Chapter 1.10, Problem 38E

a

To determine

To calculate: To find the number of butterflies after migration and after reproduction using the given values

Answer to Problem 38E

There are $160-48=112$ butterflies that started on the second island and 38 that started on the second. Now the 112 reproduce, giving the total number of butterflies on the first island as $y_1=112\times 2+38=262$

Explanation of Solution

Given information: Each year, 20% of the butterflies from the first island fly to the second and 30% of the butterflies from the second fly to the first.

Calculation:

Let us consider the following situation. There are two nearby islands which have populations of butterflies, with $x_t$ on the first island and $y_t$ on the second.

Let $p_t$ represents the fraction of butterflies on the first island.

Consider the first island. In the beginning there are 100 butterflies out of which 20 migrate to the second island and 30 migrate from the second island to the first island.

Hence, there are $100-20=80$ butterflies that started on the first island and 30 that started on the second. Now the 80 reproduce, giving the total number of butterflies on the first island as $x_1=80\times 2+30=190$

Now consider the second island. In the beginning there are 100 butterflies out of which 30 migrate to the first island and 20 migrate from the first island to the second island.

Hence, there are $100-30=70$ butterflies that started on the second island and 20 that started on the second. Now the 70 reproduce, giving the total number of butterflies on the first island as $y_1=70\times 20+2=160$

Again consider the first island. Now there are 190 butterflies out of which 38 migrate to the second island and 33 migrate from the second island to the first island.

Hence, there are $190-38=152$ butterflies that started on the first island and 33 that started on the second. Now the 152 reproduce, giving the total number of butterflies on the first island as $x_2=152\times 2+33=337$

Now consider the second island- In the beginning there are 110 butterflies out of which 33 migrate to the first island and 38 migrate from the first island to the second island.

Hence, there are $160-48=112$ butterflies that started on the second island and 38 that started on the second. Now the 112 reproduce, giving the total number of butterflies on the first island as $y_1=112\times 2+38=262$

b

To determine

To calculate: To find the equations of $x_t$ and $y_t$ using the given information

Answer to Problem 38E

  $y_{t+1}=1.4y_t+0.2x_t$

Explanation of Solution

Given information: Each year, 20% of the butterflies from the first island fly to the second and 30% of the butterflies from the second fly to the first.

Calculation:

Now the equation for $x_{t+1}$ in terms of $x_t$ and $y_t$ will be as follows:

  $\begin{array}{l}{x}_{t+1}=2\left(x_t-\frac{20}{100}{x}_t\right)+\frac{30}{100}{y}_t\\ =2\left(x_t-0.2x_t+0.3y_t\right)\\ x_{t+1}=1.6x_t+0.3y_t\end{array}$

Also, the equation for $y_{t+1}$ in terms of $x_t$ and $y_t$ will be as follows:

  $\begin{array}{l}{y}_{t+1}=2\left(y_t-\frac{30}{100}{y}_t\right)+\frac{20}{100}{x}_t\\ =2\left(y_t-0.3x_t\right)+\left(0.2x_t\right)\\ y_{t+1}=1.4y_t+0.2x_t\end{array}$

c

To determine

To calculate: To find the discrete-time dynamical system for $p_{t+1}$ in terms of $p_t$ using the given values

Answer to Problem 38E

  $p_{t+1}=\frac{1.6p_t+0.3\left(1-p_t\right)}{1.8p_t+1.7\left(1-p_t\right)}$

Explanation of Solution

Given information: Each year, 20% of the butterflies from the first island fly to the second and 30% of the butterflies from the second fly to the first.

Calculation:

To find the discrete-time dynamical system for $p_{t+1}$ in terms of $p_t$ consider

  $\begin{array}{l}{p}_{t+1}=\frac{x_{t+1}}{x_{t+1}+y_{t+1}}\\ =\frac{1.6x_t+0.3y_t}{\left(1.6x_t+0.3y_t\right)+\left(1.4y_t+0.2x_1\right)}\\ =\frac{1.6x_t+0.3y_t}{1.8x_t+1.7y_t}\end{array}$

  $\begin{array}{l}=\frac{1.6\frac{x_t}{x_t+y_t}+0.3\frac{y_t}{x_t+y_t}}{1.8\frac{x_t}{x_t+y_t}+1.7\frac{y_t}{x_t+y_t}}\\ p_{t+1}=\frac{1.6p_t+0.3\left(1-p_t\right)}{1.8p_t+1.7\left(1-p_t\right)}\end{array}$

d

To determine

To calculate: To find the equilibrium of $p^{•}$ using the given information

Answer to Problem 38E

Using the quadratic formula, we get

  $p^{•}=0.6457$

This is larger because the butterflies from the first island reproduce.

Explanation of Solution

Given information: Each year, 20% of the butterflies from the first island fly to the second and 30% of the butterflies from the second fly to the first.

Calculation:

To find the equilibrium $p^{•}$ consider,

  $\begin{array}{l}{p}^{•}=\frac{1.6p^{•}+0.3\left(1-p^{•}\right)}{1.8p^{•}+1.7\left(1-p^{•}\right)}\\ p^{•}=\frac{0.3+1.3p^{•}}{1.7+0.1p^{•}}\\ 0.1{\left(p^{•}\right)}^2-0.4p^{•}-0.3=0\\ {\left(p^{•}\right)}^2+4{\left(p^{•}\right)}^2-3=0\end{array}$

Now using the quadratic formula, we get

  $p^{•}=0.6457$

This is larger than the result in exercise 36 because the butterflies from the first island reproduce.

e

To determine

To calculate: To draw the graph using the obtained values

Answer to Problem 38E

Hence the graph is drawn using the obtained values.

Explanation of Solution

Given information: Each year, 20% of the butterflies from the first island fly to the second and 30% of the butterflies from the second fly to the first.

Calculation:

The graph along with the cobweb is shown below