(A Predator-Prey Model) Consider the predator-prey model governed by the system
\begin{aligned}&x_{1}^{\prime}(t)=x_{1}(t)+x_{2}(t), \\&x_{2}^{\prime}(t)=-x_{1}(t)+x_{2}(t) .\end{aligned}If the initial populations are x_{1}(0)=x_{2}(0)=1000, determine the populations of the two species for t>0.
Here A=\left(\begin{array}{rr}1 & 1 \\ -1 & 1\end{array}\right) with characteristic equation \lambda^{2}-2 \lambda+2=0, complex roots \lambda_{1}=1+i and \lambda_{2}=1-i, and eigenvectors \mathbf{v}_{1}=\left(\begin{array}{l}1 \\ i\end{array}\right) and \mathbf{v}_{2}=\left(\begin{array}{c}1 \\ -i\end{array}\right) .^{\dagger} Then
C=\left(\begin{array}{cc}1 & 1 \\i & -i\end{array}\right) \quad C^{-1}=-\frac{1}{2 i}\left(\begin{array}{rr}-i & -1 \\-i & 1\end{array}\right)=\frac{1}{2}\left(\begin{array}{rr}1 & -i \\1 & i\end{array}\right) \quad D=\left(\begin{array}{cc}1+i & 0 \\0 & 1-i\end{array}\right)and
e^{D t}=\left(\begin{array}{cc}e^{(1+i) t} & 0 \\0 & e^{(1-i) t}\end{array}\right)Now, by Euler’s formula (see Appendix 3), e^{i t}=\cos t+i \sin t. Thus e^{(1+i) t}=e^{t} e^{i t}= e^{t}(\cos t+i \sin t) . Similarly, e^{(1-i) t}=e^{t} e^{-i t}=e^{t}(\cos t-i \sin t). Thus
e^{D t}=e^{t}\left(\begin{array}{cc}\cos t+i \sin t & 0 \\0 & \cos t-i \sin t\end{array}\right)and
\begin{aligned}e^{A t} &=C e^{D t} C^{-1}=\frac{e^{t}}{2}\left(\begin{array}{cc}1 & 1 \\i & -i\end{array}\right)\left(\begin{array}{cc}\cos t+i \sin t & 0 \\0 & \cos t-i \sin t\end{array}\right)\left(\begin{array}{cc}1 & -i \\1 & i\end{array}\right) \\&=\frac{e^{t}}{2}\left(\begin{array}{cc}1 & 1 \\i & -i\end{array}\right)\left(\begin{array}{cc}\cos t+i \sin t & -i \cos t+\sin t \\\cos t-i \sin t & i \cos t+\sin t\end{array}\right) \\&=\frac{e^{t}}{2}\left(\begin{array}{cc}2 \cos t & 2 \sin t \\-2 \sin t & 2 \cos t\end{array}\right)=e^{t}\left(\begin{array}{cc}\cos t & \sin t \\-\sin t & \cos t\end{array}\right)\end{aligned}Finally,
\mathbf{x}(t)=e^{A t} \mathbf{x}(0)=e^{t}\left(\begin{array}{rr}\cos t & \sin t \\-\sin t & \cos t\end{array}\right)\left(\begin{array}{l}1000 \\1000\end{array}\right)=\left(\begin{array}{l}1000 e^{t}(\cos t+\sin t) \\1000 e^{t}(\cos t-\sin t)\end{array}\right)The prey species is eliminated when 1000 e^{t}(\cos t-\sin t)=0 or when \sin t=\cos t. The first positive solution of this last equation is t=\pi / 4 \approx 0.7854 year \approx 9.4 months.