Stability of the fixed points of the system
In this subsection, we study the asymptotic stability of the fixed points of system (4) which has the same fixed points of system (4). First, we need the following two definitions.
Definition 2
[26] (Local stability when all eigenvalues are real)
Consider the discrete, nonlinear dynamical system in (4) with a steadystate equilibrium x̄. The linearized system is given by (4). The associated Jacobian matrix has three real eigenvalues \(\lambda_{i}\) (\(i=1,2,3\)).
Lemma 1

(i)
The steadystate equilibrium
x̄
is called a stable node if
\(\vert \lambda_{i} \vert <1\)
for all
\(i=1,2,3\).

(ii)
The steadystate equilibrium
x̄
is called a twodimensional saddle if one
\(\vert \lambda_{i} \vert >1\).

(iii)
The steadystate equilibrium
x̄
is called a onedimensional saddle if one
\(\vert \lambda_{i} \vert <1\).

(iv)
The steadystate equilibrium
x̄
is called an unstable node if
\(\vert \lambda_{i} \vert >1\)
for all
\(i=1,2,3\).

(v)
The steadystate equilibrium
x̄
is called hyperbolic if one
\(\vert \lambda_{i} \vert =1\).
Definition 3
[26] (Local stability when complex eigenvalues)
Consider the discrete, nonlinear dynamical system in (4) with a steadystate equilibrium x̄. The linearized system is given by (4). The associated Jacobian matrix has a pair of complex eigenvalues \(\lambda_{1,2}=\rho+\omega i\) and one real eigenvalue \(\lambda_{3}\).

(i)
The steadystate equilibrium x̄ is called a sink if \(\vert \lambda_{i} \vert <1\) for all \(i=1,2,3\).

(ii)
The steadystate equilibrium x̄ is called a twodimensional saddle if one \(\vert \lambda_{3} \vert >1\).

(iii)
The steadystate equilibrium x̄ is called a onedimensional saddle if one \(\vert \lambda_{3} \vert <1\).

(iv)
The steadystate equilibrium x̄ is called a source if \(\vert \lambda_{i} \vert >1\) for all \(i=1,2,3\).

(v)
The steadystate equilibrium x̄ is called hyperbolic if one \(\vert \lambda_{i} \vert =1\).
Now, the Jacobian matrix \(J ( E_{0} ) \) for system given in (4) evaluated at \(E_{0}(0,0,0)\) is as follows:
$$ J ( E_{0} ) =\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1+\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) } & 0 & 0 \\ 0 & 1\frac{\delta h^{\alpha}}{\Gamma ( 1+\alpha ) } & 0 \\ 0 & 0 & 1\frac{\eta h^{\alpha}}{\Gamma ( 1+\alpha ) }\end{array}\displaystyle \right ) . $$
(5)
Theorem 1
The trivialequilibrium point
\(E_{0}\)
has at least three different topological types for its all values of parameters as follows:

(i)
\(E_{0}\)
is a source if
\(h>\max \{ \sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\delta}},\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}} \} \),

(ii)
\(E_{0}\)
is a twodimensional saddle if
\(0< h<\min \{ \sqrt [\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\delta}},\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}} \} \),

(iii)
\(E_{0}\)
is a onedimensional saddle if
\(\sqrt[\alpha]{\frac {2\Gamma ( 1+\alpha ) }{\delta}}< h<\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}}\)
or
\(\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}}< h<\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\delta}}\).
Proof
The eigenvalues corresponding to the equilibrium point \(E_{0} \) are \(\lambda_{01}=1+\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) }>0\), \(\lambda_{02}=1\frac{\delta h^{\alpha}}{\Gamma ( 1+\alpha ) } \), and \(\lambda_{03}=1\frac{\eta h^{\alpha }}{\Gamma ( 1+\alpha ) }\), where \(\alpha\in( 0,1 ] \) and \(h, \frac{h^{\alpha}}{\Gamma ( 1+\alpha ) }>0\). Hence, applying the stability conditions using Definition 2, one can obtain the results (i)(iii). □
The Jacobian matrix \(J ( E_{1} ) \) for system (4), evaluated at \(E_{1}= ( 1,0,0 ) \), is given by
$$ J ( E_{1} ) =\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) } & \frac {h^{\alpha}}{ ( 1+\beta ) \Gamma ( 1+\alpha ) } & 0 \\ 0 & 1+\frac{\delta ( R_{0}1 ) h^{\alpha}}{\Gamma ( 1+\alpha ) } & 0 \\ 0 & 0 & 1\frac{\eta h^{\alpha}}{\Gamma ( 1+\alpha ) }\end{array}\displaystyle \right ) . $$
(6)
Theorem 2
If the semitrivial equilibrium point
\(E_{1}\)
exists, then it has at least three different topological types for its all values of parameters.

