Fig. 8

Averaged phase planes. Averaged phase planes, corresponding to system (6a)–(6b), with superimposed trajectories of the fast-slow system, for \(l=0.94\) and three different values of \(c_{\mathrm{tot}}\) as in Fig. 6. Throughout this figure, the curve of HC bifurcations is green, the nullclines \(ca_{\mathrm{av}}\), \(na_{\mathrm{av}}\) are blue and red, respectively, and the yellow symbols mark the fixed points of (6a)–(6b). The oscillatory trajectories (black) from Fig. 6A–C are projected to \((na_{i}, ca_{i})\)-space in (A), (B), and the lower two panels in this figure, respectively. For (A) \(c_{\mathrm{tot}}=0.6\) and (B) \(c_{\mathrm{tot}}=0.7\), there are three fixed points of (6a)–(6b) above HC where the average nullclines intersect, namely, \(\mathrm{FP}_{\mathrm{av}1}\), \(\mathrm{FP}_{\mathrm{av}3}\), which are stable, and \(\mathrm{FP}_{\mathrm{av}2}\), which is unstable. Not all fixed points are visible since some of them lie at larger \(( na_{i}, ca_{i})\) values than those shown. (C) and (D) Enlarged views of the phase plane for \(c_{\mathrm{tot}}=0.74\) showing that both \(\mathrm{FP}_{\mathrm{av}1}\) and \(\mathrm{FP}_{\mathrm{av}2}\) vanish and the only average fixed point left (\(\mathrm{FP}_{\mathrm{av}3}\), with high \(na_{i}\) and \(ca_{i}\)) is stable. Specifically, panel (C) (resp. (D)) shows the enlarged lower left (resp. upper right) region of the phase plane; note differences in values on the axes