MATH1013 Quizzes

Quiz 5: Logistic difference equations
Question 1 Questions
Which of the following are logistic difference equations ?
 a) ${X}_{n+1}=0.1{X}_{n}$ b) ${X}_{n+1}=2\left(1-{X}_{n}\right){X}_{n}$ c) ${X}_{n+1}=\left(2+3{X}_{n}\right){X}_{n}$ d) ${X}_{n+2}-2{X}_{n+1}+3{X}_{n}=20$

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!
1. False This is a first order difference equation.
2. True
3. True
4. False This is a second order difference equation.
Why are logistic difference equations used for modelling populations ?
 a) Logistic difference equations model populations whose growth rate is constant. b) Logistic difference equations model populations whose decay rate is constant. c) Logistic difference equations model populations whose growth rate changes depending on the size of the population. d) Logistic difference equations model populations whose behaviour is chaotic.

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 ?
 a) b) c) d)

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 ${X}_{n+1}=a\left(1-{X}_{n}\right){X}_{n}\phantom{\rule{0.3em}{0ex}}?$
 a) The population approaches a stable value. It can do so by approaching from one or both sides. b) The population alternates between 2 or more fixed values. It can do so by approaching in one direction or from opposite sides in an alternating manner c) The population will eventually visit everywhere in the interval $0$ to $1\phantom{\rule{0.3em}{0ex}}.$ d) The population will grow exponentially.

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
${P}_{n+1}=\left(1.4-0.008{P}_{n}\right){P}_{n}\phantom{\rule{0.3em}{0ex}}.$
 a) $500$ b) $50$ c) $175$ d) $1750$

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 ?
 a) ${P}_{n+1}=\left(0.8-0.0002{P}_{n}\right){P}_{n}$ b) ${P}_{n+1}=\left(1.6-0.0002{P}_{n}\right){P}_{n}$ c) ${P}_{n+1}=0.8\left(1-{P}_{n}\right){P}_{n}$ d) ${P}_{n+1}=\left(1.8-0.0002{P}_{n}\right){P}_{n}$

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.
 a) $400$ b) $900$ c) $4000$ d) $9000$

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 ?
 a) ${P}_{n+1}=\left(2.8-0.006{P}_{n}\right){P}_{n}$ b) ${P}_{n+1}=\left(2.6-0.004{P}_{n}\right){P}_{n}$ c) ${P}_{n+1}=\left(1.8-0.006{P}_{n}\right){P}_{n}$ d) ${P}_{n+1}=\left(1.8-0.004{P}_{n}\right){P}_{n}$

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.
 a) $4667$ b) $467$ c) $3000$ d) $300$

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 ?
 a) For $a=0.5$ there are two attractors, $0.5130$ and $0.7995\phantom{\rule{0.3em}{0ex}}.$ b) For $a=1.5$ and $a=2.5$ there is a single attractor, $0\phantom{\rule{0.3em}{0ex}}.$ c) For $a=3.2$ there is a single attractor, $0\phantom{\rule{0.3em}{0ex}}.$ d) For $a=3.45$ there are 4 attractors.

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!