In this section, we obtain a condition under which the cross-diffusion system (1.1) loses stability and the stable positive equilibrium becomes Turing unstable.
Let \(\lbrace \mu _{n} \rbrace _{n=0}^{\infty } \) be the eigenvalues of −△ in Ω with zero Neumann boundary conditions. The linearized system of (1.1) at \(U_{*} = (u_{*},v_{*}) \) is
$$\begin{aligned} \begin{pmatrix} u_{t} \\ v_{t} \end{pmatrix} = \bigl(D\Delta + J(U_{*})\bigr) \dbinom{u}{v}, \end{aligned}$$
where J is defined by (2.3) and
$$\begin{aligned} D:= \begin{pmatrix} (1+\alpha v_{*}) & \alpha u_{*} \\ -\frac{\beta v_{*}}{(1+\beta u_{*})^{2}} & (\mu + \frac{1}{1+\beta u_{*}}) \end{pmatrix} . \end{aligned}$$
(3.1)
Then λ is an eigenvalue of \(L:= D\Delta + J(U_{*}) \) on
$$\begin{aligned} X = \bigl\lbrace (u,v) \in \mathbb{H}^{2} (\Omega ) \times \mathbb{H}^{2} ( \Omega ): \partial _{\nu } u = \partial _{\nu } v =0 \bigr\rbrace , \end{aligned}$$
if and only if λ is an eigenvalue of the matrix \(-\mu _{n} D + J(U_{*}) \) for \(n\geq 0 \). Therefore λ satisfies the characteristic equation
$$\begin{aligned} \lambda ^{2} -T_{n}(d) \lambda +D_{n}(d) =0, \end{aligned}$$
(3.2)
where
$$\begin{aligned} T_{0}(d) ={}& {-}\frac{ru_{*}}{k} + \frac{bu_{*} r^{2}(k-u_{*})^{2}}{ak^{2}v_{*}} - d, \\ T_{n}(d ) ={}&{ -}\biggl[(1 + \alpha v_{*}) + \mu + \frac{1}{1+\beta u_{*}}\biggr] \mu _{n} + T_{0} \end{aligned}$$
(3.3)
$$\begin{aligned} ={}&{ -} \biggl(1 + \alpha v_{*} + \mu + \frac{1}{1+\beta u_{*}}\biggr)\mu _{n} + \biggl(- \frac{ru_{*}}{k} + \frac{bu_{*}r^{2}(k-u_{*})^{2}}{ak^{2}v_{*}} - d\biggr), \\ D_{0}(d) ={}& \frac{d r u_{*}}{k}, \\ D_{n}(d) ={}& \biggl[\biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) (1+\alpha v_{*})+ \frac{\alpha \beta u_{*}v_{*}}{(1+\beta u_{*})^{2}}\biggr]\mu _{n} ^{2} + \biggl[d(1+ \alpha v_{*}) \\ & {}- \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) (T_{0} + d ) + \frac{\alpha d b u_{*}}{e} + \frac{e r^{2}\beta u_{*}(k-u_{*})^{2}}{ak^{2}(1+\beta u_{*})^{2}}\biggr] \mu _{n} + D_{0} \\ ={}& \biggl[\biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) (1+\alpha v_{*})+ \frac{\alpha \beta u_{*}v_{*}}{(1+\beta u_{*})^{2}}\biggr]\mu _{n} ^{2} + \biggl[ d(1+\alpha v_{*}) + \frac{e r^{2}\beta u_{*}(k-u_{*})^{2}}{ak^{2}(1+\beta u_{*})^{2}} \\ & {}- \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) \biggl(-\frac{ru_{*}}{k} + \frac{bu_{*} r^{2}(k - u_{*})^{2}}{ak^{2}v_{*}}\biggr) + \frac{\alpha d b u_{*}}{e} \biggr] \mu _{n} + \frac{rdu_{*}}{k}. \end{aligned}$$
(3.4)
Let
$$\begin{aligned} M_{n} :={}& \biggl[\mu + \frac{1}{1+\beta u_{*}}\biggr]\mu _{n} ^{2} + \biggl[ d - \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) \biggl(- \frac{ru_{*}}{k} + \frac{bu_{*} r^{2}(k-u_{*})^{2}}{ak^{2}v_{*}}\biggr) \\ &{}+\frac{e r^{2}\beta u_{*}(k-u_{*})^{2}}{ak^{2}(1+\beta u_{*})^{2}}\biggr] \mu _{n} + \frac{rdu_{*}}{k}, \\ N_{n} :={}& \biggl(\mu + \frac{1}{1+\beta u_{*}} + \frac{\beta u_{*}}{(1+\beta u_{*})^{2}} \biggr)\mu _{n} ^{2} + \biggl(d + \frac{ d b u_{*}}{ev_{*}}\biggr) \mu _{n}. \end{aligned}$$
According to (3.4),
$$\begin{aligned} D_{n}(d) = \alpha v_{*} N_{n} + M_{n}. \end{aligned}$$
In the next theorem, we summarized the asymptotic behavior of the equilibrium \(U_{*} \) for system (1.1).
