Question

Repeat Exercise 47 for the system

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

where$\mathit{k}$ is a positive constant.

Short answer

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

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

(c) Graph for the cases in the explanation. 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.

Step-by-step solution

(a) Given in the question.

The given system is written as follows:

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

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

(b) Calculate the Eigen values.

The Eigen values of the given system as follows:

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

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 $k>4$.

(c) Graph of the given system

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.