(A Model of Species Cooperation-Symbiosis) Consider the symbiotic model governed by the system
\begin{aligned}&x_{1}{ }^{\prime}(t)=-\frac{1}{2} x_{1}(t)+x_{2}(t) \\&x_{2}{ }^{\prime}(t)=\frac{1}{4} x_{1}(t)-\frac{1}{2} x_{2}(t)\end{aligned}In this model the population of each species increases proportionally to the population of the other and decreases proportionally to its own population. Suppose that x_{1}(0)= 200 and x_{2}(0)=500. Determine the population of each species for t>0.
Here A=\left(\begin{array}{rr}-\frac{1}{2} & 1 \\ \frac{1}{4} & -\frac{1}{2}\end{array}\right) with eigenvalues \lambda_{1}=0 and \lambda_{2}=-1 and corresponding eigenvectors \mathbf{v}_{1}=\left(\begin{array}{l}2 \\ 1\end{array}\right) and \mathbf{v}_{2}=\left(\begin{array}{r}2 \\ -1\end{array}\right). Then
C=\left(\begin{array}{rr}2 & 2 \\1 & -1\end{array}\right), \quad C^{-1}=-\frac{1}{4}\left(\begin{array}{rr}-1 & -2 \\-1 & 2\end{array}\right), \quad D=\left(\begin{array}{rr}0 & 0 \\0 & -1\end{array}\right)and \begin{aligned}e^{D t} &=\left(\begin{array}{cc}e^{0 t} & 0 \\0 & e^{-t}\end{array}\right)=\left(\begin{array}{cc}1 & 0 \\0 & e^{-t}\end{array}\right) \text {. Thus } \\e^{A t} &=-\frac{1}{4}\left(\begin{array}{rr}2 & 2 \\1 & -1\end{array}\right)\left(\begin{array}{cc}1 & 0 \\0 & e^{-t}\end{array}\right)\left(\begin{array}{rr}-1 & -2 \\-1 & 2\end{array}\right) \\&=-\frac{1}{4}\left(\begin{array}{rr}2 & 2 \\1 & -1\end{array}\right)\left(\begin{array}{ll}-1 & -2 \\-e^{-t} & 2 e^{-t}\end{array}\right) \\&=-\frac{1}{4}\left(\begin{array}{ll}-2 & -2 e^{-t} & -4+4 e^{-t} \\-1+ & e^{-t} & -2-2 e^{-t}\end{array}\right)\end{aligned}
and
\begin{aligned}\mathbf{x}(t)=e^{A t} \mathbf{x}(0) &=-\frac{1}{4}\left(\begin{array}{ll}-2-2 e^{-t} & -4+4 e^{-t} \\-1+e^{-t} & -2-2 e^{-t}\end{array}\right)\left(\begin{array}{l}200 \\500\end{array}\right) \\&=-\frac{1}{4}\left(\begin{array}{l}-2400+1600 e^{-t} \\-1200-800 e^{-t}\end{array}\right) \\&=\left(\begin{array}{l}600-400 e^{-t} \\300+200 e^{-t}\end{array}\right)\end{aligned}We have e^{-t} \rightarrow 0 as t \rightarrow \infty. This means that as time goes on, the two cooperating species approach the equilibrium populations 600 and 300 , respectively. Neither population is eliminated.