Theorem 3.1
Assume \(d > d_{0} \) and let (2.2) and (2.5) be satisfied. Then
-
(i)
\(U_{*} \) is unstable for system (1.1) if
$$\begin{aligned} 0 < \alpha < \frac{-M_{1}}{v_{*}N_{1}}. \end{aligned}$$
(3.5)
-
(ii)
\(U_{*} \) is locally asymptotically stable for system (1.1) if
-
(1)
\(0 < \mu < \frac{\beta m - b}{b (1+\beta u_{*})^{2}} \),
or
-
(2)
\(\alpha > \frac{-M_{1}}{v_{*}N_{1}} > 0 \) and
$$\begin{aligned} \mu _{1}^{2}\biggl(\mu + \frac{1}{1+\beta u_{*}} \biggr) + M_{1} - \frac{rdu_{*}}{k}>0. \end{aligned}$$
(3.6)
-
(iii)
Under condition (1) in (ii) system (1.1) has a Hopf bifurcation at \(U_{*} \) when \(d=d_{0}\).
Proof
(i). According to (3.3), \(T_{n}(d) > T_{n+1}(d) \) for \(n \in \mathbb{N} \cup \lbrace 0\rbrace \). Since \(T_{0}(d) < 0 \) for \(d > d_{0} \), we conclude that \(T_{n}(d) < 0 \) for \(n \in \mathbb{N} \cup \lbrace 0\rbrace \). By (3.4) and condition (3.5), we have \(D_{1}= \alpha v_{*} N_{1} +M_{1} < 0 \). So equation (3.2) has at least one root with positive real part. Then \(U_{*} \) is unstable for system (1.1).
(ii). Since \(\frac{ev_{*}}{m+bu_{*}} = 1 \), from (1) we conclude
$$\begin{aligned} \mathcal{A}&:=\mu + \frac{1}{1+\beta u_{*}} - \frac{\beta ev_{*}}{b(1+\beta u_{*})^{2}} = \mu - \frac{\beta (m+bu_{*})}{b(1+\beta u_{*})^{2}} + \frac{1}{1+\beta u_{*}} \\ & = \mu - \frac{\beta m-b}{b(1+\beta u_{*})^{2}} < 0. \end{aligned}$$
(3.7)
Furthermore, \(D_{0}(d)>0\) for \(d>0\), and we can rewrite \(D_{n}(d) \) as
$$\begin{aligned} D_{n}(d) ={}& \alpha v_{*}\biggl[\biggl(\mu + \frac{1}{1+\beta u_{*}} + \frac{\beta u_{*}}{(1+\beta u_{*})^{2}}\biggr)\mu _{n} ^{2} + \biggl(d + \frac{bd u_{*}}{ev_{*}}\biggr)\mu _{n}\biggr] \\ &{}+ \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr)\mu _{n} ^{2} + \biggl[d + \frac{ru_{*}}{k} \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) \\ &{} -\frac{bu_{*} r^{2}(k-u_{*})^{2}}{ak^{2}v_{*}}\mathcal{A}\biggr] \mu _{n} + \frac{d r u_{*}}{k}. \end{aligned}$$
(3.8)
By (3.7), we have \(D_{n}(d) > 0 \) for \(n \in \mathbb{N} \) and \(d>0\). Since \(T_{n}(d) < 0 \) for \(n \in \mathbb{N} \cup \lbrace 0\rbrace \) when \(d > d_{0} \), we deduce that the real part of every eigenvalue of L is negative. Therefore, \(U_{*} \) is locally asymptotically stable for system (1.1).
