Chapter 7, Problem 23RE

a.

To determine

that what happen to the insect population in the absence of birds

Answer to Problem 23RE

If the initial population of insects is $0$ .Then the population will remain zero

Explanation of Solution

Given information : The question is

Population of birds and insects are modeled by equation

  $\begin{array}{l}\frac{dx}{dt}=0.4x-0.002xy\mathrm{......}\left(1\right)\\ \frac{dy}{dt}=-0.2y+0.000008xy\mathrm{......}\left(2\right)\end{array}$

Calculation :

Absence of birds implies that $y=0$

When $y=0$

  $\frac{dx}{dt}=0.4x-\left(1-0.000005x\right)$

This is a logistic equation

  $\frac{dx}{dt}=0.4x-\left(1-\frac{x}{200,000}\right)$

So, the carrying capacity is $200,000$

Therefore, in the absence of birds

If the initial population of insects is $0$ .Then the population will remain zero

b.

To determine

To find : the equilibrium equation and express their significance

Answer to Problem 23RE

There are three equilibrium solution

  $\begin{array}{l}x=0\&y=0\\ x=25000\&y=175\\ x=200,000\&y=0\end{array}$

Explanation of Solution

Given information : The question is

Population of birds and insects are modeled by equation

  $\begin{array}{l}\frac{dx}{dt}=0.4x-0.002xy\mathrm{......}\left(1\right)\\ \frac{dy}{dt}=-0.2y+0.000008xy\mathrm{......}\left(2\right)\end{array}$

Calculation :

The equilibrium solution is the set of values of $x\&y$ for which both $\frac{dx}{dt}$ and $\frac{dy}{dt}$ are zero

  $\begin{array}{l}\frac{dx}{dt}=0\\ 0.4x\left(1-0.000005x\right)-0.002xy=0\\ x=0\\ 0.4\left(1-0.000005x\right)-0.002xy=0\\ 0.4\left({10}^6-5x\right)-2000y=0\\ 4\times {10}^5-2x-2000y=0\\ 2x+2000y=4\times {10}^5\\ x+1000y=2\times {10}^5\\ \frac{dy}{dt}=0\\ -0.2y+0.000008xy=0\\ y=0\\ -2500+x=0\\ x=2500\\ x=0\&y=0\\ x=25000\&y=175\\ x=200,000\&y=0\end{array}$

c.

To determine

To describe : what eventually happens to the bird and insects population

Answer to Problem 23RE

The final population of birds will be $175$ and that of insects will be $25000$

Explanation of Solution

Given information : the question is

The phase trajectory starts with $100$ birds and $40,000$ insects

Calculation :

The eventually $x$ tends to $25000$ and $y$ tends to $175$

The equilibrium condition is met the population of the two species will remain the same

The final population of birds will be $175$ and that of insects will be $25000$