### Overview of: Controlling a rabbit population

#### Project summary

Develop and evaluate strategies for controlling a rabbit population that will maintain a steady population size of around one thousand rabbits. Each strategy will be formulated in terms of a discrete dynamical system model of the form \begin{align*} p_{t+1}-p_t &= r p_t-h(p_t)\\ p_0 &= p_0, \end{align*} where $p_t$ is the rabbit population size after $t$ months from the starting point, $r$ is the reproduction rate of the rabbits, and $p_0$ is the initial population size. You do not know the exact values of $r$ or $p_0$, but you know that $r$ is close to 0.2.

The function $h(p_t)$ gives the number of rabbits harvested in each month as a function of the rabbit population size $p_t$. A given choice of $h(p_t)$ defines a *strategy* for controlling the number of rabbits. Hence the task of developing and evaluating rabbit control strategies is the same thing as developing and evaluating possible choices for $h(p_t)$. To be a valid strategy $h(p_t)$ should not be negative (at least for values of $p_t$ that you are considering).

There are two main components to the project. The first component is to evaluate the rabbit harvesting strategies proposed in the controlling rabbit population page. The second component is the challenge of developing your own strategy and then evaluating that strategy.

#### Requirements for project report

You must work in a group of 3 or 4. Your group's project report must be no longer than six pages typed (10-12 point font, 1 inch margins), with original computer-generated graphs. Please turn in your group's report via Gradescope by the date and time it is due.

Your report needs to contain the following sections:

**Background:**Give a short description of the rabbit problem. Explain what your goals are. (No math in this section.)**The general model:**Describe the discrete dynamical system model (shown above) that you used to model the rabbit growth and your harvesting of the rabbits. Explain the parameters as well as what the function $h(p_t)$ means.**Fixed removal strategy:**Describe and evaluate the fixed removal strategy, where you tried to remove a fixed number $a$ of rabbits each month. Give the form of the function $h(p_t)$. Follow the above procedure on showing how a strategy doesn't work until you can demonstrate that this strategy will not work. Include some type of graph in your explanation, such as a cobwebbing plot or a graph of the solution versus time.**Proportional removal strategy:**Describe and evaluate the proportional removal strategy. Give the form of the function $h(p_t)$ (for the case without the parameter $a$ or with $a=0$). Follow the above procedure on showing how a strategy doesn't work until you can demonstrate that this strategy will not work. You may find that you will need to treat separately the cases when $b \ne r$ and when $b=r$.**The challenge:**Describe your results in attempting to complete the challenge of coming up with a successful rabbit control strategy.If you found a function $h(p_t)$ that gives a stable population size of around a thousand rabbits, explain how you came up with that function. Follow the above procedure on demonstrating that a strategy will work. For the case when $r=0.2$, estimate the basin of attraction of the stable fixed point. For what range of $r$ does the basin of attraction contain $10 < p_0 < 5000$? Does it seem like a reasonable strategy to test out? Is it perfect or does it have some flaws?

If you cannot find a function $h(p_t)$ that gives a stable population of around a thousand rabbits, give an example of a strategy $h(p_t)$ that you tried. Follow the above procedure on showing how a strategy doesn't work to demonstrate that it does not work.

**Conclusion:**Wrap-up your report. Briefly describe what you tried. Explain which method worked the best.

#### Suggestions to approach the problem

Begin by going over the detailed discussion of the different unsuccessful rabbit control possibilities. That page contains many hints and suggestions throughout, some of which we summarize here.

Even though you don't know what the reproduction rate $r$ is, a good place to start is by setting it to 0.2. See if you can get the model to work right when $r=0.2$. After that, worry about what happens if you change $r$ (but don't change anything about your harvesting strategy).

As done with the attempts described in the rabbit control page, a good place to start is looking at the equilibria of the dynamical system either analytically or graphically. Unlike what you saw in the fixed removal strategy, you'll want to have $p_t=0$ be an equilibrium, so you don't have negative rabbits or beam in rabbits. A second equilibrium would be a good idea, too.

Another important aspect is stability of the equilibria. Do you want your equilibrium at zero to be stable? Or maybe you want the second equilibrium to be stable. Cobwebbing is a good way to investigate stability of equilibria.

