In this section, we discuss the flip bifurcation and the N-S bifurcation of system (3.1), and we choose r as a bifurcation parameter for studying bifurcations.
4.1 Flip bifurcation
From Theorem 3.1(4)(a), system (3.1) has a unique positive equilibrium \(E_{*}\), the corresponding eigenvalues are \(\lambda _{1} = - 1\), \(\lambda _{2} = 2 + rx_{*}G - H + L\) with \(\vert \lambda _{2} \vert \ne 1\).
By selecting arbitrary parameters \(( r_{1},m,K,c,b,T,T_{n},n )\), we write system (3.1) in the form
$$ \textstyle\begin{cases} x \to x\exp [ \frac{r_{1} ( 1 - \frac{x}{K} ) ( x - n )}{x + m} - \frac{bTxy}{1 + cx + bT_{n}x^{2}} ], \\ y \to x [ 1 - \exp ( - \frac{bTxy}{1 + cx + bT_{n}x^{2}} ) ]. \end{cases} $$
(4.1)
Let \(u = x - x_{*}\), \(v = y - y_{*}\), \(\delta = r - r_{1}\), system (4.1) becomes
$$ \textstyle\begin{cases} u \to a_{100}u + a_{010}v + a_{001}\delta + a_{200}u^{2} + a_{110}uv + a_{101}u\delta + a_{011}v\delta \\ \phantom{u \to}{}+ a_{020}v^{2} + a_{002}\delta ^{2} + a _{300}u^{3} \\ \phantom{u \to}{}+ a_{210}u^{2}v + a_{201}u^{2}\delta + a_{111}uv\delta + a_{102}u \delta ^{2} + a_{120}uv^{2} + a_{021}v^{2}\delta\\ \phantom{u \to}{} + a_{012}v\delta ^{2} + a_{003}\delta ^{3} + a_{030}v^{3} + O ( 4 ), \\ v \to b_{100}u + b_{010}v + b_{200}u^{2} + b_{110}uv + b_{300}u^{3} + b_{210}u^{2}v + b_{020}v^{2}\\ \phantom{v \to}{} + b_{120}uv^{2} + b_{030}v^{3} + O ( 4 ), \end{cases} $$
(4.2)
where
$$\begin{aligned} &a_{100} = 1 + r_{1}x_{*}G - H,\qquad a_{010} = - \frac{bTx_{*}^{2}}{M},\\& a_{001} = \frac{x_{*} ( 1 - \frac{x_{*}}{K} ) ( x _{*} - n )}{x_{*} + m}, \\ &a_{200} = \frac{r_{1}G}{2} + r_{1}x_{*} \biggl[ ( m + n ) \biggl( \frac{\frac{x_{*} - m}{K} - 2}{2 ( x_{*} + m ) ^{3}}- \frac{1}{2K ( x_{*} + m )^{2}} \biggr) \biggr] \\ &\phantom{a_{200} =}{}- \frac{bTy_{*} - 3b^{2}TT_{n}x_{*}^{2}y^{*}}{2M^{2}} + \frac{ ( bTx ^{*}y^{*} - b^{2}TT_{n}x_{*}^{3}y_{*} ) ( 2c + 4bT_{n}x _{*} )}{2M^{3}} \\ &\phantom{a_{200} = }{} + ( 1 + r_{1}x_{*}G - H ) \biggl( \frac{r_{1}G}{2} - \frac{H}{2x _{*}} \biggr),\\ & a_{110} = \frac{ - bTx_{*} ( 1 + r_{1}x_{*}G - H )}{2M} - \frac{bTx_{*} - b^{2}TT_{n}x_{*}^{3}}{2M^{2}},\\ & a_{101} = \frac{x_{*}G}{2} + \frac{ ( 1 + r_{1}x_{*}G - H ) ( x_{*} - n ) ( 1 - \frac{x_{*}}{K} )}{2 ( x_{*} + m )}, \\ &a_{011} = - \frac{bTx_{*}^{2} ( x_{*} - n ) ( 1 - \frac{x _{*}}{K} )}{2M ( x_{*} + m )},\qquad a_{002} = \frac{x_{*} ( 1 - \frac{x_{*}}{K} )^{2} ( x _{*} - n )^{2}}{2 ( x_{*} + m )^{2}}, \\ &a_{300} = r_{1} ( m + n ) \biggl[ \frac{ \frac{x_{*} - m}{K} - 2}{3 ( x_{*} + m )^{3}} - \frac{1}{3K ( x_{*} + m )^{2}} + \frac{x_{*}}{K ( x_{*} + m ) ^{3}} - \frac{x_{*} ( \frac{x_{*} - m}{K} - 2 )}{ ( x _{*} + m )^{4}} \biggr] \\ &\phantom{a_{300} =}{}+ \frac{b^{2}TT_{n}x_{*}y_{*}}{M^{2}} \\ &\phantom{a_{300} =}{} + \frac{2bcTy_{*} + 6b^{2}TT_{n}x_{*}y_{*} - 6b^{2}cTT_{n}x_{*} ^{2}y_{*} - 14b^{3}TT_{n}x_{*}^{3}y_{*}}{3M^{3}} \\ &\phantom{a_{300} =}{}- \frac{ ( c + 2bT _{n}x_{*} )^{2} ( bTx_{*}y_{*} - b^{2}TT_{n}x_{*}^{3}y _{*} )}{M^{4}} \\ &\phantom{a_{300} =}{} + \biggl( \frac{r_{1}G}{3} - \frac{H}{3x} \biggr) \biggl\{ r_{1}G + rx_{*} \biggl[ ( m + n ) \biggl( \frac{ \frac{x_{*} - m}{K} - 2}{ ( x_{*} + m )^{3}} - \frac{1}{K ( x_{*} + m )^{2}} \biggr) \biggr]\\ &\phantom{a_{300} ={}} - \frac{bTy_{*} - 3b ^{2}TT_{n}x_{*}^{2}y^{*}}{M_{2}} \biggr\} \\ &\phantom{a_{300} =}{} + \biggl( r_{1}G - \frac{H}{x} \biggr) \biggl[ \frac{ ( bTx ^{*}y^{*} - b^{2}TT_{n}x_{*}^{3}y_{*} ) ( 2c + 4bT_{n}x _{*} )}{3M^{3}} - \frac{a_{200}}{6} \biggr] \\ &\phantom{a_{300} =} {}+ ( 1 + r_{1}x_{*}G - H ) \biggl[ \frac{r_{1}}{6} \biggl[ \frac{\frac{x_{*} - m}{K} - 2}{ ( x_{*} + m )^{3}} - \frac{1}{K ( x_{*} + m )^{2}} \biggr] \\ &\phantom{a_{300} =}{}+ \frac{b^{2}TT_{n}x_{*}y _{*}}{3M^{2}} + \frac{ ( bTy^{*} - b^{2}TT_{n}x_{*}^{2}y_{*} ) ( c + 2bT_{n}x_{*} )}{3M^{3}} \biggr] , \\ &a_{102} = \frac{ ( 1 - \frac{x_{*}}{K} ) ( x_{*} - n )}{x_{*} + m} \biggl[ \frac{x_{*}G}{3} + \frac{ ( 1 + r _{1}x_{*}G - H ) ( x_{*} - n ) ( 1 - \frac{x _{*}}{K} )}{6 ( x + m )} \biggr], \qquad a_{020} = \frac{b^{2}T^{2}x_{*}^{3}}{6M^{2}}, \\ &a_{201} = \frac{G}{6} + x_{*} ( m + n ) \biggl( \frac{\frac{x _{*} - m}{K} - 2}{6 ( x_{*} + m )^{3}} - \frac{1}{6K ( x_{*} + m )^{2}} \biggr) \\ &\phantom{a_{201} = }{}+ \frac{ ( 1 - \frac{x_{*}}{K} ) ( x_{*} - n )}{6 ( x_{*} + m )} \biggl\{ r _{1}G + r_{1}x_{*} \biggl[ ( m + n ) \biggl( \frac{\frac{x _{*} - m}{K} - 2}{ ( x_{*} + m )^{3}} - \frac{1}{K ( x_{*} + m )^{2}} \biggr) \biggr] \\ &\phantom{a_{201} =}{} - \frac{bTy_{*} - 3b^{2}T^{2}x_{*}^{2}y^{*}}{6M^{2}} - \frac{ ( bTx^{*}y^{*} - b^{2}T^{2}x_{*}^{3}y_{*} ) ( c + 2bTx _{*} )}{3M^{3}} \biggr\} \\ &\phantom{a_{201} =}{} + \biggl( \frac{r_{1}G}{6} - \frac{H}{3x_{*}} \biggr) \biggl[ x _{*}G + \frac{ ( 1 - \frac{x_{*}}{K} ) ( x_{*} - n ) ( 1 + r_{1}x_{*}G - H )}{x_{*} + m} \biggr] + \frac{1}{6}G ( 1 + r_{1}x_{*}G - H ),\\ & a_{111} = - \frac{ ( bTx_{*} - b^{2}TT_{n}x_{*}^{3} ) ( 1 - \frac{x _{*}}{K} ) ( x_{*} - n )}{6M^{2} ( x_{*} + m )} \\ &\phantom{a_{111} =}{}- \frac{bTx_{*}}{6M} \biggl[ x_{*}G + \frac{ ( 1 - \frac{x _{*}}{K} ) ( x_{*} - n ) ( 1 + r_{1}x_{*}G - H )}{x_{*} + m} \biggr], \\ &a_{120} = \frac{bTx_{*}}{M} \biggl[ \frac{bTx_{*} - b^{2}TT_{n}x_{*} ^{3}}{3M^{2}} + \frac{bTx_{*} ( 1 + r_{1}x_{*}G - H )}{6M} \biggr], \qquad a_{030} = - \frac{b^{3}T^{3}x_{*}^{4}}{6M^{3}}, \\ &a_{003} = \frac{x_{*} ( 1 - \frac{x_{*}}{K} )^{3} ( x_{*} - n )^{3}}{6 ( x_{*} + m )^{3}}, \qquad b_{100} = 1 - \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) + H \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr), \\ &b_{010} = L, \qquad b_{110} = \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) \biggl[ \frac{bTx _{*} - b^{2}TT_{n}x_{*}^{3}}{M^{2}} + \frac{bTx_{*} ( 1 - H )}{M} \biggr], \\ &b_{200} = \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) \biggl[ \frac{H - H^{2}}{2x_{*}} + \frac{bTy_{*} - 3b^{2}TT_{n}x_{*}^{2}y^{*}}{2M^{2}} \\ &\phantom{b_{200} = }{}- \frac{ ( bTx ^{*}y^{*} - b^{2}TT_{n}x_{*}^{3}y_{*} ) ( c + 2bT_{n}x _{*} )}{M^{3}} \biggr], \\ &b_{110} = \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) \biggl[ \frac{bTx _{*} - b^{2}TT_{n}x_{*}^{3}}{2M^{2}} + \frac{bTx_{*} ( 1 - H )}{2M} \biggr], \\ &b_{020} = - \frac{b^{2}T^{2}x_{*}^{3}}{2M^{2}}\exp \biggl( - \frac{bTx _{*}y_{*}}{M} \biggr),\\ & b_{120} = \frac{bTx_{*}}{2M}\exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) \biggl[ - \frac{H}{y_{*}} - \frac{bTx_{*}}{M} ( 1 - H ) \biggr], \\ &b_{300} = \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) ( 1 - H )\\ &\phantom{b_{300} = }{}\times \biggl( - \frac{H^{2}}{6x_{*}^{2}} - \frac{b^{2}TT_{n}x_{*}y _{*}}{3M^{2}} - \frac{ ( bTx^{*}y^{*} - b^{2}TT_{n}x_{*}^{2}y _{*} ) ( c + 2bT_{n}x_{*} )}{3M^{3}} \biggr) \\ &\phantom{b_{300} = }{} + \frac{H}{x_{*}}\exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) ( 1 - H )\\ &\phantom{b_{300} = }{}\times \biggl( \frac{ ( bTx^{*}y^{*} - b^{2}TT _{n}x_{*}^{3}y_{*} ) ( c + 2bT_{n}x_{*} )}{3M^{3}} - \frac{bTy_{*} - 3b^{2}TT_{n}x_{*}^{2}y^{*}}{6M^{2}} \biggr), \\ &b_{210} = \exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) \biggl( 1 - \frac{bTx _{*}y_{*}}{M} \biggr)\\ &\phantom{b_{210} =} {}\times\biggl( \frac{bT - 3b^{2}TT_{n}x_{*}^{2}}{6M ^{2}} - \frac{ ( bTx^{*} - b^{2}TT_{n}x_{*}^{3} ) ( c + 2bT_{n}x_{*} )}{3M^{3}} \biggr) \\ &\phantom{b_{300} = }{} - \frac{1}{6}\exp \biggl( - \frac{bTx_{*}y_{*}}{M} \biggr) \biggl( \frac{bTH}{M} ( 1 - H ) - \frac{H^{2}}{x_{*}y_{*}} + \frac{H ( 1 - H )}{x_{*}y_{*}} \biggr), \\ &b_{030} = \frac{b^{3}T^{3}x_{*}^{4}}{6M^{3}}\exp \biggl( - \frac{bTx _{*}y_{*}}{M} \biggr). \end{aligned}$$
