Lab 5: Modeling Competition Using the Lotka-Volterra Equations
Goals
 Understand how various factors influence competition and, ultimately, fish community structure
 Apply the model to exotic species scenarios in Wisconsin
Giving Credit Where It’s Due
The COMPETE computer simulation and large chunks of this lab exercise were developed by Robert P.
Gordon, Indiana University of Pennsylvania. Wisconsin-specific fish adaptations were implemented by your
friendly neighborhood TAs.
Introduction – Why Model?
Mathematical models play an important role in modern ecology. You became familiar with one such model
in Bioenergetics3.0 last week. Ecologists use models because the systems they study are often so complex
that they defy easy analysis. Models simplify reality, allowing us to isolate and study the effect of one or two
variables at a time. Although it is important to keep in mind that most models are inherently "unrealistic"
they can yield valuable insight into important ecological processes. A good illustration of this is the LotkaVolterra model described below. Although this model is quite simple it continues to be very influential in
stimulating research on interspecific competition.
Exercise One – Intraspecific Competition
Intraspecific competition (competition between organisms of the same species) can occur through
interference (direct, behavioral competition), exploitation (resources are limited), or apparent (predation).
Intraspecific competition was actually already introduced to you during week three under the guise of
“Density Dependence”. In other words, intraspecific competition occurs when each individual added to the
population tends to inhibit further population growth.
Here is the simple logistic model for intraspecific competition:
∆N=net growth rate
N=Population density
∆N = rN (K-N)/K
r=intrinsic growth rate (growth rate w/o density dependence)
K= carrying capacity
Note the “(K-N)/K” portion of the equation. This chunk of the equation reflects the intraspecific competition
factor: as the population size (N) approaches the carrying capacity (K) the term (K-N)/K, and thus the growth
rate, approaches zero.
Within the simulation, you will be able to manipulate the following parameters:
Number of time intervals
Starting population size (No),
Carrying capacity (Max population size) (K),
Average number of offspring per generation (Ro),
Generation time (time from birth to sexual maturity) (T),
Competition coefficient (alpha, ).
The intrinsic growth rate (r) is determined from Ro and T.
The simulation plots the growth curve for two species simultaneously. Since we are not concerned with
interspecific competition in this exercise, set the competition coefficients for both species to 0 (No
competition between A and B). For right now, don’t worry about understanding competition coefficients…that
comes in part 2. View species A and B as two different populations of the same organism to reference
against one another.
1.) a.) According to a recent DNR survey, walleye abundances in WI lakes range from 0.5 to 12.5 fish/acre,
with most lakes containing 1.5 to 4.5 fish per acre. You are the manager of 2 popular walleye lakes in
northern Wisconsin, Lake Muir and Lake Aldo (species 1 and 2 in the simulation). These two lakes are
almost identical in every way, and they contain the same strain of walleye. These walleye are
characterized by the following parameters:
Population = 2fish/acre K=13 fish/acre Ave. # offspring reaching maturity = 4fish/acre/generation
Generation time=24 months
One day, a walleye sport-fishing club calls you to complain that there aren’t enough walleyes in Lake
Muir. They thinks you should introduce more to increase the population. You decide to look into it at get
back to them.
P o p .
S i z e
Simulate a stocking event for Lake Muir by increasing the initial population size (No) for Lake Muir. Start
with an increase of 2 to 5 fish/acre, but feel free to try other reasonable values (up to carrying capacity).
Graph the results of the simulation.
T im e
Does this change the final population size?
Run the simulation for varying amounts of time (units in months). How many months does it take for
Lake Muir to reach carrying capacity?
For Lake Aldo?
All other things being equal, does the population with the higher initial size always reach its carrying
capacity sooner?
P o p .
S i z e
You decided that stocking was too expensive for the expected payoff (return starting densities to 2
fish/acre). However, a recent study has been published documenting pristine habitat for young-of-year
(YOY) walleye in Lake Aldo. As a result, walleyes in Lake Aldo have an increased number of
offspring/generation (R increases from 4 to 7). Conversely, Lake Muir has recently been invaded by the
spiny water flea, which has reduced native zooplankton densities, reducing forage for YOY walleye (R
decreases from 4 to 2) Graph the results…is there any change the shape of the growth curve?
