EXERCISE SHEET FOR CONCEPTS OF EXPONENTIAL AND LOGISTIC GROWTH
Basics: Remember that we have discussed 3 basic forms of growth: Geometric
(N(t) = N(0)lambda^t); Exponential (N(t) = N(0)e^rt); and Logistic N(t) =
[N(0)e^rt][[K-N(t)]/K]. *Note that lambda is expressed here as e^r, not e^rt; The latter gives you [N(t)/N(0)].
To increase your understanding of these models requires that you practice
with them. Use the following data set to answer some questions. If you can
handle these questions you have a good understanding of how we use the
growth models.
Time (t) |
Number in Population (N) |
0 |
5 |
1 |
10 |
2 |
20 |
3 |
60 |
4 |
150 |
5 |
400 |
Exercises:
1. Plot N as a function of t.
2. Is the growth exponential or logistic?
3. Assuming exponential growth, calculate r for t = 1. How about lambda fort = 1?
4. Assuming exponential growth, calculate r for t = 5. Again, how about
lambda?
5. What would N be at t = 10 if growth continued as before?
6. Assume the carrying capacity is 450. Use the logistic growth model to
recalculate r for t = 1 and t = 5.
7. If K = 450, what population size would maximize yield?
Answers:
1. Plot Age (t) on the x-axis and N (could be called f(x)) on the y-axis.
2. Exponential
3. t=1, then r = ln(10/5)/1 = ln(2) = 0.69
t = 1, then lambda = 10/5 = 2
4. t = 5, then r = ln(400/5)/5 = ln(80)/5 = 0.88
t = 5, then lambda = 400/5 = 80
5. Use your estimate of r based on all 5 years (0.88) and the exponential
formula that relates N(10) to N(0)e^rt. Thus, N(10) = (5)(e^(0.88)(10) =
(5)(e^8.8) = 33171.22 This is unlikely to be an endangered species!!
6. Lets rearrange the logisitic equation like this: N(t) =
N(0)e^rt((K-N(t))/K). To get this as a function of r, you can rearrange as
N(t)/{N(0)(K-N(t))/K} = e^rt. Go a bit further to get this as
ln(N(t)/{N(0)(K-N(t))/K})/(t) = r.
Now we can plug N(0) = 5, N(1) = 10, K = 450, and t =1 into this to get:
{ln(10)/5(440/450)}/1= 0.71.
Using N(0) = 5, N(5) = 400, K = 450, and t = 5 we get:
{ln(400/[(5)(50/450)]}/5 = 1.32.
7. MSY is produced when N = 1/2K or 225 for the example.