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Theory and Modern Applications

Table 2 Stationary states and their stability in model (1.3)

From: Impact of the fear and Allee effect on a Holling type II prey–predator model

Equilibrium

Existence

Type

\(B_{0}(0,0)\)

Always exists

Locally asymptotically stable;

 

\(a>\max \{a_{1},a_{3}\}\), Globally asymptotically stable.

\(B_{1}(x_{1},0)\)

\(0< a< a_{1}\)

\(0< a< a_{1}\) and \(0< b< b_{1}\), Unstable;

 

\(0< a< a_{1}\) and \(b>b_{1}\), Saddle.

\(B_{2}(x_{2},0)\)

\(0< a< a_{1}\)

\(0< a< a_{1}\) and \(0< b< b_{2}\), Saddle;

 

\(0< a< a_{1}\) and \(b>b_{2}\), Locally asymptotically stable.

\(B^{*}(x^{*},y^{*})\)

\(r>r_{1}\) and \(ce>bd_{2}\)

\(0< a< a_{2}\) and \(r_{1}< r\leq r_{2}\), Locally asymptotically stable;

 

\(0< a< a_{2}\) and \(r>r_{2}\) and \(f>f_{1}\), Locally asymptotically stable;

 

\(a\geq a_{2}\) and \(r>r_{1}\), Unstable;

 

\(0< a< a_{2}\) and \(r>r_{2}\) and \(0< f< f_{1}\), Unstable.