Bifurcation and chaos in a discrete Holling–Tanner model with Beddington–DeAngelis functional response

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 Properties of the fixed point \(B(u^{},v^{})\)

Conditions	Eigenvalues	Properties
G<s<G + 4	\(-1<\lambda _{1,2}<1\)	stable node
\(\max \{-\frac{Gs}{H}, \frac{(2+G)(2-s)}{H} \}<\frac{1}{h}\leqslant -\frac{(G+s)^{2}}{4H}\)	\(-1<\lambda _{1,2}<1\)	stable node
G>s	\(\lambda _{1,2}>1\)	unstable node
\(-\frac{Gs}{H}<\frac{1}{h}\leqslant -\frac{(G+s)^{2}}{4H}\)	\(\lambda _{1,2}>1\)	unstable node
G + 4<s	\(\lambda _{1,2}<-1\)	unstable node
\(\frac{(2+G)(2-s)}{H}<\frac{1}{h}\leqslant -\frac{(G+s)^{2}}{4H}\)	\(\lambda _{1,2}<-1\)	unstable node
G<s	\(\lambda _{1}<-1\), \(-1<\lambda _{2}<1\)	unstable saddle
\(-\frac{Gs}{H}<\frac{1}{h}<\frac{(2+G)(2-s)}{H}\)
\(\frac{1}{h}\leqslant -\frac{(G+s)^{2}}{4H}\)
\(\frac{1}{h}>-\frac{(G+s)^{2}}{4H}\)	conjugate complex roots	stable focus
\(\frac{(1+G)(1-s)}{H}<\frac{1}{h}<\frac{G-s-Gs}{H}\)	\(\|\lambda _{1,2}\|<1\)	stable focus
\(\frac{1}{h}>\max \{-\frac{(G+s)^{2}}{4H},\frac{G-s-Gs}{H} \}\)	conjugate complex roots	unstable focus
	\(\|\lambda _{1,2}\|>1\)	unstable focus
\(\frac{1}{h}=\frac{(2+G)(2-s)}{H}\)	\(\lambda _{1}=-1\), \(\lambda _{2}\neq -1\)	nonhyperbolic
s ≠ G + 4	\(\lambda _{1}=-1\), \(\lambda _{2}\neq -1\)	nonhyperbolic
\(\frac{1}{h}=\frac{(2+G)(2-s)}{H}\)	\(\lambda _{1,2}=-1\)	nonhyperbolic
s = G + 4	\(\lambda _{1,2}=-1\)	nonhyperbolic
\(\frac{1}{h}=\frac{G-s-Gs}{H}\)	conjugate complex roots	nonhyperbolic
G<s<G + 4	\(\|\lambda _{1,2}\|=1\)	nonhyperbolic