Chapter 9.1, Problem 45E

Two herds of vicious animals are fighting each other to the death. During the fight, the populations $x(t)$ and $y(t)$ of the two species can be modeled by the following system ${:}^2$
$\left|\begin{array}{c}\frac{dx}{dt}=-4y\\ \frac{dy}{dt}=-x\end{array}\right|$
a. What is the significance of the constants −4 and −1 in these equations? Which species has the more vicious (or more efficient) fighters?
b. Sketch a phase portrait for this system.
c. 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?
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²This is the simplest in a series of combat models developed by by F. W. Lanchester during World War I (F. W. Lanchester, Aircraft in Warfare, the Dwan of the Fourth Arm, Tiptree, Constable and Co., Ltd., 1916).