91影视

The given system is written as follows:

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

Hence, this is a symbiotic system and$k$ denotes rate of increase in one population because of the other.

The Eigen values of the given system as follows:

$$\begin{gathered} \text{det}\left|\begin{array}{c}\left[\begin{array}{cc}-1& k\\ 1& -4\end{array}\right]\text{-}\end{array}\left[\begin{array}{cc}位& 0\\ 0& 位\end{array}\right]\right|\text{=0} \\ \text{det}\left[\begin{array}{cc}-1-位& k\\ & -4-位\end{array}\right]\text{=0} \\ {位}^2+5位+4-k=0 \\ 位=\frac{-5卤\sqrt{25-4\left(4-k\right)}}{2} \end{gathered}$$

As observe that there are two negative Eigen values if $k<4$and one negative, one positive if$k>4$ .

Hence, there are two negative Eigen values if $k<4$and one negative, one positive if $k>4$.

The graph is plotted below:

Case 1: for $k<4$,

Case 2: for $k>4$,

Hence,$k$denotes rate of increase in one population because of the other. From above two graphs, the first graph show the future of population is decrease when it is less than 4 and the second graph show that population increases when more than 4.