Competition Models
Today's exercise uses the program POPULUS to refresh and extend your understanding of the basic deterministic model of
competition, the Lotka-Volterra model. We're going to be moving on to spatial models, but it is good to have these results under your belt.
Lotka-Volterra Competition
At the main menu choose "Multi-Species Interactions". On the multi-species
interactions menu choose "Lotka-Volterra Competition". This will get you
into the model that we will use this week. Take the time to read the introductory
material to re-familiarize yourself with the concepts associated with competition
models. If the material is not familiar to you from BIO113, ask the instructor, and/or read the help documentation included with populus, and your text book.
Exercise #1 - Basics - Competition Coefficients
The Lotka-Volterra model of competition is a simple extension of the logistic
model of intraspecific competition and population growth, with the added
complications of equations for two (or more) species, and the addition
of interspecific competition coefficients. In the two species case Populus
designates the competition coefficients as a and
b, where a is the
effect of species 2 on species 1, and b is the
effect of species 1 on species 2. Note that in the book the equivalent
terminology for a is a 12
and for b is a 21.
The remaining terminology (r, K, N) is the same as in the logistic model.
To help you understand the concept of competition coefficients, conduct
a series of runs in which you vary the competition coefficients. Begin
by setting the parameters for both species to identical values (N = 10,
K = 500, r = .5). Now set a = 0
and b = 0 and begin the simulation by pressing
the enter key. Repeat the simulation for values of a
= b = .5, 1.0, and 2. In each case, view
the results as density versus time. What effect does the increase in
the competition coefficient have upon equilibrium density? The competition
coefficients can be interpreted as the effect of a second species on the
first relative to the effect of the first upon itself; does this interpretation
make sense in light of the results of these runs?
Exercise #2 - Basics - Isoclines
Reset the parameters of the model as follows: for species 1 N = 10, r =
.5, K = 500, and a = 0, and for
species 2 N = 10, r = 1, K = 500, and b = 0.
Now, in a series of runs, allow a to take on
values of 0, .5, 1, 2, and 10. For each run examine the isocline plot (N1
vs. N2), and notice how the position of the species 1 isocline
varies. Does the intercept on the N1 axis vary as a
varies? How does the position of the intercept on the N2
vary as a varies?
Reset the parameters of the model so that a
= .5 and b = .5. Rerun the model for
a series of runs in which you vary the initial number of individuals for
each species. Start some near to the origin and others at or above the
carrying capacity of one or both species. For each run examine the isocline
graph, paying particular attention to the population trajectory line relative
to the isoclines. How does the density of species 1 change when the
trajectory is above (or to the right of) the species 1 isocline? What is
the relationship between the trajectory and the species 2 isocline? What
do these relationships tell you about the nature of isoclines?
Exercise #3 - Competition coefficients and the coexistence of species
Reset the parameters of the model as follows: for species 1, N = 10, r
= .5, K = 500, and a = 0, and
for species 2, N = 10, r = 1, K = 500, and b =
0. Now, in a series of runs, set a = b
= 0, .5, .9, 1.1, 2, and 10. Do the species coexist under all conditions?
If not, what are the conditions which lead to competitive exclusion?
Do another series of runs in which a
is not equal to b. Let the values range
between 0 and 10 or so. For each run, write down the values of a
and b and then note whether the species
coexisted. What are the conditions which lead to competitive exclusion
versus coexistence?
Exercise #4 - Importance of r for competition
Reset the parameters of the model as follows: for species 1, N = 10, r
= .5, K = 500, and a = .5, and
for species 2, N = 10, r = 1, K = 500, and b =
.5. Now determine the effect of r on the success of a species in competition.
Vary the value of r for species 1 from low values (ca. .1) to high values
(ca. 10). Does the value of r affect the trajectory of densities? Does
the value of r affect the outcome of competition?
Change the parameters of the model so that a
= b = 2, and repeat the last experiment.
Does the value of r affect the trajectory of densities? Does the value
of r affect the outcome of competition?
Exercise #5 - Importance of initial conditions
Reset the parameters of the model as follows: for species 1, N = 10, r
= .5, K = 500, and a = .5, and
for species 2, N = 10, r = 1, K = 500, and b =
.5. Now determine the effect of initial densities on the outcome of competition.
Vary the values of N for each species between 1 and K, sometimes giving
species 1 an advantage, other times giving species 2 an advantage. Do
the initial densities affect the outcome of competition?
Change the parameters of the model so that a
= b = 2, and repeat the last experiment.
Do the initial densities affect the outcome of competition?
General Questions
What biological conditions would lead to high values of competition
coefficients?
What biological conditions would lead to competition in which the
outcome of competition in indeterminant?
Under what conditions will intrinsic growth rate of a population
affect the outcome of competition?
Thanks to John Addicott at the University of Alberta!