(i)
\(E_{1}\)
is a source if
\(h>\max \{ \sqrt[\alpha]{2\Gamma ( 1+\alpha ) },\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\mu}} \} \),

(ii)
\(E_{1}\)
is a twodimensional saddle if
\(0< h<\min \{ \sqrt [\alpha]{2\Gamma ( 1+\alpha ) },\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}} \} \),

(iii)
\(E_{1}\)
is a onedimensional saddle if
\(\sqrt[\alpha]{2\Gamma ( 1+\alpha ) }< h<\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}}\)
or
\(\sqrt[\alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta}}< h<\sqrt[\alpha]{2\Gamma ( 1+\alpha ) }\).
Proof
The eigenvalues corresponding to the equilibrium point \(E_{0} \) are \(\lambda_{11}=1\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) }\), \(\lambda_{12}=1+\frac{\delta ( R_{0}1 ) h^{\alpha}}{\Gamma ( 1+\alpha ) }>0\), and \(\lambda_{13}=1\frac{\eta h^{\alpha}}{\Gamma ( 1+\alpha ) }\). Hence, applying the stability conditions using Definition 2, one can obtain the results (i)(iii). □
For investigating the stability of \(E_{2}= ( \frac{\beta\delta }{\gamma \delta},\frac{\gamma ( \beta+1 ) ( R_{0}1 ) x_{2}^{2}}{\beta\delta},0 ) \), let \(J ( E_{2} ) \) be the Jacobian matrix for the system given in (4) evaluated at \(E_{2}\), then
$$ J ( E_{2} ) =\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) } [ x_{2}\frac{ x_{2}y_{2}}{ ( \beta+x_{2} ) ^{2}} ] & \frac {x_{2}h^{\alpha}}{ ( \beta+x_{2} ) \Gamma ( 1+\alpha ) } & 0 \\ \frac{\beta\gamma y_{2}h^{\alpha}}{ ( \beta+x_{2} ) ^{2}\Gamma ( 1+\alpha ) } & 1 & \frac{y_{2}h^{\alpha}}{\Gamma ( 1+\alpha ) } \\ 0 & 0 & 1\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) } ( \eta \sigma y_{2} )\end{array}\displaystyle \right ) . $$
(7)
The characteristic equation of the Jacobian matrix (7) is
$$ \bigl[ \lambda_{21} \bigl( 1H [ \eta\sigma y_{2} ] \bigr) \bigr] \bigl[ \lambda^{2}\mathit{Tr}_{2} \lambda+\mathit{Det}_{2} \bigr] =0, $$
(8)
where \(H=\frac{h^{\alpha}}{\Gamma ( 1+\alpha ) }\), \(\mathit{Tr}_{2}=2B_{2}H\), \(B_{2}=\frac{ ( 1+\beta ) x_{2}^{2}}{ ( \beta+x_{2} ) } [ 1R_{0} ( 1\beta ) ] \), \(\mathit{Det}_{2}=A_{2}H^{2}B_{2}H+1 \), and \(A_{2}=\frac{\beta\gamma x_{2}y_{2}}{ ( \beta+x_{2} ) ^{3}}\).
Theorem 3
If
\(1< R_{0}<\frac{\beta\delta\eta}{\gamma\sigma ( 1+\beta ) x_{2}^{2}}+1\), then the semitrivial equilibrium point
\(E_{2}\)
has at least four different topological types for its all values of parameters

(i)
\(E_{2}\)
is asymptotically stable (sink) if one of the following conditions holds:

(i.1)
\(\Delta\geq0\)
and
\(0< h<\min \{ h_{1},h_{2} \} \),

(i.2)
\(\Delta<0\)
and
\(0< h< h_{2}\);

(ii)
\(E_{2}\)
is unstable (source) if one of the following conditions holds:

(ii.1)
\(\Delta\geq0\)
and
\(h>\max \{ h_{2},h_{3} \} \),

(ii.2)
\(\Delta<0\)
and
\(h>h_{2}\);

(iii)
\(E_{2}\)
is a twodimensional saddle if the following case is satisfied:
\(\Delta\geq0\)
and
\(h<\min \{ h_{1},h_{2} \} \);

(iv)
\(E_{2}\)
is a onedimensional saddle if one of the following cases is satisfied:

(iv.1)
\(\Delta\geq0\)
and
\(h_{3}< h< h_{2}\)
or
\(\max \{ h_{1},h_{2} \} < h< h_{3}\),

(iv.2)
\(\Delta<0\)
and
\(0< h< h_{2}\);

(v)
\(E_{2}\)
is nonhyperbolic if one of the following conditions holds:

