# Math Insight

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

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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} = 3 x_{n} e^{- \frac{x_{n}}{5}} \end{align*}

The equilibria of this model are $x_n=0$ and $x_n= 5 \ln{\left (3 \right )}$. The function $g(x)=3 x e^{- \frac{x}{5}}$ and the diagonal are plotted below.

1. Calculate the stability of the equilibria using calculus.
2. Confirm the stability through cobwebbing the above graph.

2. A population of termites is growing out of control, so that each year the number of termites is increasing by 320%. In attempt to limit the population, each year a number of termites is killed that is proportional to the square of the population. If $m_t$ is the number of termites in year $t$, the population follows the dynamical system \begin{align*} m_{t+1}=4.2 m_t - d m_t^2 \end{align*} where $d$ is a positive parameter that determines the amount of harvesting.
1. Find the equilibria of this system.
2. Calculate the stability of the equilibria.
3. How does the stability of the equilibria depend on the parameter $d$?
4. Will this strategy succeed in bringing the number of termites to a stable small population?

3. A population of mice would be increasing at the rate of 60% per year, except that due to a scarcity of cheese, there is only enough food to support 690 mice. Therefore, if $m_t$ is the number of mice in year $t$, the population evolves according to the dynamical system \begin{align*} m_{t+1} - m_{t} = 0.6 m_t\left(1 - \frac{ m_t}{ 690 }\right). \end{align*}

Calculate the equilibria of the dynamical system and their stability. If the initial population is $m_0=500$, what will happen to the population after a long time?

4. A population of cheetahs would be increasing at the rate of $a$ per year, except that due to a scarcity of antelope, there is only enough food to support $W$ cheetahs. Therefore, if $c_t$ is the number of cheetahs in year $t$, the population evolves according to the dynamical system \begin{align*} c_{t+1} - c_{t} = a c_t\left(1 - \frac{ c_t}{ W }\right). \end{align*}
1. Calculate the equilibria of the dynamical system.
2. If $a = 0.9$, determine the stability of the equilibria.
3. If $a = 0.9$, $W = 490$, and the initial population is $c_0=770$, what will happen to the population after a long time?
4. If $a = 3.3$, determine the stability of the equilibria.
5. For what range of $a$ is there a stable nonzero equilibrium?

5. A population of mice is growing exponentially so that in each year the population increases by 140%. In order to keep the population from exploding, a number of mice is harvested annually that is proportional to the square of the population. If $m_t$ is the number of mice in year $t$, the population follows the dynamical system \begin{align*} m_{t+1}=2.4 m_t - c m_t^2 \end{align*} where $c$ is a positive parameter that determines the amount of harvesting.
1. Find the equilibria of this system.
2. Calculate the stability of the equilibria.
3. How does the stability of the equilibria depend on the parameter $c$?
4. For what value of $c$ is there a stable equilibrium of 490 mice?