Question

The interaction of two populations of animals is modelled by the differential equations:

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

For some positive constant $\mathit{k}$.

1. What kind of interaction do we observe? What is the pratical significance of the constant $\mathit{k}$?
2. Find the eigenvalues of the coefficient matris of the system. What can you say about the signs of these eigen values? How does your answer depend on the value of the constant$\mathit{k}$ ?
3. For each case you discussed in part (b), sketch a rough phase portrait. What does each phase portrait tell you about the future of the two populations?

Short answer

(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 $k<2$and one negative, one positive if $k>2$.

(c) Graph for the cases in the explanation. The first graph show the future of population is decrease when it is less than 2 and the second graph show that population increases when more than 2.

Step-by-step solution

(a) Given in the question.

The given system is written as follows:

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

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:

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

As observe that there are two negative Eigen values if $k<2$and one negative, one positive if $k>2$.

Hence, there are two negative Eigen values if $k<2$and one negative, one positive if $k>2$.

(c) Graph of the given system.

The graph is plotted below:

Case 1: for $k<2$,

Case 2: for $k>2$

Thus,$k$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 2 and the second graph show that population increases when more than 2.