(v.1)
\(\Delta\geq0\)
and
\(h=h_{1}\)
or
\(h_{3}\),

(v.2)
\(\Delta<0\)
and
\(h=h_{2}\),
where
$$\begin{aligned}& h_{1} =\sqrt[\alpha]{\frac{4\Gamma ( 1+\alpha ) }{B_{2}+\sqrt{\Delta}}},\qquad h_{3}=\sqrt[ \alpha]{\frac{4\Gamma ( 1+\alpha ) }{B_{2}\sqrt{\Delta}}} , \\& \Delta=B_{2}^{2}4A_{2},\qquad h_{2}=\sqrt[ \alpha]{\frac{2\Gamma ( 1+\alpha ) }{\eta\sigma y_{2}}}. \end{aligned}$$
Proof
The eigenvalues corresponding to the equilibrium point \(E_{2} \) are the roots of the characteristic equation (8), which is \(\lambda_{21}=1H [ \eta\sigma y_{2} ] \) and \(\lambda _{22,23}=\frac{1}{2} ( \mathit{Tr}_{2}\pm\sqrt{\mathit{Tr}_{2}^{2}4\mathit{Det}_{2}} ) =1\frac {H}{2} ( B_{2}\pm\sqrt{B_{2}^{2}4A_{2}} ) \). Hence, applying the stability conditions using Lemma 1, one can obtain the results (i)(v). □
The fourth and fifth equilibrium points are \(E_{j}= ( x_{j},y_{j},z_{j} ) \), \(j=3,4\), where
$$\begin{aligned}& x_{3} =\frac{1}{2} \biggl[ 1\beta\sqrt{ ( 1\beta ) ^{2}4 \biggl( \frac{\eta}{\sigma}\beta \biggr) } \biggr] , \\& x_{4} =\frac{1}{2} \biggl[ 1\beta+\sqrt{ ( 1\beta ) ^{2}4 \biggl( \frac{\eta}{\sigma}\beta \biggr) } \biggr] , \\& y_{j} =\frac{\eta}{\sigma}, \\& z_{j} =\frac{\beta\gamma\sigma}{\eta} ( x_{j}1 ) +\gamma \delta. \end{aligned}$$
For the dynamical properties of the interior (positive) equilibrium point \(E_{j}\) (\(j=3,4 \)) we need to state these lemmas.
Lemma 2
[27]
Let the equation
\(x^{3}+bx^{2}+cx+d=0\), where
\(b,c,d\in R \). Let further
\(A=b^{2}3c\), \(B=bc9d\), \(C=c^{2}3bd\), and
\(\Delta =B^{2}4AC\). Then

(1)
The equation has three real roots if and only if
\(\Delta\leq0\).

(2)
The equation has one real root
\(x_{1}\)
and a pair of conjugate complex roots if and only if
\(\Delta>0\). Furthermore, the conjugate complex roots
\(x_{2,3}\)
are
$$x_{2,3}=\frac{1}{6} \bigl[ \sqrt[3]{y_{1}}+ \sqrt[3]{y_{2}}2b\pm\sqrt{3} i \bigl( \sqrt[3]{y_{1}} \sqrt[3]{y_{2}} \bigr) \bigr], $$
where
$$y_{1,2}=bA+\frac{3}{2} \bigl( B\pm\sqrt{B^{2}4AC} \bigr) . $$
Lemma 3
[28–31]
Let
\(F(\lambda)=\lambda^{2}\mathit{Tr}\lambda+\mathit{Det}\). Suppose that
\(F(1)>0\), \(\lambda_{1}\)
and
\(\lambda_{2}\)
are the two roots of
\(F(\lambda)=0\). Then

(i)
\(\vert \lambda_{1} \vert <1\)
and
\(\vert \lambda _{2} \vert <1\)
if and only if
\(F(1)>0\)
and
\(\mathit{Det}<1\),

(ii)
\(\vert \lambda_{1} \vert <1\)
and
\(\vert \lambda _{2} \vert >1\) (or
\(\vert \lambda_{1} \vert >1\)
and
\(\vert \lambda_{2} \vert <1\)) if and only if
\(F(1)<0\),

(iii)
\(\vert \lambda_{1} \vert >1\)
and
\(\vert \lambda _{2} \vert >1\)
if and only if
\(F(1)>0\)
and
\(\mathit{Det}>1\),

(iv)
\(\lambda_{1}=1\)
and
\(\lambda_{2}\neq1\)
if and only if
\(F(1)=0\)
and
\(\mathit{Tr}\neq0,2\),

(v)
\(\lambda_{1}\)
and
\(\lambda_{2}\)
are complex and
\(\vert \lambda _{1} \vert = \vert \lambda_{2} \vert \)
if and only if
\(\mathit{Tr}^{2}4\mathit{Det}<0\)
and
\(\mathit{Det}=1\).
The necessary and sufficient conditions ensuring that \(\vert \lambda _{1} \vert <1\) and \(\vert \lambda_{2} \vert <1\) are as follows [28]:
$$ \begin{aligned} \text{(i)}&\quad 1\mathit{Tr} J+\det J>0, \\ \text{(ii)}&\quad 1+\mathit{Tr} J+\det J>0, \\ \text{(iii)}&\quad \det J< 1. \end{aligned} $$
(9)
If one of conditions (9) is not satisfied, then we have one of the following cases [25].

