot about every 10th point.  Be sure to include points to identify starting and equilibrium situations. This will show a reasonably accurate curve and take much less time than plotting every point!

The assignment:

1.  Determine values of K and alpha for the two species that result in each of the following possible outcomes (these are the four possible outcomes of competition according to the Lotka-Volterra models):

a. species 1 always competitively excludes species 2
b. species 2 always competitively excludes species 1
c. one species will always competitively exclude the other, but the "winner" of competition cannot be determined just from the K and a values
d. species 1 and species 2 will coexist

Note that you should be able to determine these BEFORE running the model by the relationships of K and K/a values you have learned in lecture.

For each set of values, make an isocline/trajectory graph AND a graph of population size versus time for both species.  To do this, use initial populations sizes of 2 for both species; let r₁=0.4 and r₂=0.9.  For situation "c" above (where the "winner" cannot be determined from just the K and a values) there is a small chance that the trajectory will lead to the unstable equilibrium point with the two species coexisting -- this is mathematically possible if you put in exactly the right combination of numbers but ecologically impossible (real world random fluctuations in population size will prevent it from ever occurring).  If this happens, change r₁ to 0.5 and try again.  You should get a different outcome (if you don't, see me for help.)

BE SURE to WRITE ALL information you typed into the computer to obtain each graph (all r, N, K, a, and time values) next to that graph!  BE SURE to IDENTIFY each graph as corresponding to situation 1a, 1b, 1c, or 1d.

2.  Consider the information from situation c above.  Use the same values of initial population sizes, K, alpha, and r₂.  Change the value of r₁ until you obtain a situation in which the species that initially was the "winner" of competition becomes the "loser."  Make isocline/trajectory and population size verus time plots for this situation. BE SURE to WRITE ALL information you typed into the computer to obtain each graph (all r, N, K, a, and time values) next to that graph!  Be sure to identify the graph as graph 2.

3.  Again, consider situation 1c.  Use the r values you used for question 1.  Change values of N₁ or N₂ until you obtain a situation in which the species that initially was the "winner" of competition becomes the "loser."  Make isocline/trajectory and population size verus time plots for this situation. BE SURE to WRITE ALL information you typed into the computer to obtain each graph (all r, N, K, a, and time values) next to that graph! Be sure to identify the graph as graph 3.

4.  Based on the changes you made throughout the exercise, determine the correct answer to each of the following questions and circle the correct answer on the answer sheet obtained by clicking here.  If you are not sure of the answer, run some more models to test out the possibilities until you figure out what the correct pattern is.  If you're working in a group, each group member must take primary responsibility for at least one question (but remember you all have to know all of the answers and are all responsible for the values filled in on the sheet.)

(a) (i) Is coexistence more likely when alpha values are both large (i.e. both greater than 1), both small (i.e. both less than 1), or when one of them is large and the other is small?
    (ii) Do the equilibrium numbers of species 1 and species 2 depend on the initial population sizes or r values for species 1 and species 2?

(b) In the situations in which one species always competitively excludes the other, and the same species will always "win" regardless of initial population sizes or r values,
    (i) is the "winner" more likely to have a higher or lower carrying capacity?
   (ii) is the "winner" more likely to have a higher or lower alpha value (in the growth equation for the "winning" species.)

(c) In the situation for which one species always competitively excludes the other, but the "winner" cannot be determined from just the K and a values,
   (i) is the "winner" more likely to have a higher or a lower r value?
  (ii) is the "winner" more likely to have a higher or lower initial population size?