Fig. 3: Eigenvalue distributions.
Eigenvalues \(\lambda _{\mathrm{i}}\) of the Jacobian or community matrix in the complex plane, for four different values of P. Imaginary parts Im(λi) versus real parts Re(λi) are plotted. The “bulk” eigenvalues (red dots) are contained in an ellipse which centers close to the mean equilibrium level, i.e., \(- N_{\mathrm{i}}^ \ast\) plotted as blue circles. Stability becomes stronger (ellipse shifts to the left) as the proportion P of mutualistic interactions increases in otherwise exploitative communities. When P > 0.4, the largest outlier eigenvalue is \(\Lambda = - 1\) (see Eq. (2)). For P = 0.8, it was necessary to stretch the scale for the x-axis and it is different to the other panels. Parameters: n = 100 species; interaction variability σ = 0.02; connectance C = 0.7; growth rates \(r_{\mathrm{i}} = 1\), as in Fig. 2.