1.
A saddlenode (often called fold bifurcation in maps), transcritical or pitchfork bifurcation if one of the eigenvalues =1 and other eigenvalues (real) ≠1. This local bifurcation leads to the stability switching between two different steady states;

2.
A flip bifurcation if one of the eigenvalues \(=1\), other eigenvalues (real) \(\neq1\). This local bifurcation entails the birth of a period 2cycle;

3.
A NeimarkSacker (secondary Hopf) bifurcation; in this case we have two conjugate eigenvalues and the modulus of each of them =1.
This local bifurcation implies the birth of an invariant curve in the phase plane. The NeimarkSacker bifurcation is considered to be an equivalent to the Hopf bifurcation in continuous time and in fact the major instrument to prove the existence of quasiperiodic orbits for the map.
Note
You can get any local bifurcation (fold, flip and NeimarkSacker) by taking specific parameter value such that one of the conditions of each bifurcation is satisfied.
The Jacobian matrix \(J ( E_{j} ) \) for system (4) evaluated at the interior equilibrium point \(E_{j}\) is as follows:
$$ J ( E_{j} ) =\left ( \textstyle\begin{array}{c@{\quad}c@{\quad}c} 1H [ x_{j}\frac{x_{j}y_{j}}{ ( \beta+x_{j} ) ^{2}} ] & \frac{x_{j}H}{\beta+x_{j}} & 0 \\ \frac{\beta\gamma y_{j}H}{ ( \beta+x_{j} ) ^{2}} & 1 & y_{j}H \\ 0 & \sigma z_{j}H & 1\end{array}\displaystyle \right ), $$
(10)
then the characteristic equation of the Jacobian matrix (10) is
$$ F(\lambda)=\lambda^{3}+b_{j}\lambda^{2}+c_{j} \lambda+d_{j}=0, $$
(11)
where
$$\begin{aligned}& b_{j}=3\varepsilon_{j}H, \quad\varepsilon_{j}= \frac{x_{j}}{\beta +x_{j}} [ 2x_{j}+\beta1 ] , \\& c_{j}=\psi_{j}H^{2}2\varepsilon_{j}H+3,\quad \psi_{j}=\frac{\beta \gamma x_{j}y_{j}}{ ( \beta+x_{j} ) ^{3}}+\sigma y_{j}z_{j}, \end{aligned}$$
and
$$d_{j}=\nu_{j}H^{3}\psi_{j}H^{2}+ \varepsilon_{j}H1,\quad\nu _{j}=\sigma\varepsilon_{j}y_{j}z_{j}. $$
By some computation, we have
$$\begin{aligned}& A =b_{j}^{2}3c_{j}= \bigl( \varepsilon_{j}^{2}3\psi_{j} \bigr) H^{2}, \\& B =b_{j}c_{j}9d_{j}= ( \varepsilon_{j} \psi_{j}9\nu_{j} ) H^{3}+ \bigl( 6 \psi_{j}2\varepsilon_{j}^{2} \bigr) H^{2}, \\& C =c_{j}^{2}3b_{j}d_{j}= \bigl( \psi_{j}^{2}3\varepsilon_{j}\nu _{j} \bigr) H^{4}+ ( 9\nu_{j}\varepsilon_{j} \psi_{j} ) H^{3}+\bigl(\varepsilon_{j}^{2}3 \psi_{j}\bigr)H^{2}, \end{aligned}$$
and
$$\Delta_{j}=\bar{\Delta}_{j}H^{6}, $$
where
$$\bar{\Delta}_{j}=3\bigl(27\nu_{j}^{2}+4 \varepsilon_{j}^{3}\nu_{j}+4\psi _{j}^{3}\varepsilon_{j}^{2} \psi_{j}^{2}18\varepsilon_{j}\psi _{j}\nu _{j}\bigr). $$
It is clear that equation \(F^{\prime}(\lambda)=0\) has the following two roots:
$$\lambda_{1,2}^{\ast}=\frac{1}{3} \Bigl( b_{j}\pm\sqrt {b_{j}^{2}3c_{j}} \Bigr) =1\frac{H}{3} \Bigl( \varepsilon_{j}\pm\sqrt{ \varepsilon _{j}^{2}3\psi_{j}} \Bigr). $$
If \(\bar{\Delta}_{j}\leq0\), then \(\Delta_{j}\leq0\); by Lemma 1, equation (10) has three real roots \(\lambda_{i}\), \(i=1,2,3\) (let \(\lambda _{1}\leq\lambda_{2}\leq\lambda_{3}\)). From this, we note that two roots \(\lambda_{1,2}^{\ast}\) (let \(\lambda_{1}^{\ast}\leq\lambda _{2}^{\ast}\)) of equation \(F^{\prime}(\lambda)=0\) also are real.
When \(\bar{\Delta}_{j}>0\), namely, \(\Delta_{j}>0\), by Lemma 2, we have that equation (11) has one real root \(\lambda_{1}\) and a pair of conjugate complex roots \(\lambda_{2,3}\). The conjugate complex roots are as follows:
$$\lambda_{2,3}=\frac{1}{6} \bigl[ \sqrt[3]{y_{1}}+ \sqrt[3]{y_{2}}2b\pm \sqrt{3}i \bigl( \sqrt[3]{y_{1}} \sqrt[3]{y_{2}} \bigr) \bigr] , $$
where
$$y_{1,2}=\frac{H^{3}}{2} \bigl( 2\varepsilon_{j}^{3}9 \varepsilon _{j}\psi _{j}27\nu_{j}\pm3\sqrt{ \bar{\Delta}_{j}} \bigr) , $$
and
$$F(1)=\nu_{j}H^{3}\quad \text{and}\quad F(1)=8+4\varepsilon_{j}H2 \psi _{j}H^{2}+\nu_{j}H^{3}. $$
Now, we will introduce the stability of \(E_{j}\), we have the following theorem.
Theorem 4
If the positive equilibrium point
\(E_{j}\)
exists, then it has the following topological types of its all values of parameters:

