Question

Repeat Exercise 45 for the system:

$\left|\begin{array}{c}\frac{dx}{dt}=-py\\ \frac{dy}{dt}=-qx\end{array}\right|$

Where$p$ and$q$ are two positive constants.

Short answer

(a) Species is the more vicious species and the constant is used to determine the rate of decreasing of the populations.

(b) The graph in the explanation is obtained.

(c) The species 1 that is $x$species win.

Step-by-step solution

Determine the species that is more vicious

The given system is written as follows:

$\left[\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right]=\left[\begin{array}{cc}0& -p\\ -q& 0\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

The rate of death of the first species is$\frac{p}{q}$ for every $y$and the rate of death of the second species is$\frac{q}{p}$ for every$x$ .

(b) Graph of the given system.

The graph is plotted below:

(c) Winner of the fight.

Species 1 that is species win if$\frac{y\left(0\right)}{x\left(0\right)}<\frac{q}{p}$ , otherwise species 2 that is$y$ species win.

Hence, species 1 that is$x$ species win.