This
is a computerbased activity using Populus software (P), followed by
EcoBeaker analyses of moose populations on Isle
Royale and the impact of wolves.
Introduction:
Some whitetailed deer are eating
lettuces in your vegetable garden and you want to understand the interaction
before you decide what to do to.
Your first step is to build a basic LotkaVolterra PredatorPrey model for a description of the
interaction using Populus:
dN/dt = r_{1}N
 CNP
and
dP/dt = gCNP – d_{2}P
where:
N = number of lettuce plants (prey),
P = number
of deer (predators),
r_{1} = the intrinsic rate of increase of lettuce
(prey),
C = the
feeding rate of deer and CN is the per capita functional
response of deer to lettuce density,
g =
the efficiency with which deer turn food
into progeny, and gCN is the numerical response
of deer (rate of new deer born as a function of prey density), and,
d_{2}_{ } = deer death rate.
which means that
lettuce would increase without limit in the absence of deer.
This is the first of the ÒContinuous
PredatorPrey modelsÓ in Populus. Starting with the default parameters
indicated below please answer the following questions:
Predator: P_{o}
= 20, d_{2}
= 0.6, g
= 0.5
Prey: N_{o}
= 20 r_{1}
= 0.9 C
= 0.1
10. Is the model
any use for solving your problem?
Given that your answer to 10 is likely
to be negative you decide to add more realism to the model and limit the
lettuce population with intraspecific competition (you sowed the seeds too
densely and foolishly forgot to thin the seedlings as your Granny always told
you to do!).
So now you add densitydependent
limitation of lettuces with logistic limitation to a carrying capacity as in
the Populus (theta) θ–logistic
predatorprey model in which:
dN/dt = r_{1}N(1(N/K))
 CNP
This means that in the absence of
predators (P = 0) the lettuces are limited only by intraspecific competition
where (1(N/K)) is the same as (KN)/K.
You now have the 2 extra parameters of K
(the carrying capacity) and θ. Set K to 100 and keep θ
at 1 (this basically ignores the densitydependent feedback provided by θ). All other parameters should be the same
as above. This model also includes
the predatorÕs functional response (numbers of prey eaten at different prey
densities). Set the model to the
following:
Predator: P_{o}
= 20, D
= 0.6, g
= 0.6
Prey: N_{o}
= 20 r_{1}
= 0.9 K
= 100 θ = 1
with a type 2
functional response (C = 0.05 and h =1) and answer the following questions:
Now change D
to 0.7:
Now change functional
response to type 3 (c = 0.05, h = 1):
10. Is the new
interaction stable or unstable?
Lastly, you decide to simplify the
theta logistic model slightly and build your own logistically limited model of
the interaction. So you go to the
Interaction Engine in Populus and add your own
logistic term to the LotkaVolterra mass action
predator–prey model.
Using the Interaction Engine build the
following model of the interaction:
dN_{1}/dt = (r*N_{1}*(K – N_{1})/K)
– f*N_{2}*N_{1}
and
dN_{2}/dt = (c*N_{2}*N_{1})(m*N_{2})
where:
N_{1} =
number
of lettuce plants (prey),
N_{2}
= number
of deer (predators),
r =
the
intrinsic rate of increase of lettuce (prey),
K = carrying
capacity of lettuce
f = the
feeding rate of deer,
c = the
numerical response of deer (efficiency with which deer turn food into progeny),
and,
m = deer
death rate.
Model these continuous, differential
equations using the POPULUS Interaction Engine and set the plot
to N
vs T, t = 100 and do not plot isoclines. Enter the equations to build the model
(uncheck N3 because you are only
interested in 2 species) and set the starting parameters to:
N_{1} = 200
N_{2} = 20
r = 0.2
m = 0.3
K = 1000
c = 0.001
f = 0.001
Run the model and turn on the gridding function .
Now that you have a functioning,
resourcelimited, plantherbivore model examine how changes in r, f,
and c influence the stability and final
population sizes of lettuce and deer after 100 time intervals.
To do this please fill in the following
table of lettuce and deer numbers after 100 time intervals :
Parameter 
Numbers of lettuce N_{1} 
No's of deer N_{2} 
r = 0.2 


r = 0.4 


r = 1.2 


With r = 0.5 examine the following: 


f =
0.005 


f =
0.001 


f =
0.0008 


With f = 0.001 examine the following: 


c =
0.0005 


c =
0.001 


c =
0.005 


Using this
table and your screen answer these questions :
1) What effect does the intrinsic rate of lettuce increase, r,
have on the equilibrium population of lettuce plants?
2) What effect
does increasing r have on deer abundance?
3) What effect
does increasing r have on the rate of return to equilibrium?
4) What effect
does f
have on the equilibrium lettuce population?
5) What effect
does increasing f have on deer abundance?
6) What effect does increasing c have on the equilibrium
lettuce
population?
7) What effect does increasing c have on the equilibrium
deer population?
8) What effect does increasing c have on the stability
of lettucedeer dynamics?
In
your lab report on the use of Populus to examine LotkaVolterra
predation/herbivory dynamics please include the
following information:
Introduction:
Give background
information on the ecological processes of predation and herbivory.
Consider functional responses and
the properties of the LotkaVolterra predation model.
Talk
about our goal for the lab of modeling an interaction between plants and
herbivores (white tailed deer and lettuce) and learning about the properties of
the LotkaVolterra model.
Methods:
State
the software used and the models and model parameters used. State which
equation parameters we varied to model the interaction.
Results:
Give
the answers to the questions listed above either in paragraphs or in question
and answer format.
Discussion:
State
what was learned from the modeling. What properties were
shown by the LotkaVolterra model? Which
parameters seemed most important for determining the stability of the
plantherbivore interaction? What are the uses and limitations of the different
models? Is modeling valuable?
Isle
Royale and the dynamics of moose and wolf populations: an example of logistic
population limitation, density dependence and the impact of a predator.
Work through
the ÒEcoBeakerÓ
student workbook for ÒIsle RoyaleÓ and while you do this keep in mind that both
resource
availability, as well as both intraspecific competition and predation
impact moose populations on a closed island in Lake Superior.
More
information about EcoBeaker
software is available at the SimBiotic Software
website at:
EcoBeakerª: Isle Royale (101)
This popular laboratory
explores basic population biology concepts including exponential and logistic
growth and carrying capacity. It is based on the textbook example of a
predatorprey system involving wolves and moose on an island in Lake Superior.
Students start out by characterizing the growth of a colonizing population of
moose in the absence of predators. Next they introduce wolves, and study the
resulting predatorprey cycles. Do predators increase or decrease the health of
their prey populations? Students investigate this question by sampling the
energy stores of moose with and without wolves present. Finally, they try
changing the plant growth rate to see how primary productivity influences
population dynamics.
PRIMARY
CONCEPTS: logistic & exponential population growth; carrying capacity;
predatorprey cycles; energy flow