(1)
\(E_{j}\)
is a sink if one of the following conditions holds:

(1.i)
\(\bar{\Delta}_{j}\leq0\), \(F(1)>0\), \(F(1)<0\)
and
\(1<\lambda _{1,2}^{\ast}<1\),

(1.ii)
\(\bar{\Delta}_{j}>0\), \(F(1)>0\), \(F(1)<0\)
and
\(\vert \lambda_{2,3} \vert <1\).

(2)
\(E_{j}\)
is a source if one of the following conditions holds:

(2.i)
\(\bar{\Delta}_{j}\leq0\)
and one of the following conditions holds:

(2.i.a)
\(F(1)>0\), \(F(1)>0\)
and
\(\lambda_{2}^{\ast}<1\)
or
\(\lambda_{2}^{\ast}>1\),

(2.i.b)
\(F(1)<0\), \(F(1)<0\)
and
\(\lambda_{2}^{\ast}<1\)
or
\(\lambda_{2}^{\ast}>1\),

(2.ii)
\(\bar{\Delta}_{j}>0\)
and one of the following conditions holds:

(2.ii.a)
\(F(1)<0\)
and
\(\vert \lambda _{2,3} \vert >1\),

(2.ii.b)
\(F(1)>0\)
and
\(\vert \lambda _{2,3} \vert >1\).

(3)
\(E_{j}\)
is a onedimensional saddle if one of the following conditions holds:

(3.i)
\(\bar{\Delta}_{j}\leq0\)
and one of the following conditions holds:

(3.i.a)
\(F(1)>0\), \(F(1)<0\)
and
\(\lambda_{1}^{\ast}<1\)
or
\(\lambda_{2}^{\ast}>1\),

(3.i.b)
\(F(1)<0\), \(F(1)>0\).

(3.ii)
\(\bar{\Delta}_{j}>0\)
and one of the following conditions holds:

(3.ii.a)
\(F(1)>0\), \(F(1)<0\)
and
\(\vert \lambda _{2,3} \vert >1\),

(3.ii.b)
\(F(1)<0\)
and
\(\vert \lambda _{2,3} \vert <1\),

(3.ii.c)
\(F(1)>0\)
and
\(\vert \lambda _{2,3} \vert <1\).

(4)
\(E_{j}\)
is a twodimensional saddle if one of the following conditions holds:

(4.i)
\(\bar{\Delta}_{j}\leq0\)
and one of the following conditions holds:

(4.i.a)
\(F(1)>0\), \(F(1)>0\)
and
\(1<\lambda_{2}^{\ast}<1\),

(4.i.b)
\(F(1)<0\), \(F(1)<0\)
and
\(1<\lambda_{1}^{\ast}<1\),

(4.ii)
\(\bar{\Delta}_{j}>0\)
and one of the following conditions holds:

(4.ii.a)
\(F(1)<0\)
and
\(\vert \lambda _{2,3} \vert <1\),

(4.ii.b)
\(F(1)>0\)
and
\(\vert \lambda _{2,3} \vert <1\).

