This
assignment is due to be handed in to Dr Malcolm by Tuesday, 30 March 2004. It may be handed in at any time before
that deadline.
This is a
computer-based activity using Populus software.
The assignment is
worth 100 points
Assignment:
Some
neighbors are stealing and eating carrots from your vegetable garden and being
a nice person you want to know whether you can afford to let them have a few
carrots or whether you need them all yourself. So you decide to predict your yield of carrots after the
impact of losses to your neighbors.
You also realize that over long periods of time carrots might even
result in an increase in the number of your neighbors. In order to make predictions you decide
to build a simple model.
Your
first step was to take a look at the basic Lotka-Volterra Predator-Prey model
for a description of the interaction because:
dP/dt
= aP - bNP
and
dN/dt
= cNP - dN
where:
P
= number of carrot plants
N
= number of neighbors,
a
= the intrinsic rate of increase of carrots,
b
= the feeding rate of neighbors and a measure of the effects of carrots
on thieving neighbors,
c
= the numerical response of neighbors (efficiency with which neighbors
turn carrots into kids), and
d
= neighbor death rate (if you decide to emigrate and leave them to grow
their own darn carrots).
which
means that carrots would increase without limit in the absence of neighbors.
This
is unreasonable because you have limited space and so you decide to add a logistic
term, (K-P)/K,
that limits your carrot population to a carrying capacity, K, through intraspecific competition. This means that,
dP/dt
= (aP(K - P)/K) - bNP
and
dN/dt
= cNP - 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 for 2 interacting
species (carrots and neighbors) and set the starting parameters to:
P = 200
N = 20
a = 0.2,
d = 0.3,
K = 1000,
c = 0.001, and
b = 0.001.
Run
the model and turn on the gridding function.
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 carrots and neighbors after 100 time
intervals.
To
do this please fill in the following table of carrot and neighbor 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.
|
Parameter |
Numbers of carrots P |
No's of neighbors 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
carrot increase, a,
have on the equilibrium population of carrots?
2)
What effect does increasing a have on neighbor abundance?
3)
What effect does increasing a have on the rate of return to equilibrium?
4)
What effect does b
have on the equilibrium carrot population?
5)
What effect does increasing b have on neighbor abundance?
6) What effect does increasing c have on the equilibrium carrot population?
7) What effect does increasing c have on the equilibrium neighbor
population?
8) What effect does increasing c have on the stability of carrot-neighbor
dynamics?
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 carrot and neighbor
numbers for these different K values after 100 time intervals as you did above and also
submit the 3 printed graphs.
|
Parameter |
Numbers of carrots P |
Numbers of neighbors N |
|
K = 500 |
|
|
|
K = 1000 |
|
|
|
K = 4000 |
|
|
b) What happens to the equilibrium carrot
density with increasing K
and why is it paradoxical?