Now, let \((2) \) be satisfied. Then \(D_{1}= \alpha v_{*} N_{1} +M_{1} > 0 \). Also, we can rewrite \(M_{n} \) as
$$\begin{aligned} M_{n} &= \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) \bigl(\mu _{n}^{2} - \mu _{n}\mu _{1}\bigr) + \frac{M_{1}\mu _{n}}{\mu _{1}} - \frac{rdu_{*}}{k}\biggl( \frac{\mu _{n}}{\mu _{1}} - 1\biggr) \\ & =\mu _{n} \biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) (\mu _{n} - \mu _{1}) - \frac{rdu_{*}}{k\mu _{1}}(\mu _{n} - \mu _{1}) + \frac{M_{1}\mu _{n}}{\mu _{1}}. \end{aligned}$$
Hence, for \(n = 2, 3,\ldots \) , using condition (3.6) we have
$$\begin{aligned} M_{n}-M_{1}={}&\biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr)\mu _{n} (\mu _{n} - \mu _{1}) + \frac{M_{1}(\mu _{n} - \mu _{1})}{\mu _{1}} - \frac{rdu_{*}}{k\mu _{1}}(\mu _{n} - \mu _{1}) \\ \geq{} &\frac{(\mu _{n} - \mu _{1})}{\mu _{1}} \biggl(\mu _{1}^{2}\biggl(\mu + \frac{1}{1+\beta u_{*}}\biggr) + M_{1} - \frac{rdu_{*}}{k} \biggr)>0. \end{aligned}$$
Since \(N_{n+1}\geq N_{n}\) for \(n \in \mathbb{N} \), the above inequalities imply \(D_{n} (d)\geq D_{1}(d)> 0 \) for \(n \geq 2 \) and \(d>d_{0}\). Again using \(T_{n}(d) < 0 \) for \(n \in \mathbb{N} \), we conclude that all roots of the characteristic equation (3.2) have negative real part for \(n\geq 0\) when \(d>d_{0}\). Therefore \(U_{*} \) is locally asymptotically stable for system (1.1).
(iii) Under condition (1) in (ii), inequality (3.7) is stratified. Then from (3.8) we get \(D_{n}(d_{0})>0\) for \(n \in \mathbb{N} \). Also \(D_{0}(d_{0})>0\). On the other hand, \(T_{0}(d_{0})=0\), then by (3.3) we have \(T_{n}(d_{0})<0\) for \(n\in \mathbb{N}\). Therefore, all roots of the characteristic equation (3.2) have negative real part for \(n\geq 1\) when \(d=d_{0}\). Finally from (2.6) we conclude that system (1.1) has a Hopf bifurcation at \(U_{*}\) when \(d=d_{0}\). □
Now, we determine the direction and stability of the Hopf bifurcation for system (1.1) in \(\Omega = (0, \mathit{l}\pi )\) for \(\mathit{l} \in \mathbb{R}^{+} \), i.e., the following system:
$$\begin{aligned} \textstyle\begin{cases} u_{t} - ((1+\alpha v)u)_{xx} = r u (1-\frac{u}{k} ) - \frac{a u v}{m+bu+cv}, & x\in \Omega, t>0, \\ v_{t} - ((\mu + \frac{1}{1+\beta u})v)_{xx} = d v (1 - \frac{ev}{m+bu}), & x\in \Omega, t>0, \\ u(x,0) = u_{0} (x)\geq 0,\qquad v(x,0)=v_{0} (x)\geq 0, & x\in \Omega, \\ \partial _{\nu } u=\partial _{\nu } v=0, & x\in \partial \Omega, t>0. \end{cases}\displaystyle \end{aligned}$$
(3.9)
The eigenvalues and the corresponding eigenfunctions of the operator \(u\longrightarrow - u_{xx} \) with zero Neumann boundary conditions on \((0,\mathit{l} \pi ) \) are given by
