(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 $t>0$ .

Question

(A Predator-Prey Model) Consider the predator-prey model governed by the system

[latex]\begin{aligned}&x_{1}^{\prime}(t)=x_{1}(t)+x_{2}(t), \&x_{2}^{\prime}(t)=-x_{1}(t)+x_{2}(t) .\end{aligned}[/latex]

If the initial populations are [latex]x_{1}(0)=x_{2}(0)=1000[/latex], determine the populations of the two species for [latex]t>0[/latex].

Accepted Answer

Here [latex]A=\left(\begin{array}{rr}1 & 1 \ -1 & 1\end{array} ight)[/latex] with characteristic equation [latex]\lambda^{2}-2 \lambda+2=0[/latex], complex roots [latex]\lambda_{1}=1+i[/latex] and [latex]\lambda_{2}=1-i[/latex], and eigenvectors [latex]\mathbf{v}_{1}=\left(\begin{array}{l}1 \ i\end{array} ight)[/latex] and [latex]\mathbf{v}_{2}=\left(\begin{array}{c}1 \ -i\end{array} ight) .^{\dagger}[/latex] Then

[latex]C=\left(\begin{array}{cc}1 & 1 \i & -i\end{array} ight) \quad C^{-1}=-\frac{1}{2 i}\left(\begin{array}{rr}-i & -1 \-i & 1\end{array} ight)=\frac{1}{2}\left(\begin{array}{rr}1 & -i \1 & i\end{array} ight) \quad D=\left(\begin{array}{cc}1+i & 0 \0 & 1-i\end{array} ight)[/latex]

and

[latex]e^{D t}=\left(\begin{array}{cc}e^{(1+i) t} & 0 \0 & e^{(1-i) t}\end{array} ight)[/latex]

Now, by Euler's formula (see Appendix 3), [latex]e^{i t}=\cos t+i \sin t[/latex]. Thus [latex]e^{(1+i) t}=e^{t} e^{i t}= e^{t}(\cos t+i \sin t) .[/latex] Similarly, [latex]e^{(1-i) t}=e^{t} e^{-i t}=e^{t}(\cos t-i \sin t)[/latex]. Thus

[latex]e^{D t}=e^{t}\left(\begin{array}{cc}\cos t+i \sin t & 0 \0 & \cos t-i \sin t\end{array} ight)[/latex]

and

[latex]\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} ight)\left(\begin{array}{cc}\cos t+i \sin t & 0 \0 & \cos t-i \sin t\end{array} ight)\left(\begin{array}{cc}1 & -i \1 & i\end{array} ight) \&=\frac{e^{t}}{2}\left(\begin{array}{cc}1 & 1 \i & -i\end{array} ight)\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} ight) \&=\frac{e^{t}}{2}\left(\begin{array}{cc}2 \cos t & 2 \sin t \-2 \sin t & 2 \cos t\end{array} ight)=e^{t}\left(\begin{array}{cc}\cos t & \sin t \-\sin t & \cos t\end{array} ight)\end{aligned}[/latex]

Finally,

[latex]\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} ight)\left(\begin{array}{l}1000 \1000\end{array} ight)=\left(\begin{array}{l}1000 e^{t}(\cos t+\sin t) \1000 e^{t}(\cos t-\sin t)\end{array} ight)[/latex]

The prey species is eliminated when [latex]1000 e^{t}(\cos t-\sin t)=0[/latex] or when [latex]\sin t=\cos t[/latex]. The first positive solution of this last equation is [latex]t=\pi / 4 \approx 0.7854[/latex] year [latex]\approx 9.4[/latex] months.

Question 12.6.3: (A Predator-Prey Model) Consider the predator-prey model gov...

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