T im e
Is final population size affected?
S i z e
You want to examine the benefits of stocking Lake Muir now that growth rate has been diminished.
Increase the starting population size in Lake Muir from 2 to 6 fish per acre (keep the changes to growth
rate). Graph the results.
P o p .
b.
T im e
Comment on the effects of growth rate and starting population size
S i z e
After a year or two, it is discovered that the generation time (T) for walleye in Lake Aldo has increased
from 24 months to 48 months. Increased abundance of walleye has led to stunted growth and intense
intraspecific competition for spawning habitat: only older, larger fish reproduce. How does increasing the
generation time (T) change the shape of the growth curve? (Hint: you will have to increase the duration
of the simulation to see the final outcome.)
P o p .
c.
T im e
Is final population size affected?
d. Which of the four population parameters (K, T, R, No) affect the final population size in these single
species simulations?
e. Step away from COMPETE for the moment and consider this real-life management example. In the
lower Milwaukee River DNR managers have recently constructed a shoal at a site just south of a
former dam location. This artificial reef represents an effort to restore spawning habitat that was lost
to walleyes when the dam was originally constructed. What effect might this effort have on walleye in
the Milwaukee River if successful?
Exercise 2: Effect of Interspecific Competition on Population Growth
Thus far we have used a logistic equation to describe intraspecific competition (competition within a
population). Lotka and Volterra modified this equation to incorporate the effect of competition between
species. Their model is based on the assumption that individuals of species 2 inhibit the growth of species 1
and vice versa. The competition coefficient, 12, is the relative inhibitory effect of species 2 on species 1.
For example, if ten individuals of species 2 inhibit species 1 to the same extent as a single member of
species 1 then 12 = 0.1. That is, the inhibitory effect of species 2 on species 1 is 1/10 that of species 1 on
itself. Although  is often less than 1.0 this does not have to be the case. For example, if individuals of
species 2 consume twice as much of a resource as individuals of species 1 then 12 might be close to 2.0.
Summary of competition coefficients: If <1, intraspecific competition is more limiting than interspecific
competition. If >1, interspecific competition is more limiting than intraspecific competition.
In the Lotka-Volterra model the inhibitory effect of species 2
dN1
(K1
 r1N1
dt

(12N2) is added to that of species 1:
(N1  12N2 ))
.
K1
Note how this equation is the SAME as the logistic equation, except that K-N is replaced by K-(N1 + α12N2).
Think of the term α12N2 as the equivalent number of N1 organisms in the system.
Similarly, members of species 1 inhibit the growth of species 2:
dN2
(K2
 r2N2
dt

(N2  21N1 ))
,
K2
where 21 is the inhibitory effect of species 1 on species 2.
There are three possible outcomes to interspecific competition: species 1 could go extinct, species 2 could
go extinct, or both species could coexist. One way to predict the outcome of competition is to solve the two
equations simultaneously and iteratively (i.e., repeatedly). For each time interval the size of each population
is calculated. These new values of N1 and N2 are then be substituted into the two equations and the
process repeated. This approach would be incredibly laborious if done by hand, but is rapidly accomplished
with the computer simulation you are using.
Throughout the exercises on the following pages keep in mind that you are dealing with an abstract, and in
many ways unrealistic, model. The usefulness of such models is usually judged on the basis of whether
they provide insight into a particular phenomenon, and whether they stimulate further inquiry. By these
criteria the Lotka-Volterra competition model is one of the most successful in ecology.
EXERCISE 2: Interspecific Competition
Back in the days of yore, Lake Michigan included a set of planktivorous ciscoes, often thought of as a
species flock (diverse group of closely related species, similar to darwin’s finches). Consider two such
species, the shortnose cisco and the bloater. Both species feed on zooplankton, crustaceans, and mollusks.
As a result, they compete for resources to some extent. The bloater, however, is common at depths greater
than 80m while the shortnose cisco is more abundant in waters less than 80m.
Do you think the competition coefficients 21 and 12 should be >1 or <1? Why?
In 1900, shortnose cisco is found at a density of 200 fish per acre while bloater are found at a density of 100
fish per acre. The carrying capacity for both fish is 1000 fish per acre. Both species produce 50
fish/individual/generation, with a generation time of 18 months. 21=12=0.4
a.