$$\begin{aligned} \mu _{n} = \frac{n^{2}}{\mathit{l}^{2}}, \qquad\varphi _{n}(x) = \cos \biggl( \frac{nx}{\mathit{l}}\biggr),\quad \text{for }n = 0,1,2,\ldots. \end{aligned}$$
The linearized system of (3.9) at \(U_{*}\) has the following form:
$$\begin{aligned} \begin{pmatrix} u_{t} \\ v_{t} \end{pmatrix} = L \dbinom{u}{v} = D \begin{pmatrix} u_{xx} \\ v_{xx} \end{pmatrix} + J \dbinom{u}{v}, \end{aligned}$$
(3.10)
where J and D are defined by (2.3) and (3.1) respectively and
$$\begin{aligned} L = \begin{pmatrix} - (1+\alpha v_{*})\mu _{n} + r (1- \frac{2 u_{*}}{k} ) - \frac{a v_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{2}} & -\alpha u_{*}\mu _{n} - \frac{au_{*}(m+bu_{*})}{(m+bu_{*}+cv_{*})^{2}} \\ \frac{\beta v_{*}\mu _{n}}{(1+\beta u_{*})^{2}} + \frac{bd}{e} & -( \mu + \frac{1}{1+\beta u_{*}})\mu _{n} - d \end{pmatrix} . \end{aligned}$$
(3.11)
Similarly, by transferring the equilibrium point \(U_{*}\) to the origin, system (3.9) becomes
$$\begin{aligned} \begin{pmatrix} u_{t} \\ v_{t} \end{pmatrix} = D \begin{pmatrix} u_{xx} \\ v_{xx} \end{pmatrix} + J \dbinom{u}{v}+ \begin{pmatrix} f_{1}(u,v,d) \\ f_{2}(u,v,d) \end{pmatrix} , \end{aligned}$$
(3.12)
where \(f_{1} \) and \(f_{2} \) are defined by (2.8) and (2.9). Define the operator \(L^{*} \) by
$$\begin{aligned} L^{*} \dbinom{u}{v}:= D \begin{pmatrix} u_{xx} \\ v_{xx} \end{pmatrix} + J^{*} \dbinom{u}{v}, \end{aligned}$$
(3.13)
where \(J^{*} \) is the conjugate of J. Then \(L^{*} \) is the adjoint operator of L. Set
$$\begin{aligned} q= \begin{pmatrix} \frac{e(d_{0}+i\beta _{0})}{bd_{0}} \\ 1 \end{pmatrix} ,\qquad q^{*} = \frac{1}{2\mathit{l}\pi } \begin{pmatrix} \frac{bd_{0}}{e\beta _{0}}i \\ - \frac{d_{0}}{\beta _{0}}i + 1 \end{pmatrix} , \end{aligned}$$
where \(\beta _{0}(d_{0}) = \sqrt{\frac{ru_{*}d_{0}}{k}} \) and i is the imaginary unit. Then
$$\begin{aligned} L q =i\beta _{0} (d_{0}) q,\qquad L^{*} q^{*} = -i\beta _{0} (d_{0}) q^{*},\qquad \bigl\langle q^{*}, q \bigr\rangle =1,\qquad \bigl\langle q^{*}, \bar{q} \bigr\rangle = 0, \end{aligned}$$
where \(\langle f, g \rangle = \int _{0}^{\mathit{l}\pi } \bar{f}^{T} g \,dx \) is the inner product in \(\mathbb{L}^{2} (0,\mathit{l}\pi ) \times \mathbb{L}^{2} (0, \mathit{l}\pi ) \). We decompose X as \(X = X^{c} \oplus X^{s} \) where
$$\begin{aligned} X^{c} = \lbrace z q + \bar{z} \bar{q}: z \in \mathbb{C} \rbrace,\qquad X^{s} = \bigl\lbrace w \in X: \bigl\langle q^{*}, w \bigr\rangle = 0 \bigr\rbrace . \end{aligned}$$
According to [8], for every \(U^{T} = (u,v) \in X \), we have
