http://animaldiversity.ummz.umich.edu/accounts/odocoileus/o._virginianus$media.html
Your
first step was to realize the inadequacy of a basic Lotka-Volterra
Predator-Prey model for a description of the interaction because:
dV/dt
= aV - bNV
and
dN/dt
= cNV - dN
where:
V =
number of milkweed ramets,
N =
number of deer,
a =
the intrinsic rate of increase of milkweed ramets,
b =
the feeding rate of deer and milkweed defensive response to feeding,
c =
the numerical response of deer (efficiency with which deer turn food into
progeny), and
d =
deer death rate.
which
means that milkweed ramets would increase without limit in the absence of deer.
This
is unreasonable and so you decide to add a logistic term, (K-V)/K, that limits your ramet population to a carrying
capacity, K, through
intraspecific (& intragenet) competition. This means that,
dV/dt
= (aV(K - V)/K) - bNV
and
dN/dt
= cNV - dN
Model
these continuous,
differential equations using the POPULUS Interaction Engine (under the "Multispecies
Interactions" category) and
set the plot to N vs T, t
= 100 and do not plot isoclines.
Enter the equations to build the model (delete N3 to N9 (N3 won't delete but set it to 0) because the engine
can handle up to 9 interacting species and you are only interested in 2
species) and set the starting parameters to:
V = 200
N = 20 (in the upper box)
and
in the lower box,
a = 0.2,
d = 0.3,
K = 1000,
c = 0.001, and
b = 0.001.
Run
the model (press <enter>) and turn on the gridding function with
<alt-G> (twice).
Now
that you have a functioning resource-limited, plant-herbivore model examine how
changes in a, b, and c
influence the stability and final population sizes of milkweed ramets
and deer after 100 time intervals.
To
do this please fill in the following table of milkweed and deer numbers after
100 time intervals and print out the 9 relevant N vs T plots of the interactions that show the data
listed in the table (NB
set the printer model under options, type <alt-O>(and set resolution at
150 dpi), and to print, type <alt-P>. You might also like to press <F4> to set the
"show previous plot" option).
If you need any help press <F1>.
|
Parameter
|
Numbers of ramets V
|
No's of deer N
|
|
a = 0.2
|
|
|
|
a = 0.4
|
|
|
|
a = 1.2
|
|
|
|
With a = 0.5 examine the following:
|
|
|
b = 0.005
|
|
|
|
b = 0.001
|
|
|
|
b = 0.0008
|
|
|
|
With b = 0.001 examine the following:
|
|
|
c = 0.0005
|
|
|
|
c = 0.001
|
|
|
|
c = 0.005
|
|
|
Using
these graph printouts and the table answer these questions (please submit
this table, the 9 printed graphs and a printout of the Interaction Engine model
formulation screen with your answers):
1) What effect does the intrinsic rate of
milkweed ramet increase, a,
have on the equilibrium population of milkweed ramets?
2)
What effect does increasing a have on deer abundance?
3)
What effect does increasing a have on the rate of return to equilibrium?
4)
What effect does b
have on the equilibrium milkweed ramet population?
5)
What effect does increasing b have on deer abundance?
6) What effect does increasing c have on the equilibrium milkweed ramet
population?
7) What effect does increasing c have on the equilibrium deer population?
8) What effect does increasing c have on the stability of milkweed-deer dynamics?
BIOS 615: Ecology
Computer Modelling
Assignment - Bonus point activity
For an additional 20
bonus points try and perform the following activity, once you have completed
your computer modelling assignment, using the Populus interaction engine and answer the questions below:
For 20 bonus
points:
Illustrate Rosenzweig's "paradox
of enrichment" (Rosenzweig,
1971 Science 171: 385-387; Berryman, 1992, Ecology 73(5): 1530-1535) by setting a = 0.5, b = 0.001, c = 0.001, and d = 0.3, and increase K from 500, to 1000, to 4000 to see the paradox.
a) Complete a table of milkweed ramet and deer
numbers for these different K values after 100 time intervals as you did above and also submit
the 3 printed graphs.
|
Parameter
|
Numbers of ramets V
|
Numbers of deer N
|
|
K = 500
|
|
|
|
K = 1000
|
|
|
|
K = 4000
|
|
|
b) What happens to the equilibrium ramet density
with increasing K and
why is it paradoxical?
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Biological Sciences
