Chapter 2

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 \(N_{0} .\) \(R_{0}=4, a=1 / 60, N_{0}=2\)

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ 5,10,17,26,37 $$

Write down a formula for the population size, \(N_{t}\), as a function of time, \(t\). Find the exponential growth equation for a population whose size increases by \(50 \%\) in each unit of time and that has 32 individuals at time 0 .

A population obeys the Beverton-Holt model. You know that \(R_{0}=3\) for this population. As \(t \rightarrow \infty\) you observe that \(N_{t} \rightarrow 100 .\) What value of \(a\) is needed in the model to fit it to these data?

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

. Find the recursion for a population that doubles in size every unit of time and that has 11 individuals at time \(0 .\)

Find the next four values of the sequence \(\left\\{a_{n}\right\\}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, a_{3}, a_{4}\). $$ 1,3,9,27,81 $$

Find the recursion for a population that triples in size every unit of time and that has 6 individuals at time \(0 .\)

In Problems 25-36, find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(0,1,2,3,4, \ldots\)

Find the recursion for a population that quadruples in size everv unit of time and that has 30 individuals at time 0 .

A population obeys the Beverton-Holt model. You know that \(R_{0}=4\) for this population. One year you measure \(N_{t}=50\). The next year you measure that \(N_{t+1}=40 .\) What value of \(a\) is needed in the model to fit these data?

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(0,2,4,6,8, \ldots\)

Find the recursion for a population that has a reproductive rate of \(1 / 3\) and that has 63 individuals at time \(0 .\)

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(1,3,5,7,9, \ldots\)

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ f u n c t i o n s ~ \(f(x)=a^{x}, x \in[0, \infty)\), \mathrm{\\{} a n d ~ \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\).$$ a=R=3 $$

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(1, \frac{1}{3}, \frac{1}{9}, \frac{1}{27}, \frac{1}{81}, \ldots\)

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ f u n c t i o n s ~ \(f(x)=a^{x}, x \in[0, \infty)\), \mathrm{\\{} a n d ~ \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=1 / 2 $$

Assume that the discrete logistic equation is used with parameters \(R_{8}\) and \(b .\) Write the equation in the dimensionless form \(x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right)\), and determine \(x_{t}\) in terms of \(\bar{N}_{t}\) \(R_{0}=2, b=\frac{1}{20}\)

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(\frac{1}{3}, \frac{2}{5}, \frac{3}{7}, \frac{4}{9}, \frac{5}{11}, \ldots\)

\mathrm{\\{} I n ~ P r o b l e m s ~ , ~ g r a p h ~ t h e ~ f u n c t i o n s ~ \(f(x)=a^{x}, x \in[0, \infty)\), \mathrm{\\{} a n d ~ \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=1 / 3 $$

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(-1,2,-3,4,-5, \ldots\)

In Problems 31-42, find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=2 N_{t} \text { with } N_{0}=3 $$

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(9,16,25,36,49\)

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=2 N_{t} \text { with } N_{0}=5 $$

Investigate the advantage of dimensionless variables. A population obeys the discrete logistic equation: $$ N_{t+1}=R_{0} \cdot N_{t}-b N_{t}^{2} $$ Find the possible fixed points of the population size (one fixed point will depend on the unknown parameters \(R_{0}\) and \(b\) ).

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(5,7,9,11,13\)

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=3 N_{t} \text { with } N_{0}=2 $$

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=3 N_{t} \text { with } N_{0}=7 $$

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(2,0,2,0,2\)

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=5 N_{t} \text { with } N_{0}=1 $$

Investigate the advantage of dimensionless variables. You are studying a population that obeys the discrete logistic equation. You know that \(R_{0}=2 .\) One year you measure \(N_{t}=10\). The next year you measure that \(N_{t+1}=15 .\) What value of \(b\) is needed in the model to fit these data?

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots\) \(0,1,2,0,1,2\)

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=7 N_{t} \text { with } N_{0}=4 $$

In Problems \(37-44\), use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=\sqrt{a_{n}+1}\) and \(a_{0}=1\), find \(a_{11}\).

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=640 $$

Investigate the advantage of dimensionless variables. You are studying a population that obeys the discrete logistic equation. You know that \(b=\frac{1}{10} .\) One year you measure \(N_{t}=15\). The next year you measure that \(N_{t+1}=20\). What value of \(R_{0}\) is needed in the model to fit these data?

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{3}{2} N_{t} \text { with } N_{0}=32 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=2, x_{0}=0.2\)

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=a_{n}-\frac{1}{a_{n}}\) and \(a_{0}=3\), find \(a_{11}\).

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=1215 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=2, x_{0}=0.1\)

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=a_{n}+\frac{1}{a_{n}}\) and \(a_{0}=1\), find \(a_{13}\)

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{1}{3} N_{t} \text { with } N_{0}=2430 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=2, x_{0}=0.9\)

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=\sqrt{\sqrt{a_{n}}+1}\) and \(a_{0}=6\) find \(a_{12}\)

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{1}{5} N_{t} \text { with } N_{0}=31250 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=2, x_{0}=0\)

Use a spreadsheet to calculate the specified term of each recursively defined sequence. If \(a_{n+1}=a_{n}+\frac{1}{2}\) and \(a_{0}=1\) find \(a_{12}\).

In Problems , find the population sizes for \(t=0,1\), 2, \(\ldots, 5\) for each recursion. Then write the equation for \(N_{t}\) as a function of \(t\). $$ N_{t+1}=\frac{1}{4} N_{t} \text { with } N_{0}=8192 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=R_{0} x_{t}\left(1-x_{t}\right) $$ Compute \(x_{t}\) for \(t=0,1,2, \ldots, 20\) for the given values of \(r\) and \(x_{0}\), and graph \(x_{t}\) as a function of \(t .\) \(R_{0}=3.1, x_{0}=0.5\)