$$\begin{aligned} \dbinom{u}{v} = z q + \bar{z} \bar{q} + \dbinom{w_{1}}{w_{2}},\quad z= \bigl\langle q^{*}, (u,v)^{T} \bigr\rangle , \end{aligned}$$
where \(z q + \bar{z} \bar{q} \in X^{c} \) and \(w^{T} = (w_{1}, w_{2}) \in X^{s} \). Then
$$\begin{aligned} & z= \bigl\langle q^{*}, U \bigr\rangle , \\ & w = U - \bigl\langle q^{*}, U \bigr\rangle q - \bigl\langle \bar{q^{*} }, U \bigr\rangle \bar{q}. \end{aligned}$$
Thus system (3.12) in \((z, w)\) coordinates is as follows:
$$\begin{aligned} \textstyle\begin{cases} \frac{dz}{dt} = i \beta _{0} (d_{0}) z + \langle q^{*},f^{\vee } \rangle, \\ \frac{dw}{dt} = Lw+ H(z, \bar{z},w ), \end{cases}\displaystyle \end{aligned}$$
(3.14)
where \(f^{\vee } = (f_{1},f_{2} )^{T} \) is defined by (2.8)–(2.9) and
$$\begin{aligned} H(z, \bar{z},w ):= f^{\vee } - \bigl\langle q^{*},f^{\vee } \bigr\rangle q- \bigl\langle \bar{q^{*}},f^{ \vee } \bigr\rangle \bar{q}. \end{aligned}$$
By calculation, we obtain
$$\begin{aligned} &\bigl\langle q^{*},f^{\vee } \bigr\rangle q= \begin{pmatrix} \frac{e(d_{0}+i\beta _{0})}{bd_{0}}(-\frac{bd_{0}i}{2e\beta _{0}}f_{1} +\frac{d_{0}i}{2\beta _{0}}f_{2} +\frac{f_{2}}{2}) \\ -\frac{bd_{0}i}{2e\beta _{0}}f_{1} +\frac{d_{0}i}{2\beta _{0}}f_{2} + \frac{f_{2}}{2} \end{pmatrix}, \\ &\bigl\langle \bar{q^{*}},f^{\vee } \bigr\rangle \bar{q} = \begin{pmatrix} \frac{e(d_{0}-i\beta _{0})}{bd_{0}}(\frac{bd_{0}i}{2e\beta _{0}}f_{1} -\frac{d_{0}i}{2\beta _{0}}f_{2} +\frac{f_{2}}{2}) \\ \frac{bd_{0}i}{2e\beta _{0}}f_{1} - \frac{d_{0}i}{2\beta _{0}}f_{2} + \frac{f_{2}}{2} \end{pmatrix}, \\ &\bigl\langle q^{*},f^{\vee } \bigr\rangle q - \bigl\langle \bar{q^{*}},f^{\vee } \bigr\rangle \bar{q} = \begin{pmatrix} f_{1} \\ f_{2} \end{pmatrix}. \end{aligned}$$
Then \(H(z, \bar{z},w ) = 0 \). Besides, the center manifold of system (3.14) is illustrated as follows:
$$\begin{aligned} w =\biggl(\frac{w_{20}}{2}\biggr) z^{2} + w_{11} z \bar{z} + \biggl(\frac{w_{02}}{2}\biggr) \bar{z} ^{2} + O \bigl( \vert z \vert ^{3} \bigr). \end{aligned}$$
Then we get
$$\begin{aligned} \textstyle\begin{cases} w_{20} =(2 i \beta _{0} (d_{0})I - L )^{-1} H_{20}, \\ w_{11} = (-L)^{-1}H_{11}, \\ w_{02} = (- 2 i \beta _{0} (d_{0}) I - L )^{-1} H_{02}. \end{cases}\displaystyle \end{aligned}$$
So \(w_{20} =w_{02} =w_{11} =0 \). Therefore, the restriction of (3.14) to the center manifold in z and z̄ coordinates is given by
$$\begin{aligned} \frac{dz}{dt} = i \beta _{0} (d_{0}) z + \bigl\langle q^{*},f^{\vee } \bigr\rangle = i \beta _{0} (d_{0})z + \frac{h_{20}}{2}z^{2} + h_{11}z \bar{z} + \frac{h_{02}}{2}\bar{z}^{2} + O \bigl( \vert z \vert ^{3} \bigr), \end{aligned}$$
