Theory and Modern Applications
Table 1 Equilibria of system (1.7) in finite planes
Possibilities of parameters | Location of equilibria | Types and stability | |
|---|---|---|---|
\(0< c< c^{**}\) | – | \(E_{0}\), \(E_{1}^{*}\) | \(E_{0}\) saddle, \(E_{1}^{*}\) stable node |
\(c=c^{**}\) | \(\frac{b}{a}-e-1>0\) | \(E_{0}\), \(E_{1}^{*}\) | \(E_{0}\) saddle node, \(E_{1}^{*}\) stable node |
\(\frac{b}{a}-e-1=0\) | \(E_{0}\) | \(E_{0}\) stable node | |
\(\frac{b}{a}-e-1<0\) | \(E_{0}\) | \(E_{0}\) saddle node | |
\(c^{**}< c< c^{*}\) | \(\frac{b}{a}-e-1>0\) | \(E_{0}\), \(E_{1}^{*}\), \(E_{2}^{*}\) | \(E_{0}\) stable node, \(E_{1}^{*}\) stable node, \(E_{2}^{*}\) saddle |
\(\frac{b}{a}-e-1<0\) | \(E_{0}\) | \(E_{0}\) stable node | |
\(c=c^{*}\) | \(\frac{b}{a}-e-1>0\) | \(E_{0}\), \(E_{3}^{*}\) | \(E_{0}\) stable node, \(E_{3}^{*}\) saddle node |
\(c>c^{*}\) | – | \(E_{0}\) | \(E_{0}\) stable node |