(5)
\(E_{j}\)
is nonhyperbolic if one of the following conditions holds:

(5.i)
\(\bar{\Delta}_{j}\leq0\)
and
\(F(1)=0\)
or
\(F(1)=0\),

(5.ii)
\(\bar{\Delta}_{j}>0\)
and
\(F(1)=0\)
or
\(F(1)=0\)
or
\(\vert \lambda_{2,3} \vert =1\).
Proof
Let \(\bar{\Delta}_{j}\leq0\). From Lemma 2, equation (11) has three real roots \(\lambda_{i}\), \(i=1,2,3\). Further, we obtain that equation \(F^{\prime}(\lambda)=0\) has also two real roots \(\lambda_{1}^{\ast }\) and \(\lambda_{2}^{\ast}\). From the expression of \(F^{\prime}(\lambda)\), we have \(F^{\prime}(\lambda)>0\) for all \(\lambda\in ( \infty ,\lambda _{1}^{\ast} ) \cup ( \lambda_{2}^{\ast},\infty ) \) and \(F^{\prime}(\lambda)<0\) for all \(\lambda\in ( \lambda_{1}^{\ast },\lambda_{2}^{\ast} ) \). Hence, \(F(\lambda)\) is increasing for all \(\lambda\in ( \infty,\lambda_{1}^{\ast} ) \cup ( \lambda _{2}^{\ast},\infty ) \) and decreasing for all \(\lambda\in ( \lambda_{1}^{\ast},\lambda_{2}^{\ast} ) \). Therefore, we finally obtain \(F(\lambda_{1}^{\ast})\geq0\), \(F(\lambda_{1}^{\ast})\leq 0\), \(\lambda_{1}\in ( \infty,\lambda_{1}^{\ast} ] \), \(\lambda _{2}\in [ \lambda_{1}^{\ast},\lambda_{2}^{\ast} ) \) and \(\lambda_{3}\in [ \lambda_{2}^{\ast},\infty ) \).
If condition (1.i) holds, then we obviously have \(\lambda_{1}\in ( 1,\lambda_{1}^{\ast} ] \), \(\lambda_{2}\in [ \lambda _{1}^{\ast},\lambda_{2}^{\ast} ] \) and \(\lambda_{3}\in [ \lambda_{2}^{\ast},1 ) \). Therefore, \(E_{j}\) is a sink.
If condition (2.i.a) holds. When \(\lambda_{2}^{\ast}<1\), we have \(\lambda_{1}<1\) and \(\lambda_{2}<1\). Since \(F(\lambda)\) is increasing for all \(\lambda\in [ \lambda_{2}^{\ast},\infty ) \) and \(F(1)>0\), we can obtain \(\lambda_{3}<1\). Therefore, \(E_{j}\) is a source. When \(\lambda_{2}^{\ast}>1\), then from \(F(1)>0\) we have \(\lambda _{1}<1\). Hence \(F(\lambda)>0\) for all \(\lambda\in ( 1,1 ) \). Consequently, \(\lambda_{2}>1\) and \(\lambda_{3}>1\). Therefore, \(E_{j}\) is a source.
By the same way, we prove that when condition (2.i.b) holds, \(E_{j}\) is also a source.
If condition (3.i.a) holds, then, when \(\lambda_{1}^{\ast}<1\) we have \(\lambda_{1}<1\) and when \(\lambda_{2}^{\ast}>1\) we have \(\lambda _{3}>1\). From \(F(1)>0\) and \(F(1)<0\) we have \(\lambda_{2}\in ( 1,1 ) \). Therefore, \(E_{j}\) is a onedimensional saddle.
If condition (3.i.b) holds, then we clearly have \(\lambda_{1}<1\), \(\lambda_{2}\in ( 1,1 ) \) and \(\lambda_{3}>1\). Therefore, \(E_{j}\) is a onedimensional saddle too.
If condition (4.i.a) holds, we have \(\lambda_{2,3}\in ( 1,1 ) \) and \(\lambda_{1}^{\ast}\in ( \infty,1 ) \), \(F(\lambda)\) is increasing for all \(\lambda\in ( \infty,\lambda_{1}^{\ast} ] \), we obtain \(\lambda_{1}\in ( \infty,1 ) \). Therefore, \(E_{j}\) is a twodimensional saddle.
By the same way, we can prove that when condition (4.i.b) holds, \(E_{j}\) is also a twodimensional saddle.
If condition (5.i) holds, then we can easily prove that \(E_{j}\) is nonhyperbolic.
Now, we let \(\bar{\Delta}_{j}>0\). From Lemma 2, equation (11) has one real root \(\lambda_{1}\) and a pair of conjugate complex roots \(\lambda_{2,3}\). If condition (1.ii) holds, then from \(F(1)>0\) and \(F(1)<0\) we have that a real root \(\lambda_{1}\in ( 1,1 ) \). Therefore, from \(\vert \lambda_{2,3} \vert <1\) we obtain that \(E_{j}\) is a sink.
If condition (2.ii.a) holds, then from \(F(1)<0\) we have a real root \(\lambda _{1}>1\). Therefore, from \(\vert \lambda_{2,3} \vert >1\) we obtain that \(E_{j}\) is a source.
By the same way, we can prove that if condition (2.ii.b) holds, then \(E_{j}\) is also a source.
If condition (3.ii.a) holds, then we have a real root \(\lambda_{1}\in ( 1,1 ) \). Therefore, from \(\vert \lambda_{2,3} \vert >1\) we have that \(E_{j}\) is a onedimensional saddle.
By the same way, we can prove that when conditions (3.ii.b) and (3.ii.c) hold, then \(E_{j}\) is also a onedimensional saddle.
If condition (4.ii.a) holds, then from \(F(1)<0\) we have a real root \(\lambda _{1}>1\). Therefore, from \(\vert \lambda_{2,3} \vert >1\) we have that \(E_{j}\) is a twodimensional saddle.
By the same way, we can prove that if condition (4.ii.b) holds, then \(E_{j}\) is also a twodimensional saddle.
Lastly, we can easily prove that if condition (5.ii) holds, then \(E_{j}\) is nonhyperbolic. □
Theorem 5
If the positive equilibrium point
\(E_{j}\)
exists, then
\(E_{j}\)
loses its stability via