(3.15)
where
$$\begin{aligned} & h_{20} = \bigl\langle q^{*}, B(q,q) \bigr\rangle \\ &\phantom{h_{20} }= \frac{re}{kbd_{0}\beta _{0}}\bigl(d_{0}^{2}i - 2\beta _{0}d_{0} - \beta _{0}^{2}i\bigr) + \frac{\beta _{0}d_{0}i + \beta _{0}^{2}}{v_{*}d_{0}} \\ &\phantom{h_{20} = }{} + \frac{am(m+bu_{*}+cv_{*}) + abcu_{*}v_{*}}{(m+bu_{*}+cv_{*})^{3}}\biggl( \frac{\beta _{0}i}{d_{0}}\biggr) \\ &\phantom{h_{20} = }{} + \frac{am}{(m+bu_{*}+cv_{*})^{2}}, \end{aligned}$$
(3.16)
$$\begin{aligned} &h_{11} = \bigl\langle q^{*},B(q,\bar{q}) \bigr\rangle \\ &\phantom{h_{11} } = \frac{re}{kbd_{0}\beta _{0}}\bigl(d_{0}^{2}i + \beta _{0}^{2}i\bigr) - \frac{\beta _{0}d_{0}i + \beta _{0}^{2}}{v_{*}d_{0}} \\ &\phantom{h_{11} =}{}+ \frac{am(m+bu_{*}+cv_{*}) + abcu_{*}v_{*}}{(m+bu_{*}+cv_{*})^{3}}\biggl(- \frac{\beta _{0}i}{d_{0}}\biggr), \end{aligned}$$
(3.17)
$$\begin{aligned} &h_{21} = \bigl\langle q^{*},C(q,q,\bar{q}) \bigr\rangle = \frac{3ab^{2}e^{2}v_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggl( \frac{d_{0}^{3}i -\beta _{0}^{3} - \beta _{0}d_{0}^{2} + \beta _{0}^{2}d_{0}i}{b^{2}d_{0}^{2}\beta _{0}}\biggr) \\ &\phantom{h_{21} = }{}+ \biggl( \frac{abm(m+bu_{*}+cv_{*})+2ab^{2}cu_{*}v_{*}-abcv_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggr) \end{aligned}$$
(3.18)
$$\begin{aligned} &\phantom{h_{21} = }{}\times \biggl( \frac{e(-3d_{0}^{2}i - \beta _{0}^{2}i + 2\beta _{0}d_{0})}{b\beta _{0}d_{0}}\biggr) \\ &\phantom{h_{21} = }{} + \biggl( \frac{acm(m+bu_{*}+cv_{*})+2abc^{2}u_{*}v_{*}-abcu_{*}(m+bu_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggr) \biggl( \frac{-3d_{0}i + \beta _{0}}{\beta _{0}}\biggr) \\ & \phantom{h_{21} = }{}+ \frac{3ac^{2}u_{*}(m+bu_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggl( \frac{bd_{0}i}{e\beta _{0}}\biggr)+ \frac{3\beta _{0}^{3}i + \beta _{0}^{2}d_{0}}{v_{*}^{2}d_{0}^{2}} \biggl(1+ \frac{d_{0}i}{\beta _{0}}\biggr), \\ &B(u,v) = \dbinom{B_{1}(u,v)}{B_{2}(u,v)},\qquad C(q,q,\bar{q}) = \dbinom{C_{1}(q,q,\bar{q})}{C_{2}(q,q,\bar{q})}, \\ &B_{1}(q,q) ={-} \frac{2re^{2}(d_{0} + i \beta _{0})^{2}}{kb^{2}d_{0}^{2}} + \frac{2av_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{3}}\biggl( \frac{e^{2}(i\beta _{0} d_{0} - \beta _{0}^{2})}{bd_{0}^{2}}\biggr) \\ &\phantom{B_{1}(q,q) =}{} - \frac{abcu_{*}v_{*}}{(m+bu_{*}+cv_{*})^{3}}\biggl( \frac{2e\beta _{0}i}{bd_{0}}\biggr), \\ &B_{2}(q,q) = \frac{2\beta _{0}^{2}}{v_{*}d_{0}}, \\ &B_{1}(q,\bar{q}) ={ -} \frac{2re^{2}(d_{0}^{2}+\beta _{0}^{2})}{kb^{2}d_{0}^{2}} + \frac{2av_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{3}}\biggl( \frac{e^{2}\beta _{0}^{2}}{bd_{0}^{2}}\biggr), \\ &B_{2}(q,\bar{q}) ={ -} \frac{2\beta _{0}^{2}}{v_{*}d_{0}}, \\ &C_{1}(q,q,\bar{q}) = {-} \frac{6ab^{2}e^{3}v_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggl( \frac{d_{0}^{3} +\beta _{0}^{3}i + \beta _{0}d_{0}^{2}i + \beta _{0}^{2}d_{0}}{b^{3}d_{0}^{3}}\biggr) \\ &\phantom{C_{1}(q,q,\bar{q}) = }{}- \frac{6ac^{2}u_{*}(m+bu_{*})}{(m+bu_{*}+cv_{*})^{4}} \\ &\phantom{C_{1}(q,q,\bar{q}) = } {}+\biggl( \frac{2abm(m+bu_{*}+cv_{*})+4ab^{2}cu_{*}v_{*}-2abcv_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggr) \\ &\phantom{C_{1}(q,q,\bar{q}) = }{}\times \biggl( \frac{e^{2}(3d_{0}^{2} + \beta _{0}^{2} + 2\beta _{0}d_{0}i)}{b^{2}d_{0}^{2}}\biggr) \\ & \phantom{C_{1}(q,q,\bar{q}) = }{}+ \biggl( \frac{2acm(m+bu_{*}+cv_{*})+4abc^{2}u_{*}v_{*} - 2abcu_{*}(m+bu_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggr) \\ &\phantom{C_{1}(q,q,\bar{q}) = }\times \biggl(\frac{e(3d_{0} + \beta _{0}i)}{bd_{0}}\biggr), \\ &C_{2}(q,q,\bar{q}) = \frac{6\beta _{0}^{3}i + 2\beta _{0}^{2}d_{0}}{v_{*}^{2}d_{0}^{2}}. \end{aligned}$$
(3.19)
According to [8],
$$\begin{aligned} \operatorname{Re} \bigl(c_{1}(0)\bigr) &= \operatorname{Re} \biggl( \frac{i}{2\beta _{0} (d_{0})}\biggl(h_{20}h_{11}-2 \vert h_{11} \vert ^{2} -\frac{1}{3} \vert h_{02} \vert ^{2} \biggr)+ \frac{h_{21}}{2}\biggr) \\ & = - \frac{1}{2 \beta _{0} (d_{0})}\operatorname{Im} (h_{20}h_{11}) + \frac{1}{2} \operatorname{Re} (h_{21}). \end{aligned}$$
Then from (3.16)–(3.18), we have
$$\begin{aligned} \operatorname{Re} c_{1}(0) = {}&\frac{1}{4}\biggl[- \frac{6av_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{4}}\biggl( \frac{e^{2}\beta _{0}^{2}}{d_{0}^{2}}\biggr) - \frac{2am}{(m+bu_{*}+cv_{*})^{2}}\biggl( \frac{re}{kbd_{0}} + \frac{red_{0}}{kb\beta _{0}^{2}}\biggr) \\ &{} - \frac{4av_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{3}} \biggl( \frac{erm}{kbd_{0}v_{*}}\biggr)+ \frac{4a^{2}mv_{*}(m+cv_{*})}{(m+bu_{*}+cv_{*})^{5}} \biggl( \frac{e}{d_{0}}\biggr) + \frac{4r^{2}em}{k^{2}b^{2}d_{0}v_{*}} \\ &{} + \frac{4r^{2}emd_{0}}{k^{2}b^{2}\beta _{0}^{2}v_{*}}\biggr]. \end{aligned}$$
Now we summarized the above results as follows.
Theorem 3.2
Assume that (2.2) and (2.5) are satisfied and let \(d = d_{0} \). Then system (3.9) has a Hopf bifurcation at \(U_{*} \). The direction of Hopf bifurcation is subcritical and the bifurcating periodic solutions are asymptotically orbitally stable when \(\operatorname{Re} c_{1}(0) < 0 \). The direction of Hopf bifurcation is supercritical and the bifurcating periodic solutions are unstable when \(\operatorname{Re} c_{1}(0) > 0 \).