These exercises allow you to practice how to solve discrete dynamical systems involving exponential growth and decay.

#### Exercise 1

Write a solution equation for the following discrete dynamical systems. In each case, compute the solution at time $t=100$. \begin{align*} {\rm{a.}} \hspace{2mm} x_0 & = 1,000 & x_{t+1} - x_t & = 0.2 \times x_t \\ {\rm{b.}} \hspace{2mm} x_0 & = 138 & x_{t+1} - x_t & = 0.05 \times x_t \\ {\rm{c.}} \hspace{2mm} B_0 & = 138 & B_{t+1} - B_t & = 0.5 \times B_t \\ {\rm{d.}} \hspace{2mm} y_0 & = 1,000 & y_{t+1} - y_t & = -0.2 \times y_t \\ {\rm{e.}} \hspace{2mm} P_0 & = 1,000 & P_{t+1} & = 1.2 \times P_t \\ {\rm{f.}} \hspace{2mm} z_0 & = 1000 & z_{t+1} - z_t & = -0.1 \times z_t \\ {\rm{g.}} \hspace{2mm} c_0 & = 1,000 & c_{t+1} & = 0.9 \times c_t \end{align*}

#### Exercise 2

The equation, $x_{t} - x_{t-1} = r x_{t-1}$, carries the same information as $x_{t+1} - x_{t} = r x_{t}$.

- Write the first four instances of $x_{t} - x_{t-1} = r x_{t-1}$ using $t=1$, $t=2$, $t=3$, and $t=4$.
- Obtain an expression for $x_4$ in terms of $r$ and $x_0$.
- Write solutions to and compute $x_{40}$ for \begin{align*} {\rm{(i.)}} \hspace{2mm} x_0 & = 50 \hspace{5mm} & x_{t} - x_{t-1} & = 0.2 \times x_{t-1} \\ {\rm{(ii.)}} \hspace{2mm} x_0 & = 50 & x_{t} - x_{t-1} &= 0.1 \times x_{t-1}\\ {\rm{(iii.)}} \hspace{2mm} x_0 & = 50 & x_{t} - x_{t-1} &= 0.05 \times x_{t-1}\\ {\rm{(iv.)}} \hspace{2mm} x_0 & = 50 & x_{t} - x_{t-1} &= -0.1 \times x_{t-1} \end{align*}

#### Exercise 3

Suppose a population is initially of size 1,000,000 and grows at the rate of 2% per year. What will be the size of the population after 50 years?

#### Exercise 4

The polymerase chain reaction is a means of making multiple copies of a DNA segment from only a minute amount of original DNA. The procedure consists of a sequence of multiple cycles. During the course of one cycle, each DNA segment present is duplicated. Introduce notation and write a discrete dynamical system with initial condition from which the amount of DNA present at the end of each cycle can be computed. Suppose you begin with 1 picogram = 0.000000000001 g of DNA. How many grams of DNA would be present after 30 cycles.

#### Exercise 5

Write a solution to the dynamic system you obtained for growth of *V. natriegens* in growth medium of pH 7.85 in exericse 5 from the bacteria growth exercises. Use your solution to compute an estimate of $B_4$.

#### Exercise 6

There is a suggestion that the world human population is growing exponentially. Shown below are the human population numbers in billions of people for the decades 1940 - 2010.

Year | Index, $t$ | Human population (billons) |
---|---|---|

1940 | 0 | 2.30 |

1950 | 1 | 2.52 |

1960 | 2 | 3.02 |

1970 | 3 | 3.70 |

1980 | 4 | 4.45 |

1990 | 5 | 5.30 |

2000 | 6 | 6.06 |

2010 | 7 | 6.80 |

- Test the equation $$P_t = 2.2 \times 1.19^t$$ against the data where $t$ is the time index in decades after 1940 and $P_t$ is the human population in billions.
- What percentage increase in human population each decade does the model for the equation assume?
- What world human population does the equation predict for the year 2050?

#### Selected answers

Once you've worked out some of these exercises, you can check your work with the answers to selected problems.