(i)
a saddlenode bifurcation, if one of the following conditions holds:

(i.1)
\(\bar{\Delta}_{j}\leq0\), \(F(1)=0\)
and
\(F(1)\neq0\),

(i.2)
\(\bar{\Delta}_{j}>0\), \(F(1)=0\)
and
\(\vert \lambda _{2,3} \vert \neq1\);

(ii)
flip bifurcation, if one of the following conditions holds:

(ii.1)
\(\bar{\Delta}_{j}\leq0\), \(F(1)=0\)
and
\(F(1)\neq0\),

(ii.2)
\(\bar{\Delta}_{j}>0\), \(F(1)=0\)
and
\(\vert \lambda _{2,3} \vert \neq1\);

(iii)
Hopf bifurcation, if the following condition holds:
Proof ([29])
(i) introduced a thorough study of the main types of bifurcations for 3D maps. In line with this study, we can see that \(E_{j}\) undergoes a saddlenode bifurcation when a single eigenvalue becomes equal to 1. Therefore \(E_{j}\) can lose its stability through the saddlenode bifurcation when one of \(\lambda_{i}=1\), \(i=1,2,3\). A saddlenode bifurcation of \(E_{j}\) may occur if the parameters vary in the small neighborhood of the following sets:
$$S_{1}= \bigl\{ ( h,\alpha,\beta,\gamma,\delta,\eta,\sigma ) :\bar{\Delta}_{j}\leq0,F(1)=0 \text{ and } F(1)\neq0 \bigr\} , $$
or
$$S_{2}= \bigl\{ ( h,\alpha,\beta,\gamma,\delta,\eta,\sigma ) :\bar{\Delta}_{j}>0,F(1)=0\text{ and }\vert \lambda _{2,3} \vert \neq1 \bigr\} . $$
By the same way, we can prove (ii).
(iii) When the Jacobian has a pair of complex conjugate eigenvalues of modulus 1, we get the Hopf bifurcation, then \(E_{j}\) can lose its stability through the Hopf bifurcation when one of \(\vert \lambda _{2,3} \vert =1 \), and then it also implies that all the parameters locate and vary in the small neighborhood of the following set:
$$S_{3}= \bigl\{ ( h,\alpha,\beta,\gamma,\delta,\eta,\sigma ) :\bar{\Delta}_{j}>0,F(1)\neq0,F(1)\neq0\text{ and } \vert \lambda_{2,3} \vert =1 \bigr\} . $$
□
Numerical simulations
In this section, we give the phase portraits, the attractor of parameter β and bifurcation diagrams to confirm the above theoretical analysis and to obtain more dynamical behaviors of the palm trees, lesser date moth and predator model. Since most of the fractionalorder differential equations do not have exact analytic solutions, approximation and numerical techniques must be used.
From the numerical results, it is clear that the approximate solutions depend on the fractional parameters h, α see Figure 1. The approximate solutions \(x_{n}\), \(y_{n}\), and \(z_{n}\) are displayed in the figures below.
We use some documented data for some parameters like \(\beta=0.5\), \(\gamma =3\), \(\delta=\eta=1\), and \(\sigma=3\), then we have \(( x_{1},y_{1},z_{1} ) = ( 0.7,0.3,0.8 ) \). Other parameters will be (a) \(h=0.05\), \(\alpha=0.95\), (b) \(h=0.07\), \(\alpha=0.95\), (c) \(h=0.09\), \(\alpha=0.95\), and (d) \(h=0.09\), \(\alpha=0.75\).
Figure 1 depicts the phase portraits of model (4) according to the chosen parameter values and for various values of the fractionalorder parameters h and α. We can see that, whenever the value of α is fixed and the value of h increases, then \(E_{4}\) moves from the stabilized to the chaotic band. Figure 1(c) depicts the phase portrait for model (4).
By computing, we have \(E_{4}\simeq ( 0.7,0.33,0.84 ) \), and we can get the critical value of flip bifurcation for model (4). In Figure 1(a) we have \(\bar{\Delta}_{4}=1\mbox{,}153\mbox{,}256\mbox{,}457>0\), \(F(1)\simeq{0.00009>0}\), \(F(1)\simeq7.872<0\), and \(\vert \lambda_{2,3} \vert \simeq 0.999<1\). In this case we get \(E_{4}\) is a sink according to case (1.ii) in Theorem 4.
In Figure 1(b) we have \(\bar{\Delta}_{4}=165\mbox{,}217\mbox{,}502.2>0\), \(F(1)\simeq 0.00024>0\), \(F(1)\simeq7.8274<0\), and \(\vert \lambda _{2,3} \vert \simeq0.999<1\). In this case we get \(E_{4}\) is a sink according to case (1.ii) in Theorem 4.
