Chapter 2

Find the exponential growth equation for a population that quadruples in size every unit of time and that has five 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=20, N_{0}=10

Find the exponential growth equation for a population that quadruples in size every unit of time and that has 17 individuals at time 0 .

Assume that the discrete logistic equation is used with parameters \(R\) and \(K .\) Write the equation in the canonical form \(x_{t+1}=r x_{t}\left(1-x_{t}\right)\), and determine \(r\) and \(x_{t}\) in terms of \(R, K\), and R=1, K=10

Find the recursion for a population that doubles in size every unit of time and that has 20 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 doubles in size every unit of time and that has 37 individuals at time \(0 .\)

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 triples in size every unit of time and that has 10 individuals at time \(0 .\)

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ 1,2,4,8,16, \ldots $$

Assume that the discrete logistic equation is used with parameters \(R\) and \(K .\) Write the equation in the canonical form \(x_{t+1}=r x_{t}\left(1-x_{t}\right)\), and determine \(r\) and \(x_{t}\) in terms of \(R, K\), and R=2, K=20

Find the recursion for a population that triples in size every unit of time and that has 84 individuals at time \(0 .\)

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

Assume that the discrete logistic equation is used with parameters \(R\) and \(K .\) Write the equation in the canonical form \(x_{t+1}=r x_{t}\left(1-x_{t}\right)\), and determine \(r\) and \(x_{t}\) in terms of \(R, K\), and \(R=2.5, K=30\)

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

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

Assume that the discrete logistic equation is used with parameters \(R\) and \(K .\) Write the equation in the canonical form \(x_{t+1}=r x_{t}\left(1-x_{t}\right)\), and determine \(r\) and \(x_{t}\) in terms of \(R, K\), and \(R=2.5, K=50\)

. Find the recursion for a population that quadruples in size every unit of time and that has 62 individuals at time \(0 .\)

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

In Problems 31-34, graph the functions \(f(x)=a^{x}, x \in[0, \infty)\), and \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=2 $$

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

Investigate the advantage of dimensionless variables. To quantify the spatial structure of a plant population, it might be convenient to introduce a characteristic length scale. This length scale might be characterized by the average dispersal distance of the plant under study. Assume that the characteristic length scale is denoted by \(L .\) Denote by \(x\) the distance of seeds from their source. Define \(z=x / L .\) Find \(z\) if \(x=100 \mathrm{~cm}\) and \(L=50 \mathrm{~cm}\), and show that \(z\) has the same value if \(x\) and \(L\) are measured in units of meters instead.

In Problems graph the functions \(f(x)=a^{x}, x \in[0, \infty)\), and \(N_{t}=R^{t}, t \in \mathbf{N}\), 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 \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ 2,-4,6,-8,10, \ldots $$

Investigate the advantage of dimensionless variables. Suppose a bacterium divides every 20 minutes, which we call the characteristic time scale and denote by \(T\). Let \(t\) be the time elapsed since the beginning of an experiment that involves this bacterium. Define \(z=t / T .\) Find \(z\) if \(t=120\) minutes, and show that \(z\) has the same value if \(t\) and \(T\) are measured in units of hours instead.

In Problems graph the functions \(f(x)=a^{x}, x \in[0, \infty)\), and \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=1 / 2 $$

Find an expression for \(a_{n}\) on the basis of the values of \(a_{0}, a_{1}, a_{2}, \ldots .\) $$ -\frac{1}{2}, \frac{1}{3},-\frac{1}{4}, \frac{1}{5},-\frac{1}{6}, \ldots $$

Investigate the advantage of dimensionless variables. The time to the most recent common ancestor of a pair of individuals from a randomly mating population depends on the population size. Let \(t\) denote the time, measured in units of generations, to the most recent common ancestor, and let \(T\) be equal to \(N\) generations, where \(N\) is the population size of the randomly mating population. Define \(z=t / T .\) Show that \(z\) is dimensionless and that the value of \(z\) does not change, regardless of whether \(t\) and \(T\) are measured in units of generations or in units of, say, years. (Assume that one generation is equal to \(n\) years.)

In Problems graph the functions \(f(x)=a^{x}, x \in[0, \infty)\), and \(N_{t}=R^{t}, t \in \mathbf{N}\), together in one coordinate system for the indicated values of a and \(R\). $$ a=R=1 / 3 $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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=2, x_{0}=0.2

In Problems 35-46, find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion. $$ N_{t+1}=2 N_{t} \text { with } N_{0}=3 $$

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

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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=2, x_{0}=0.1

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

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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=2, x_{0}=0.9

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

In Problems \(37-44\), write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{1}{n+2} $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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=2, x_{0}=0

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

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{2}{n+1} $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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=3.1, x_{0}=0.5

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

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{n}{n+1} $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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 \(t\). Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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 \(t\). \(r=3.1, x_{0}=0.1\)

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

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{2 n}{n+2} $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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 \(t\). \(r=3.1, x_{0}=0.9\)

In Problems , find the population sizes for \(t=0,1,2, \ldots, 5\) for each recursion.$$ N_{t+1}=\frac{1}{2} N_{t} \text { with } N_{0}=1024 $$

Write the first five terms of the sequence \(\left\\{a_{n}\right\\}\), \(n=0,1,2,3, \ldots\), and find \(\lim _{n \rightarrow \infty} a_{n}\). $$ a_{n}=\frac{1}{n^{2}+1} $$

Investigate the behavior of the discrete logistic equation $$ x_{t+1}=r x_{t}\left(1-x_{t}\right) $$ 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 \(t\). \(r=3.1, x_{0}=0\)