Theory and Modern Applications
Table 4 Equilibria \(E_{i}, i=0,1,2\), of system (4) with \(a>d\)
From: Stability and bifurcation in a Holling type II predator–prey model with Allee effect and time delay
\(Q_{0}\leq 0\) | \(0< A< A_{1}\) | \(E_{0}\) LAS, \(E_{1}\) LAS, \(E_{2}\) unstable | ||
\(Q_{0}>0\) | \(x^{*}\geq \frac{a-d}{b}\) | \(A<\frac{b(x^{*})^{2}}{d}\) | \(0< A< A_{1}\) | \(E_{0}\) LAS, \(E_{1}\) LAS, \(E_{2}\) unstable |
\(A\geq \frac{b(x^{*})^{2}}{d}\) | \(0< A< A_{1}\) | \(E_{0}\) LAS, \(E_{1}\) unstable, \(E_{2}\) unstable | ||
\(x^{*}<\frac{a-d}{b}, x^{*}\neq x_{0}^{*}\) | \(A<\frac{b(x^{*})^{2}}{d}\) | \(0< A< A_{2}\) | \(E_{0}\) LAS, \(E_{1}\) unstable, \(E_{2}\) unstable | |
\(A_{2}< A< A_{1}\) | \(E_{0}\) LAS, \(E_{1}\) LAS, \(E_{2}\) unstable | |||
\(A\geq \frac{b(x^{*})^{2}}{d}\) | \(0< A< A_{1}\) | \(E_{0}\) LAS, \(E_{1}\) unstable, \(E_{2}\) unstable | ||
\(x^{*}=x_{0}^{*}\) | \(0< A< A_{1}\) | \(E_{0}\) LAS, \(E_{1}\) unstable, \(E_{2}\) unstable | ||
\(A> A_{1}\) | \(E_{0}\) GAS, \(E_{1}\) and \(E_{2}\) do not exist | |||