Chapter 2

You are building a mathematical model for the population of cod fish in a North Atlantic fishery. Write a word equation relating the population \(N_{t}\) in one year to the population \(N_{t+1}\) in the next year. Your word equation should include the following terms: \- Number of cod fish born during the year \- Cod fish dying of old age during the year \- Cod fish killed by predators during the year \- Cod fish removed by fishing boats during the year

In Problems 1-16, determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=n+1 $$

In Problems \(1-4\), produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=3^{t} $$

You are building a mathematical model for the human population of a small Southern California town. Write a word equation relating the population \(N_{t}\) in one year to the population \(N_{t+1}\) in the next year. Your word equation should include the following terms: \- Number of children born during the year \- People dying from any cause during the year \- People moving into the town from other towns during the year \- People leaving the town to live in other towns during the year

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=3 n^{2} $$

In Problems , produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=6 \cdot 2^{t} $$

You are building a math model for the size of the wild population of kakapo (rare ground dwelling flightless parrots) in New Zealand. Write a word equation relating the population \(N_{t}\) in one year to the population \(N_{t+1}\) in the next year. Your word equation should include the following terms: \- Number of kakapo births in the wild during the year \- Kakapo removed for captive breeding during the year \- Kakapo reintroduced into the wild from captive breeding during the year \- Kakapo killed by predators during the year \- Kakapo deaths from disease during the year

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{n+2}{n} $$

You are building a mathematical model for the spread of Sudden Oak Death Syndrome - a disease that has wiped out over one million oak and tanoak trees in Coastal California. Write a word equation relating number of oak trees in one year, \(N_{t}\), to the number, \(N_{t+1}\), in the next year. Your word equation will include the following terms: \- Number of trees seeded in the wild during the year \- Trees planted by people during the year \- Trees killed by the disease during the year \- Trees cut down by loggers during the year

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{n}{n+2} $$

In Problems , produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\). $$ N_{t}=0.2(0.8)^{t} $$

You are building a mathematical model for the size of coral reefs in a patch of the Pacific Ocean. Rather than directly measuring the number of corals, you measure the area of living coral in each coral reef. Write a word equation relating the area, \(A_{t}\), of living coral in one year, to the area, \(A_{t+1}\), in the next year. Your word equation will include the following terms: \- Area killed by ocean acidification during the year \- Area killed by fishing during the year \- Area of reef restored or rebuilt during the year

In Problems 5-7, give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=2\); population doubles every 20 minutes; one unit of time is 20 minutes

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{1}{\sqrt{n+1}} $$

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=4\); population doubles every 40 minutes; one unit of time is 40 minutes

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=1\); population doubles every 40 minutes; one unit of time is 80 minutes

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. Suppose \(N_{t}=20 \cdot 4^{t}, t=0,1,2, \ldots\), and one unit of time corresponds to 3 hours. Determine the amount of time it takes the population to double in size.

Saving the Kakapo You are modeling the size of the population of kakapo (a rare flightless parrot) in an island reserve in New Zealand. You want to use the mathematical model to predict the size of the population. The data in this question are taken from Elliot et al. (2001) (a) You start by writing a word equation relating the population size \(N_{t}\) in year \(t\), that is, \(t\) years after the study began, to the population size \(N_{t+1}\) in the next year. umber of birds number of birds that \(N_{t+1}=N_{t}+\begin{array}{l}\text { number } 6 \pi \text { in } \\ \text { born in one vear }\end{array}\) die in one ve We will derive together formulas for each of these terms (i) To estimate the number of birds born, assume that half of the birds are female. A female bird lays one egg every four years. However, because of the large numbers of predators (mostly rats) only \(29 \%\) of hatchlings survive their first year. Explain how based on this data our prediction for the number of births is: \(N_{t} \cdot 0.5 \cdot 0.25 \cdot 0.29=0.03625 \cdot N_{t}\) (ii) Kakapo life expectancy is not well understood, but we will assume that they live around 50 years. That is, in a given year, one in fifty kakapo will die. What is the corresponding number of deaths? (iii) Assume that the starting population size on this island is 50 birds (i.e., \(N_{0}=50\). Calculate the predicted population size over the next five years (i.e., calculate \(N_{1}, N_{2}, \ldots, N_{5}\) ). (iv) When (if ever) will the population size reach 100 birds? What about 200 birds? (You will find it helpful to derive an ex plicit formula for the size of the population \(N_{\mathrm{t}}\). (b) Using your model from part (a) you want to evaluate the effectiveness of two different conservation strategies: (Strategy 1) If the kakapo are given supplementary food, then they will breed more frequently. If given supplementary food, then rather than laying an egg every four years, a female will lay an egg every two years. (Strategy 2) By hand-rearing kakapo chicks, it is possible to increase their one year survival rate from \(29 \%\) to \(75 \%\). (i) Write down a recurrence equation for the population size \(N_{t}\) if strategy 1 is implemented. Assuming \(N_{0}=50\), calculate \(N_{1}\), \(N_{2}, \ldots, N_{5}\) (ii) Write down a recurrence equation for the population size \(N_{t}\) if strategy 2 is implemented. Assuming \(N_{0}=50\), calculate \(N_{1}\), \(N_{2}, \ldots, N_{5}\) (iii) Which conservation strategy gives the biggest increase in population size?

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=(-1)^{n}+(-1)^{n+1} $$

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. Suppose \(N_{t}=100 \cdot 2^{t}, t=0,1,2, \ldots\), and one unit of time corresponds to 2 hours. Determine the amount of time it takes the population to triple in size.

