Fish and Fishing
An Ecological Model of Fish Harvesting
Version 3.141
Background
The Discovery Channel’s Deadliest Catch has introduced many people to the seemingly
strange rules and laws governing the way we harvest food from the world’s oceans.
Fishing and crabbing industries are very tightly controlled with the government
rigorously dictating when fishing can take place, where and how fish may be harvested,
and how many fish can be taken from the sea in any one season. In this activity, you can
investigate a very simple yet insightful mathematical model of a fish population that is
subjected to “harvesting”.
Setting up the Model
We will start with a Logistic Model. Recall that the logistic model begins with two basic
assumptions:
1. The change in the population is proportional to the current population (this is
basically exponential growth) for small populations, but
2. The change in population is negative if the population exceeds the carrying
capacity of the environment.
Thus, we have two parameters which govern how the population will change. We’ll use
the following notation:

Pn , the fish population (at time interval n) measured in thousands
of tons.
 r, the exponential growth rate (for small populations)
 L, the carrying capacity of the population
In an earlier activity you began to explore the logistic model which interprets the basic
assumptions mathematically by the statement
P
Change in population = rP (1  )
L
Warm Up
Before we add in a “harvesting” element to our model, let’s first re-familiarize ourselves
with the logistic model. Let’s assume that r = .17, L = 2000, and P0  500 .
1. Using the initial population figure and our formula for how the population will
change during each time interval, compute the population for the next 40 time
steps. Graph your results.
2. As time goes on, what is the “limiting” fish population?
3. Is the limiting fish population still reached even when the initial fish population is
very small? Explain.
4. What happens if the initial fish population is 2000?
5. What happens if the initial fish population is greater then 2000?
Connecting the Algebra with the Numbers
The answers to the last several questions could have been answered simply by
understanding the “algebra” of the model. By “model” we just mean our formula for the
change in population:
P
)
L
6. If we think of the right hand side of this equation as a function of P, what kind of
function is it?
7. It is easy to find the two horizontal intercepts. Do so, and sketch a plausible
graph of the function.
Change in population = rP (1 
To interpret this graph you need to pay attention to the axes. The horizontal axis
represents P, the fish population. The vertical axis represents the change in the fish
population.
8. Using this graph,
a. when would you expect no change in the fish population?
b. when would you expect to see positive change in the fish population?
c. when would you expect to see negative change in the fish population?
d. Do your answers to these questions agree with what you saw in questions
2-5 above?
Thus, we can sometimes avoid a lot of numerical calculations if we are just interested in
the overall behavior of the model. Of course, if we want specific values for the fish
population at different times, we will have to resort to some calculations.
Adding in the Fishing Industry
The model above takes into account the natural growth rate of the fish population as well
as the limited access to resources the fish have (food, natural streams for reproduction,
etc.). We will now modify the model very slightly to take into account the amount of fish
harvested each year (from now on, let’s assume that each time step represents one year).
If the fishing industry is allowed to harvest C thousand tons of fish each year we might
adjust our model to become:
P
)C
L
For the following let’s keep r = .17 and L = 2000. Let’s start with C = 10
9. Starting again with P0  500 , investigate the overall behavior of the fish
population under harvesting. What is the “limiting” fish population? How does
this compare to the limiting population without any harvesting?
10. Compare the limiting fish population for different choices for C. Explain your
findings.
Change in population = rP (1 
Depending of the type of technology one has access to, recalculating 40+ years of fish
populations for different values of C (or P0 or r or L or …) can become tedious. As we
did above, we can see if studying the underlying algebra of the model can provide any
insight into this model. In particular, let’s focus on the question when are the fisheries
taking too much from the sea?
Focus on the Mathematics
When are the fisheries taking too much from the sea? Clearly, if too many fish are
harvested each year one might expect that the overall fish population could decline to the
point of extinction or near extinction. For example, we saw above that for C = 10, the
fish population continued to increase towards a healthy limiting value. However, if we
start with 500 thousand tons of fish and allow 500 thousand tons to harvested, we can
probably assume that this will decimate the fish population. Therefore, one could argue
that there is some magic value of C, a threshold, so that if one harvests less than that
amount we have a sustainable source of fish while if one harvests more than that amount
overfishing (and extinction) will result. For our fish population, this threshold value of C
is probably between 10 and 500. To help determine the exactly value of this threshold,
let’s return to our harvesting model:
P
)C
L
It is probably reasonable to assume that if this change in population (the right hand side
of the above equation) is positive, we have a healthy fish population. If the change is
negative, then the fish population is not healthy (since if it stays negative for too long,
this would imply that the population may reach zero!).
Change in population = rP (1 
11. If one thinks of the right hand side of the above equation as a function of P, what
kind of function is it? Hint: Refer to your answer to question 7 above.
12. How does the graph of this function compare to the graph from question 7? Label
P = L on the horizontal axis.
13. How many roots does this function have? What is the significance of the roots to
the harvesting model we are studying?
14. For C = 10, what is the value of the leftmost root?
15. If P0 were less than this value, describe the resulting long term behavior of the
fish population.
16. Determine a value for this root that depends only on C (you may use r = .17 and L
= 2000).
17. Use your formula to produce a graph of the value of the root vs. C.
18. If our fish population begins with 500 thousand tons of fish, use the above graph
to determine the “magic” threshold for C.
19. Double check your work by showing that the initial fish population will increase
if C is less than your magic threshold and will decrease if C is greater then this
threshold.
Determining a legal fishing limit in the real world is very tricky! As we have seen above,
one needs to have a good idea of the current fish population to know if C is too large (of
course if C is too small this is good for the fish but makes for a lot of unhappy captains).
Slight changes in C can have drastic effects on the fish population (healthy vs.
extinction). Thus one can begin to see why fishing industries are so tightly regulated and
why harvesting limits change from year to year.