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### Harvesting Models Labs

#### Background

Consider a population of (say) fish, with a growth rate of 18% in an environment with a carrying capacity of 50,000. We take a standard simple model of population growth, the logistic model (see the section on population models). We then consider different ways of harvesting the fish, with a goal of maximizing the long-term yield (that is we want to harvest the largest number of fish without driving the population to extinction). We will experiment with this numerically and then analyze the situation graphically to find the optimum harvesting. It is worth noting that while we could solve the differential equations exactly, it will be easier to find the answer to the question of optimum harvesting from the geometric analysis without explicitly solving the equation.
There are two standard approaches to harvesting from a population. We can
harvest a set number of individuals every time (constant harvesting), or
we can harvest a set percentage of the population every time
(proportional harvesting). The basic model of the *unharvested*
population is
$$
P' = .18P(1-P/50000)
$$
where $P$ is the population and the derivative is taken with
respect to time $t$. If we harvest $h$ units of fish
every unit of time, we get the *constant harvesting model*
$$
P' = .18P(1-P/50000) - h
$$
In the other model, if we harvest a fraction $E$ of the
population every unit of time, we get the *proportional harvesting
model*
$$
P' = .18P(1-P/50000) - EP
$$
You should think about how the final terms in the two harvesting
models reflect the different harvesting strategies. You will be asked to
explain this in your lab reports.

In real life, constant harvesting can be enforced by setting a quota on all harvesters and then counting the harvest. Proportional harvesting is often enforced by limiting the number of days that harvesting is permitted, with the assumption that in a fixed period of time it is only possible to catch a certain percentage of the fish available. Another way to enforce proportional harvesting is to do a periodic census, and then adjusting quota values for harvesters according to the current population figures.

#### Lab Instructions

- Launch the harvesting models lab (see link at top of this page). The
applet already includes the basic logistic model, $P'=.18P(1-P/50000)$,
all you need to do is
type in the harvesting function $h$ or $E*P$ (you don't have to type in a
minus sign;
the applet already knows to subtract the harvest). Type in the harvesting
function
`1000`

and press enter to see how the population evolves through time if you harvest 1000 units of fish per unit time. Note that the yield in year 100 is listed below the graph, which we will take as an approximation to the long-term sustainable yield. - Try several different values for the constant harvesting model. How does changing the constant change the population curve. What seems to be the maximum sustainable harvest?
- For the constant harvesting model, draw the $dP/dt$ vs. $P$ graph and identify the stable equilibrium. Find the value of this point as a function of $h$.
- What is the maximum value of $h$ (that is the maximum amount to harvest), subject to the condition that the stable equilibrium must stay larger than 0 (otherwise, the population goes extinct). Note your answer here is probably a bit smaller than it appeared from the numerical experiments since it is possible to harvest in such a way that you have a good harvest in year 100 and have the population crash immediately thereafter.
- Now try the proportional harvesting model. Enter the harvesting
function
`.05p`

and see how the population evolves. The long-term yield is still listed at the lower right. How does changing the constant of proportionality (the`.05`

in the example) change the populaton curves and the yield. What value gives the maximum long-term yield according to your experiments? - For the proportional harvesting model, draw the $dP/dt$ vs. $P$ graph and identify the stable equilibrium. Find the value of this point as a function of $E$.
- If $p(E)$ is the stable equilibrium in the proportional harvesting model, explain why $E*p(E)$ is the long-term yield.
- Find the maximum long-term yield for the proportional harvesting model, by maximizing $E*p(E)$ as a function of $E$. How does this compare to the maximum long-term yield for the constant harvesting model. Can you explain why these two are related in the way they are?

#### Lab Report

Write up a report which includes- A justification for the mathematical models we used for the constant and proportional harvesting strategies.
- A discussion of how changing the constant $h$ changes the population and the yield for the constant harvesting model. Include a sketch of the different population curves.
- A mathematical computation of the optimal $h$ for the constant harvesting model to maximize long-term yield.
- A discussion of how changing the constant $E$ changes the population and the yield for the proportional harvesting model. Include a sketch of the different population curves.
- A mathematical computation of the optimal $E$ for the proportional harvesting model to maximize long-term yield.
- A discussion of which model is best for long-term stability of the population. You may want to consider what happens if, through a miscalculation, you allow slightly too many or too few fish to be harvested (i.e. $h$ or $E$ are a little above or below the optimum values) or issues with enforcing limits on harvesting in real life.

If you have any problems with this page, please contact bennett@math.ksu.edu.

©2010 Andrew G. Bennett