Question

Two herds of vicious animals are fighting each other to the death. During the fight, the populations and of the two species can be modelled by the following system:

$\left|\begin{array}{c}\frac{dx}{dt}=-4y\\ \frac{dy}{dt}=-x\end{array}\right|$

1. What is the significance of the constants -4and -1in these equations? Which species has the more vicious (or more efficient) fighters.
2. Sketch a phase portrait for this system.
3. Who wins the fight (in the sense that some individuals of that species are left while the other herd is eradicated)? How does your answer depend on the initial populations?

Short answer

(a)The$y$ is the more vicious species and the constants determine the rate of decrease of populations.

(b) The graph is obtained and explained.

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

Step-by-step solution

(a) Given in the question.

The given system is written as follows:

$\left[\begin{array}{c}\frac{dx}{dt}\\ \frac{dy}{dt}\end{array}\right]=\left[\begin{array}{cc}0& -4\\ -1& 0\end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]$

The rate of death of the first species is 4 times that of the second thus the -4 and -1. For every y spices the rate of population decline of x goes down by 4.

Thus,$y$ is the more vicious species.

(b) Determine the graph of the given system.

The graph is plotted below:

(c) Determine the winner of the fight.

Species 1 that is$x$ species win if$\frac{y\left(0\right)}{x\left(0\right)}<\frac{1}{4}$, otherwise species 2 that isspecies win.

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