Environmental Studies 100
Fall 2004
Fisheries Models Exercise
Due: in lecture, Monday November 22nd. You will work on this assignment in section. Any
unfinished portions you should complete as homework.
Instructions: Please answer the following on a separate sheet of paper.
Use simple logistic growth for your fishery:
dX
X
= r X * (1 - )
dt
K
)
X= fish population (in pounds)
r = growth rate constant = 0.5/year
K = carrying capacity = 100,000 pounds of fish
1. Calculate the growth rate (dX/dt, expressed in pounds of fish per year) at intervals of 0.1K. For
example, to find the growth rate at 0.1K, calculate what the growth rate would be at a population
size of 10,000 lbs of fish.
2. Plot a growth rate vs. population size curve from the data above.
3. What is the maximum growth rate of the fish population (in pounds/year)?
4. At what fraction of K do you get the highest growth rate?
5. Are the following harvest regimes stable? Consider how stable each harvest regime would be if
continued indefinitely.
a.
b.
c.
d.
15,000 pounds of fish per year at a population of 50,000 pounds
7,500 pounds of fish per year at a population of 80,000 pounds
9,000 pounds of fish per year at a population of 80,000 pounds
9,000 pounds of fish per year at a population of 20,000 pounds
6. Economics of Fishing:
i. You are allowed to harvest only the “sustainable yield” which is equal to the fish growth rate.
ii. Fish sell for $1/pound (wholesale) so income = $1 * catch in pounds.
iii. The effort to catch the “allowable” harvest is:
Effort = 1- X/K
The effort is 0 when the population is at K because you wouldn’t be allowed to take any fish
(since growth rate is 0). The effort needed to catch the legal fish limit increases in inverse
proportion to the standing stock:
iv. A 100% effort costs $15,000 so cost = $15,000 * effort.
At what fish population size would you balance cost and income? Solve this mathematically and
show it graphically.
7. Is this fishery likely to be sustainable? Why or why not?