Mountain Gorilla Conservation You are trying to build a mathematical model for the size of the population of mountain gorillas in a national park in Uganda. The data in this question are taken from Robbins et al. (2009). (a) You start by writing a word equation relating the population of gorillas \(t\) years after the study begins, \(N_{t}\), to the population \(N_{t+1}\) in the next year: \(N_{t+1}=N_{t}+\begin{array}{l}\text { number of gorillas } \\ \text { born in one year }\end{array}-\begin{array}{l}\text { number of gorillas } \\\ \text { that die in one yeai }\end{array}\) We will derive together formulas for the number of births and the number of deaths. (i) Around half of gorillas are female, \(75 \%\) of females are of reproductive age, and in a given year \(22 \%\) of the females of reproductive age will give birth. Explain why the number of births is equal to: \(\quad 0.5 \cdot 0.75 \cdot 0.22 \cdot N_{t}=0.0825 \cdot N_{t}\) (ii) In a given year \(4.5 \%\) of gorillas will die. Write down a formula for the number of deaths. (iii) Write down a recurrence equation for the number of gorillas in the national park. Assuming that there are 300 gorillas initially (that is \(N_{0}=300\) ), derive an explicit formula for the number of gorillas after \(t\) years. (iv) Calculate the population size after \(1,2,5\), and 10 years. (v) According to the model, how long will it take for population size to double to 600 gorillas? (b) In reality the population size is almost totally stagnant (i.e., \(N_{t}\) changes very little from year to year). Robbins et al. (2009) consider three different explanations for this effect: (i) Increased mortality: Gorillas are dying sooner than was thought. What percentage of gorillas would have to die each year for the population size to not change from year to year? (ii) Decreased female fecundity: Gorillas are having fewer offspring than was thought. Calculate the female birth rate (percentage of reproductive age females that give birth) that would lead to the population size not changing from year to year. Assume that all other values used in part (a) are correct. (iii) Emigration: Gorillas are leaving the national park. What number of gorillas would have to leave the national park each year for the population to not change from year to year?

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=(-1)^{n}+1 $$

In Problems , give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. Suppose you measure the following data for the size of a population of bacteria: $$ \begin{array}{lcccc} \hline \boldsymbol{t} & 0 & 1 & 2 & 3 \\ \hline \boldsymbol{N}_{t} & 10 & 30 & 90 & 270 \\ \hline \end{array} $$

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{4 N_{t}}{1+N_{t} / 30}\)

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=\frac{n^{2}}{n+1} $$

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). 11\. A strain of bacteria reproduces asexually every hour. That is, every hour, each bacterial cell splits into two cells. If, initially, there is one bacterium, find the number of bacterial cells after 1 hour, 2 hours, 3 hours, 4 hours, and 5 hours.

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{2 N_{t}}{1+N_{t} / 60}\)

Determine the values of the sequence \(\left|a_{n}\right|\) for \(n=0,1,2, \ldots, 5\) $$ a_{n}=n^{3} \sqrt{n+1} $$

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). . A strain of bacteria reproduces asexually every 30 minutes. That is, every 30 minutes, each bacterial cell splits into two cells. If, initially, there is one bacterium, find the number of bacterial cells after 1 hour, 2 hours, 3 hours, 4 hours, and 5 hours.

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{2 N_{t}}{1+N_{t} / 90}\)

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). A strain of bacteria reproduces asexually every 42 minutes. That is, every 42 minutes, each bacterial cell splits into two cells. If, initially, there is 1 bacterium, how long will it take until there are 1024 bacteria?

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{3 N_{t}}{1+N_{t} / 100}\)

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). A strain of bacteria reproduces asexually every 24 minutes. That is, every 24 minutes, each bacterial cell splits into two cells. If, initially, there is 1 bacterium, how long will it take until there are 512 bacteria?

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{3 N_{t}}{1+N_{t} / 30}\)

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). A strain of bacteria reproduces asexually every 10 minutes. That is, every 10 minutes, each bacterial cell splits into two cells. If, initially, there are 5 bacteria, how long will it take until there are 320 bacteria?

Assume that the population growth is described by the Beverton-Holt model. Find all fixed points. \(N_{t+1}=\frac{5 N_{t}}{1+N_{t} / 240}\)

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). A strain of bacteria reproduces asexually every 12 minutes. That is, every 12 minutes, each bacterial cell splits into two cells.

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=2, a=0.01, N_{0}=2\)

In Problems 17-24, find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ 1,4,9,16,25 $$

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population that doubles in size every unit of time and that has 40 individuals at time 0 .

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=2, a=0.1, N_{0}=2\)

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential decay equation for a population that halves in size every unit of time and that has 1024 individuals at time \(0 .\)

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=3, a=1 / 20, N_{0}=7\)

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population that has a reproductive rate of 4 and has 20 individuals at time \(0 .\)

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=3, a=1 / 10, N_{0}=3\)

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ -1, \frac{1}{4},-\frac{1}{9}, \frac{1}{16},-\frac{1}{25} $$

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population that triples in size every unit of time and that has 72 individuals at time 0 .

Assume that the population growth is described by the Beverton-Holt recruitment curve with parameters \(R_{0}\) and a. Find the population sizes for \(t=1,2, \ldots, 5\) and find \(\lim _{t \rightarrow \infty} N_{t}\) for the given initial value \(N_{0} .\) \(R_{0}=4, a=1 / 40, N_{0}=2\)

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ \frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6} $$

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population that quadruples in size every unit of time and that has five individuals at time 0 .