Relationships between predator and prey can be quite complex. You may recall hearing about the relationship between owls and rodents during your stay at Camp Shamineau this fall. Today you will explore the relationship between some wolves and some rabbits.
R = number of rabbits
M = rabbit population growth rate if there were no wolves (.05)
K = rabbit population death rate because of wolves (.001 per wolf)
W = number of wolves
G = wolf population growth rate thanks to plenty of rabbits to eat (.0002 per rabbit)
D = wolf population death rate if there are not enough rabbits to eat (.03)
n = time in months
Rn = Rn-1(1 + M - KWn-1)
Wn = Wn-1(1+GRn-1 - D)
Press Y= and enter the following sequences:
nMin = 1
u(n) = u(n-1) * (1+.05 - .001 * v(n-1))
u(nMin) = {200}
v(n) = v(n-1) * (1 + .0002 * u(n-1) - .03)
v(nMin) = {50}
Press FORMAT (2nd and ZOOM). Choose Time in the first row:
Time
Web
uv
vw
uw
Press WINDOW and enter the following:
nMin = 0
Xmin = 0
Ymin = 0
nMax = 400
Xmax = 400
Ymax = 300
PlotStart = 1
Xscl = 100
Yscl = 100
PlotStep = 1
Press GRAPH. Carefully sketch the resulting graph. From this graph, how would you say the number of wolves and rabbits are related?
We can better understand the relationship of the number of wolves of rabbits by graphing the number of wolves against the number of rabbits, and ignoring the time. Press FORMAT (2nd and ZOOM). Choose uv in the first row:
Time
Web
uv
vw
uw
Press WINDOW and change:
Xmin = 0
Ymin = 0
Xmax = 250
Ymax = 100
Xscl = 50
Yscl = 25
Press GRAPH. Watch the spiraling graph as it is displayed. Carefully sketch the graph on a piece of graph paper. Describe it. Which way does it spiral (clockwise or counter-clockwise)? What does this say about the relationship between the wolves and rabbits?
What if the wolves get a little more aggressive? Press Y= and replace .001 with .002 in u(n). Do both a line graph and the uv graph again. How are they different this time? Do you think the populations will be able to survive for a long time?
Most real populations in nature involve more complex relationships between several predators and prey, as well as other factors such as the climate. Some scientists believe that these complex relationships by themselves may be enough to cause some species to become extinct, without the assistance of wayward asteroids.