Chapter 2

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. R=2, K=15

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=2, K=50\)

produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\) $$ N_{t}=10 \cdot 2^{t} $$

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=1.5, K=40\)

produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\) $$ N_{t}=\frac{25}{4^{t}} $$

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=3, K=120\)

produce a table for \(t=0,1,2, \ldots, 5\) and graph the function \(N_{t}\) $$ N_{t}=(0.3)(0.9)^{t} $$

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

\(5-10\), 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 \(n=0,1,2, \ldots, 5\) $$ f(n)=\frac{1}{(1+n)^{2}} $$

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) For the given values of \(R\) and \(K\), graph \(N_{t} / N_{t+1}\) as a function of \(N_{t}\) and find the recursion for the BevertonHolt recruitment curve. \(R=2, K=150\)

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

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

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

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

give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=6\); population doubles every 40 minutes; one unit of time is 60 minutes

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find \(R\) and \(K .\) N_{t+1}=\frac{1.5 N_{t}}{1+0.5 N_{t} / 30}

give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. . \(N_{0}=2\); population quadruples every 30 minutes; one unit of time is 15 minutes

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

give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. \(N_{0}=10 ;\) population quadruples every 20 minutes; one unit of time is 10 minutes

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

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.

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

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find all fixed points. N_{t+1}=\frac{4 N_{t}}{1+N_{t} / 30}

In Problems give a formula for \(N(t), t=0,1,2, \ldots\), on the basis of the information provided. . 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.

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find all fixed points. N_{t+1}=\frac{3 N_{t}}{1+N_{t} / 60}

. 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 recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find all fixed points. N_{t+1}=\frac{2 N_{t}}{1+N_{t} / 30}

A strain of bacteria reproduces asexually every 23 minutes. That is, every 23 minutes, each bacterial cell splits into two cells. If, initially, there is 1 bacterium, how long will it take until there are 128 bacteria?

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find all fixed points. N_{t+1}=\frac{2 N_{t}}{1+N_{t} / 100}

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 512 bacteria?

Determine the values of the sequence \(\left\\{a_{n}\right\\}\) for \(n=0,1,2, \ldots, 5\) $$ f(n)=2^{0.2 n} $$

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find all fixed points. N_{t+1}=\frac{3 N_{t}}{1+N_{t} / 30}

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 3 bacteria, how long will it take until there are 96 bacteria?

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) Find all fixed points. N_{t+1}=\frac{5 N_{t}}{1+N_{t} / 120}

A strain of bacteria reproduces asexually every 50 minutes. That is, every 50 minutes, each bacterial cell splits into two cells. If, initially, there are 10 bacteria, how long will it take until there are 640 bacteria?

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 growth parameter \(R\) and carrying capacity \(K .\) 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=2, K=20, N_{0}=5

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) 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=3, K=15, N_{0}=1

. Find the exponential growth equation for a population that triples in size every unit of time and that has 20 individuals at time \(0 .\)

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

Assume that the population growth is described by the Beverton-Holt recruitment curve with growth parameter \(R\) and carrying capacity \(K .\) 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=3, K=30, N_{0}=0

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 growth parameter \(R\) and carrying capacity \(K .\) 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=4, K=40, N_{0}=3