Which of the following are logistic difference equations ?

For example, choice (a) should be False.

For example, choice (b) should be True.

For example, choice (c) should be True.

For example, choice (d) should be False.

*There is at least one mistake.*

For example, choice (a) should be False.

This is a first order
difference equation.

*There is at least one mistake.*

For example, choice (b) should be True.

*There is at least one mistake.*

For example, choice (c) should be True.

*There is at least one mistake.*

For example, choice (d) should be False.

This
is a second order difference equation.

*Correct!*

*False*This is a first order difference equation.*True**True**False*This is a second order difference equation.

Why are logistic difference equations used for modelling populations ?

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

*Choice (d) is incorrect*

Consider the logistic difference equation
${X}_{n+1}=a\left(1-{X}_{n}\right){X}_{n}\phantom{\rule{0.3em}{0ex}}.$
Which of the graphs below would represent a population dying out ?

*Choice (a) is correct!*

No matter what starting point is chosen the population goes to zero.

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Which of the statements below describes the periodic behaviour
of a population which satisfies the logistic difference equation
${X}_{n+1}=a\left(1-{X}_{n}\right){X}_{n}\phantom{\rule{0.3em}{0ex}}?$

*Choice (a) is incorrect*

*Choice (b) is correct!*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Find the positive equilibrium value for the logistic difference equation

$${P}_{n+1}=\left(1.4-0.008{P}_{n}\right){P}_{n}\phantom{\rule{0.3em}{0ex}}.$$

*Choice (a) is incorrect*

*Choice (b) is correct!*

We need to solve
$$\begin{array}{llll}\hfill & X=\left(1.4-0.008X\right)X,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & 0.008{X}^{2}-0.4X=0,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & X\left(X-50\right)=0.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
So $X=50\phantom{\rule{0.3em}{0ex}}.$

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Questions 6 and 7 use the same information.

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual rate of change of the population is $0.8-0.0002{P}_{n}\phantom{\rule{0.3em}{0ex}},$ that is, it is partly constant and partly dependent upon the size of the population.

What is the logistic difference equation that models this population ?

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual rate of change of the population is $0.8-0.0002{P}_{n}\phantom{\rule{0.3em}{0ex}},$ that is, it is partly constant and partly dependent upon the size of the population.

What is the logistic difference equation that models this population ?

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*

$$\begin{array}{llll}\hfill \frac{{P}_{n+1}-{P}_{n}}{{P}_{n}}& =0.8-0.0002{P}_{n},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {P}_{n+1}-{P}_{n}& =\left(0.8-0.0002{P}_{n}\right){P}_{n},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {P}_{n+1}& =\left(1.8-0.0002{P}_{n}\right){P}_{n}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

Questions 6 and 7 use the same information.

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual rate of change of the population is $0.8-0.0002{P}_{n}\phantom{\rule{0.3em}{0ex}},$ that is, it is partly constant and partly dependent upon the size of the population.

Determine the positive equilibrium value for this population.

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual rate of change of the population is $0.8-0.0002{P}_{n}\phantom{\rule{0.3em}{0ex}},$ that is, it is partly constant and partly dependent upon the size of the population.

Determine the positive equilibrium value for this population.

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is correct!*

We need to solve
$$\begin{array}{llll}\hfill & X=\left(1.8-0.0002X\right)X,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & 0.0002{X}^{2}-0.8X=0,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & X\left(X-4000\right)=0.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
So $X=4000\phantom{\rule{0.3em}{0ex}}.$

*Choice (d) is incorrect*

Questions 8 and 9 use the same information.

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual birth rate is $2.2-0.001{P}_{n}$ and the death rate is $0.4+0.005{P}_{n}\phantom{\rule{0.3em}{0ex}}.$

What is the logistic difference equation that models this population ?

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual birth rate is $2.2-0.001{P}_{n}$ and the death rate is $0.4+0.005{P}_{n}\phantom{\rule{0.3em}{0ex}}.$

What is the logistic difference equation that models this population ?

*Choice (a) is correct!*

The growth rate is $2.2-0.001{P}_{n}-\left(0.4+0.005{P}_{n}\right)=1.8-0.006{P}_{n}$
$$\begin{array}{llll}\hfill \frac{{P}_{n+1}-{P}_{n}}{{P}_{n}}& =1.8-0.006{P}_{n},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {P}_{n+1}-{P}_{n}& =\left(1.8-0.006{P}_{n}\right){P}_{n},\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {P}_{n+1}& =\left(2.8-0.006{P}_{n}\right){P}_{n}.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is incorrect*

Questions 8 and 9 use the same information.

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual birth rate is $2.2-0.001{P}_{n}$ and the death rate is $0.4+0.005{P}_{n}\phantom{\rule{0.3em}{0ex}}.$

Determine the positive equilibrium value for this population.

The size of a population at the beginning of the $n$th year is given by ${P}_{n}\phantom{\rule{0.3em}{0ex}}.$ Suppose that the annual birth rate is $2.2-0.001{P}_{n}$ and the death rate is $0.4+0.005{P}_{n}\phantom{\rule{0.3em}{0ex}}.$

Determine the positive equilibrium value for this population.

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*

We need to solve
$$\begin{array}{llll}\hfill & X=\left(2.8-0.006X\right)X,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & 0.006{X}^{2}-1.8X=0,\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & X\left(X-300\right)=0.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$$
So $X=300\phantom{\rule{0.3em}{0ex}}.$

Consider the function $f\left(x\right)=a\left(1-x\right)x\phantom{\rule{0.3em}{0ex}}.$
Which of the following statements are true ?

*Choice (a) is incorrect*

*Choice (b) is incorrect*

*Choice (c) is incorrect*

*Choice (d) is correct!*