Explain what a competition coefficient, 21, of 0.4 means.
b.
Run the simulation. What is the outcome, competitive exclusion or coexistence? Did either species
reach its carrying capacity? You must include the population trajectory (the white line) in the left graph.
Be sure you understand what this trajectory shows.
N
N2
N1
T im e
c.
Can you change the final outcome (i.e., final population sizes) of a stable equilibrium by varying No 1
and No2? Show at least two examples to support your conclusion. You can also use some of the extra
graphs in the back of this chapter.
N
N2
T im e
N1
N
N2
T im e
N1
d.
Reset the parameters to their original values. Several other cisco species exist in the lake (kiyi,
deepwater, and longjaw). These three species have share the bloaters values for N, R, T, and alpha.
They all have different K values, however. Can you change the final outcome, e.g. from coexistence to
competitive exclusion, by varying the carrying capacity of the bloater’s competitor species? (Try a wide
range of values for K1 to make sure that you have explored all possibilities.)
N
N2
T im e
N1
N
N2
N1
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f. Forget about bloaters for the time being. Use the model to simulate an unstable equilibrium. (This will
occur when K1/12 < K2 and K2/21 < K1.) Set both population sizes to 10 fish per acre. Let
K1=100 and K2=250. Let alpha12=0.5 and alpha21=3.0. Number of offspring =5 and generation
time = 18 for both species. Here we have a situation where there is a highly competitive species with
a low carrying capacity against a low competitive species with a high carrying capacity.
Do the isoclines cross?
Do the species coexist?
Can you change the final outcome of an unstable equilibrium by varying No 1 and No2? Graph your results
N
N2
T im e
N1
N
N2
N1
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g.) Ok, back to our ciscoes. The year is 1949: Alewives are introduced into Lake Michigan. Alewives are
voracious zooplanktivores and enter into a system where sea lamprey has decimated the only top predator
(Lake Trout). Use the simulation to model the fall of the shortnose cisco. Assume the fish has reached
carrying capacity: population and K both = 1000 fish per acre, with generation time equal to 18 months and
intrinsic growth rate at 50fish/acre/generation. The alewife has just entered the system: Population = 1
fish/acre. Since alewives are more opportunistic foragers, they can occupy many niche spaces and have a
correspondingly higher K value of 2000 fish/acre. They only reproduce at a rate of 20fish/acre/generation,
with generation time of 18 months, but are more efficient feeders, with a competition coefficient of 1.5 on
shortnose cisco. The cisco, on the other hand, only affect the alewife to a competition coefficient of 0.1.
Graph the results of the simulation over 8 years (96 months), and explain what happened!
N
N2
N1
T im e
Thought exercises: Put away COMPETE
1.) Recent publications document that the lone surviving species of native cisco in Lake Michigan, the
bloater, underwent a morphological shift (reduction in gill raker number and length) and a behavioral
shift (benthic foraging) since the invasion of the alewife. It is theorized that selective pressure drove
the bloater to become a benthic forager. Which parameters will be affected by this development?
2.) Asian carp (silver and big-head carp) are not presently in Wisconsin, but they are the most talkedabout invasive currently threatening to invade the state’s waters (both Lake Michigan and the
Mississippi River/Wisconsin River are threatened). Asian carp have been found in the Illinois River,
which connects the Mississippi River to Lake Michigan. Due to their large size (no predators) and
rapid rate of reproduction, these fish pose a significant risk to the Great Lakes Ecosystem. The Asian
carp are characterized as super efficient plankton feeders: they filter suspended food particles from
the water current through their gills (both phyto and zooplankton). This style of feeding is similar to
the paddlefish, but carp are much more efficient. Using what you have learned, answer the following.
a. Which species of fish are most likely to suffer? How would Asian carp compete with these
species?
b. Paint a picture of what Lake Michigan could look like if Asian Carp made it into the system.
ECOLOGICAL MODELING
a.
Discuss at least one way that the Lotka-Volterra model is unrealistic.
b.
Suggest some reasons why the model has been so popular among ecologists.
EXTRA GRAPHS
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N2
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N1
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N2
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N1
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