### Elementary discrete dynamical systems biology problems, part 2

#### Problem 1

Let $x_n$ be the number of fish in generation $n$ in a lake. We'll model the evolution of the fish using Ricker's model of the form \begin{align*} x_{n+1} = 5x_n e^{-x_n}, \end{align*}

The equilibria of this model are $x_n=0$ and $x_n= \ln(5)$.

- Calculate the stability of the equilibria using calculus.
- Confirm the stability through cobwebbing the below graph.
- For any positive initial fish population $x_0$, what will happen to the fish population $x_n$ as $n$ gets large?

#### Problem 2

Let $x_n$ be the number of fish in generation $n$ in a lake. We'll model the evolution of the fish using Ricker's model of the form \begin{align*} x_{n+1} = 8x_n e^{-x_n}, \end{align*}

The equilibria of this model are $x_n=0$ and $x_n= \ln(8)$.

- Calculate the stability of the equilibria using calculus.
- Confirm the stability through cobwebbing the below graph.
- If the initial fish population $x_0$ is close to (but not equal to) $\ln(8)$, what will happen to the fish population $x_n$ as $n$ increases?

#### Problem 3

A population of squirrels would be increasing at rate of 50% per year, except that due to a scarcity of acorns, there is only enough food to support 300 squirrels. Therefore, if $s_t$ is the number of squirrels in year $t$, the population evolves according to the dynamical system \begin{align*} s_{t+1} - s_{t} = 0.5 s_t\left(1 - \frac{s_t}{300}\right). \end{align*}

Calculate the equilibria of the dynamical system and their stability. If the initial population is $s_0=400$, what will happen to the population after a long time?

#### Problem 4

A population of squirrels would be increasing at rate of $r$ per year, except that due to a scarcity of acorns, there is only enough food to support $N$ squirrels. Therefore, if $s_t$ is the number of squirrels in year $t$, the population evolves according to the dynamical system \begin{align*} s_{t+1} - s_{t} = r s_t\left(1 - \frac{s_t}{N}\right). \end{align*}

- Calculate the equilibria of the dynamical system.
- If $r=1.5$, determine the stability of the equilibria.
- If $r=1.5$, $N=500$, and the initial population is $s_0=400$, what will happen to the population after a long time?
- If $r=3$, determine the stability of the equilibria.
- For what range of $r$ is there a stable nonzero equilibrium?

#### Problem 5

A population of deer is growing exponentially so that in each year the population increases by 20%. In order to keep the population from exploding, a number of deer is harvested annually that is proportional to the square of the population. If $d_t$ is the number of deer in year $t$, the population follows the dynamical system \begin{align*} d_{t+1}=1.2 d_t - c d_t^2 \end{align*} where $c$ is a positive parameter that determines the amount of harvesting.

- Find the equilibria of this system.
- Calculate the stability of the equilibria.
- How does the stability of the equilibria depend on the parameter $c$?
- For what value of $c$ is there a stable equilibrium of 200 deer?

#### Problem 6

A population of locusts is growing out of control, so that the number of locusts is quadupling each year. In attempt to limit the population, each year a number of locusts is killed that is proportional to the square of the population. If $l_t$ is the number of locusts in year $t$, the population follows the dynamical system \begin{align*} l_{t+1}=4 l_t - c l_t^2 \end{align*} where $c$ is a positive parameter that determines the amount of harvesting.

- Find the equilibria of this system.
- Calculate the stability of the equilibria.
- How does the stability of the equilibria depend on the parameter $c$?
- Will this strategy succeed in bringing the number of locusts to a stable small population?

One you have worked on a few problems, you can compare your solutions to the ones we came up with.

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