In Figure 1(c) we have \(\bar{\Delta}_{4}=38\mbox{,}548\mbox{,}982.44>0\), \(F(1)\simeq 0.00049>0\), \(F(1)\simeq7.79<0\), and \(\vert \lambda_{2,3} \vert \simeq1.0002>1\). In this case we get \(E_{4}\) is a onedimensional saddle according to case (3.ii.a) in Theorem 4. We see that the fixed point \(E_{4}\) loses its stability at the Hopf bifurcation parameter value \(h\simeq0.086\). For \(h= [ 0,0.15 ] \), there is a cascade of bifurcations. When r increases at certain values, for example, at \(h=0.09\), independent invariant circles appear. When the value of h is increased (Figure 1(c)), the circles break down and some cascades of bifurcations lead to chaos.
Figures 1(c) and 1(d) explain the effect of the parameter α on the behavior of x, y, and z. Figure 2 demonstrates the sensitivity to initial conditions of system (4). We compute two orbits with initial points \(( x_{1},y_{1},z_{1} ) \) and \(( x_{1},y_{1}+0.01,z_{1} ) \), respectively. The compositional results are shown in Figure 2. From this figure it is clear that at the beginning the time series are indistinguishable; but after a number of iterations, the difference between them builds up rapidly, which shows that the model has sensitive dependence on the initial conditions of model (4), ycoordinates of the two orbits, plotted against time; the ycoordinates of initial conditions differ by 0.01, and the other coordinates do not change. In Figure 3 we use some documented data for some parameters like \(\beta=0.5\), \(\gamma=3\), \(\delta=\eta=1\), \(\sigma=3\), \(\alpha=0.95\), then we have \(( x_{1},y_{1},z_{1} ) = ( 0.7,0.3,0.8 ) \), another parameter will be \(h=0.001:0.47\). Figure 3 describes the Hopf bifurcation diagram with respect to h, we note that, as h increases, the behavior of this model (4) becomes very complicated, and the changes of parameter h has an effect on the stability of system (4). The Hopf bifurcation diagrams of system (4) in the \((hx)\), \((hy)\) and \((hz)\) planes are given in Fig. 3.
After calculation for the fixed point \(E_{4}\) of map (4), the Hopf bifurcation emerges from the fixed point \(( 0.7,0.33,0.84 ) \) at \(h=0.086\) and \(( h,\alpha,\beta,\gamma,\delta,\eta,\sigma ) \in HB_{E_{4}}\). From Figure 3, we observe that the fixed point \(E_{4}\) of map (4) loses its stability through a discrete Hopf bifurcation for \(h= [ 0,0.15 ] \). In Figure 4 we use some documented data for some parameters like \(\gamma=3\), \(\delta=\eta=1\), \(\sigma=3\), \(h=0.85\), \(\alpha=0.95\), other parameters will be (a) \(\beta=1.55\), (b) \(\beta=1.4\), (c) \(\beta=1.2\), and (d) \(\beta=1.15\).
Figure 4(a) describes the stable equilibrium of model (4) according to the values of the parameters set out above. From Figure 4(f), 4(g) we can see that reducing and decreasing β causes disappearance of firstperiodic orbits and increase in the chaotic attractors.
In Figure 5, we introduce the 01 test for detecting the chaos. Figure 5(a) indicates that, for \(\beta=1.55\), one obtains \(K=0.066\approx0\). Then the dynamics is regular. Moreover, Figure 5(b) depicts bounded trajectories in the \(( P_{c}(n),Q_{c}(n) ) \) plane. Figure 5(c) indicates that, for \(\beta=1.15\), one obtains \(K=0.9658\approx1\). Then the dynamics is chaotic. Moreover, Figure 5(d) depicts Brownianlike (unbounded) trajectories in the \(( P_{c}(n),Q_{c}(n) ) \) plane.
In Figure 5(e)5(f), we show the asymptotic growth rate K as a function of c for regular (chaotic) dynamics. In the case of regular (chaotic) dynamics, most values of c yield \(K\approx0\) (\(K\approx1\)) as expected.
Figure 5(e)5(f) show the two mean square displacements \(M_{c}\) for system (4) with \(\beta=1.55\) (\(\beta=1.15 \)), which corresponds to regular (chaotic) dynamics.