Week 9 - Lotka-Volterra competition model - Visualization of dynamics with complex eigenvalues

Consider a linear system \[ \dfrac{d \vec{\pmb\varepsilon}}{d t} = \mathcal{J}\vec{\pmb\varepsilon} \] where \(\vec{\pmb\varepsilon} = (\varepsilon_1, \varepsilon_2)^T\) and \(\mathcal{J} = \begin{pmatrix} -1 & 1\\ -2 & -1 \end{pmatrix}\). Or, we can write the linear system by two ODEs: \[\begin{align*} \dfrac{d \varepsilon_1}{d t} &= (-1)\times \varepsilon_1 + (1)\times\varepsilon_2\\ \dfrac{d \varepsilon_2}{d t} &= (-2)\times \varepsilon_1 + (-1)\times\varepsilon_2\\ \end{align*}\]

Note that the element in the first row and second column is positive.

##              time           e1            e2
##  [99996,]  9.9995 3.192479e-05 -6.443030e-05
##  [99997,]  9.9996 3.191516e-05 -6.443024e-05
##  [99998,]  9.9997 3.190552e-05 -6.443018e-05
##  [99999,]  9.9998 3.189589e-05 -6.443011e-05
## [100000,]  9.9999 3.188626e-05 -6.443005e-05
## [100001,] 10.0000 3.187663e-05 -6.442998e-05