#### Evaluating a strategy

A strategy for rabbit control is a function $h(p_t)$. A strategy can have parameters such as the $a$ and $b$ in $h(p_t) = a +b p_t$, but in order to implement a strategy, you must choose numerical values for the parameters that can never change. To show that a strategy will never work, you must show that it will not work no matter what values you choose for the parameters. To show that a strategy will work, you need to give specific numerical values for the parameters and show that for those parameter values, the strategy will control the rabbit population.

##### Showing a strategy will not work

To show a strategy will not work, do the following. In most cases, you will just need to follow the first few steps of this procedure.

- First, evaluate the case when $r=0.2$.
- Find the equilibria. The values of the equilibria and/or the numbers of the equilibria might depend on the parameters. Be sure to explain how the equilibria depend on the parameters.
- Evaluate the stability of the equilibria. The stability may also depend on the parameters.
- If you can show that you do not get a stable equilibrium at around $p_t=1000$, no matter what values of the parameters you choose, you are done. The strategy will not work.
- If you found parameter values for which you have a stable equilibrium at around $p_t = 1000$, then check if the basin of attraction contains all initial conditions in $10 < p_0 < 5000$. If all values of $p_0$ in the range do not lead to population sizes $p_t$ that approach the equilibrium, then you are done. The strategy will not work.
- If you found parameter values where the basic of attraction contains $10 < p_0 < 5000$ when $r=0.2$, then you need to go on and check different values of $r$.

- If you found that the strategy might work for some parameters when $r=0.2$, then check other values of $r$.
- Repeat the above procedure with $r=0.18$. If you can show that the strategy will break for either $r=0.2$ or $r=0.18$, then you are done. The strategy will not work.
- If you found some parameters that work for both $r=0.18$ and $r=0.2$, check if those same parameters will also work for $r=0.22$. If they don't work for all three values of $r$, then you are done. The strategy will not work.
- If you found some parameters that work for $r=0.18$, $r=0.2$, and $r=0.22$, then you probably want to instead follow the procedure about showing that a strategy will work.

##### Demonstrating that a strategy will work

To demonstrate that a strategy will work, you need to first find the strategy $h(p_t)$. That's the hard part. The easier part is demonstrating that it will work, using the following steps. Note that, if you happen to have any parameters in your definition of $h(p_t)$, then you must choose particular numerical values for those parameters before documenting that it works using this procedure.

- Demonstrate that the strategy works for $r=0.2$.
- Find the equilibria.
- Evaluate the stability of the equilibria.
- Show that you have a stable equilibrium at around $p_t=1000$ (values between 500 and 2000 are OK).
- Show that you have an unstable equilibrium at $p_t=0$.
- Show that the basin of attraction for the stable equilibrium around $p_t=1000$ contains all initial conditions in $10 < p_0 < 5000$. All values of $p_0$ in the range should lead to population sizes $p_t$ that approach the equilibrium. For the competition (see below), the larger the range of $p_0$ that work, the better you'll do.

- Demonstrate that the strategy works for a larger range of $r$.
- Repeat the above procedure for $r=0.18$ and $r=0.22$, finding a stable equilibrium around $p_t=1000$.
- If, for both $r=0.18$ and $r=0.22$, your basin of attraction for this equilibrium is contains $10 < p_0 < 5000$, then you are done.
- For the competition (see below), the larger the range of $r$ for which the basin attraction contains $10 < p_0 < 5000$, the better you'll do.

##### Grading Notes

Background: 2 points

General Model: 3 points

Fixed Removal Model: 7 points

Proportional Removal Model: 7 points

Challenge: 9 points

Conclusion: 2 points

You will be graded on both your analysis and your explanations.
Points will be deducted for groups beyond the acceptable 3-4 person range and exceeding the page limit. Late submissions will not be accepted.

Values of $r$ and $p_0$ must be positive. Each group may earn up one bonus point as follows:

- 0.5 bonus points will be awarded to the team or teams that demonstrate a strategy giving the largest basin of attraction for the case when $r=0.2$.
- 0.5 bonus points will be awarded to the team or teams that demonstrate a strategy with a basin of attraction containing $10 < p_0 < 5000$ for the largest range of the parameter $r$.