We construct an invertible matrix:
using translation
system (4.2) becomes
(4.3)
where
$$\begin{aligned} &f ( U,V,\delta ) \\ &\quad = \frac{a_{001} ( \lambda _{2} - a _{100} )}{a_{010} ( 1 + \lambda _{2} )}\delta + \frac{a _{200} ( \lambda _{2} - a_{100} ) - a_{010}b_{200}}{a_{010} ( 1 + \lambda _{2} )}u^{2} + \frac{a_{110} ( \lambda _{2} - a_{100} ) - a_{010}b_{110}}{a_{010} ( 1 + \lambda _{2} )}uv \\ &\qquad {}+ \frac{a_{101} ( \lambda _{2} - a_{100} )}{a _{010} ( 1 + \lambda _{2} )}u\delta + \frac{a_{011} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}v\delta + \frac{a_{020} ( \lambda _{2} - a_{100} ) - a_{010}b_{020}}{a_{010} ( 1 + \lambda _{2} )}v^{2} \\ &\qquad {}+ \frac{a_{002} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}\delta ^{2} + \frac{a _{300} ( \lambda _{2} - a_{100} ) - a_{010}b_{300}}{a_{010} ( 1 + \lambda _{2} )}u^{3} \\ &\qquad {}+ \frac{a_{210} ( \lambda _{2} - a_{100} ) - a_{010}b _{210}}{a_{010} ( 1 + \lambda _{2} )}u^{2}v + \frac{a_{201} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}u^{2} \delta\\ &\qquad {} + \frac{a_{111} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}uv\delta + \frac{a _{102} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}u\delta ^{2} \\ &\qquad {} + \frac{a_{120} ( \lambda _{2} - a_{100} ) - a_{010}b _{120}}{a_{010} ( 1 + \lambda _{2} )}uv^{2} + \frac{a_{021} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}v^{2} \delta\\ &\qquad {} + \frac{a_{012} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )}v\delta ^{2} + \frac{a _{003} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )} \delta ^{3} \\ &\qquad {} + \frac{a_{030} ( \lambda _{2} - a_{100} ) - a_{010}b _{030}}{a_{010} ( 1 + \lambda _{2} )}v^{3} + O \bigl( \bigl( \vert u \vert + \vert v \vert + \vert \delta \vert \bigr)^{4} \bigr), \\ &g ( U,V,\delta ) \\ &\quad = \frac{a_{001} ( 1 + a_{100} )}{a _{010} ( 1 + \lambda _{2} )}\delta + \frac{a_{200} ( 1 + a_{100} ) + a_{010}b_{200}}{a_{010} ( 1 + \lambda _{2} )}u ^{2} + \frac{a_{110} ( 1 + a_{100} ) + a_{010}b_{110}}{a _{010} ( 1 + \lambda _{2} )}uv \\ &\qquad {}+ \frac{a_{101} ( 1 + a _{100} )}{a_{010} ( 1 + \lambda _{2} )}u\delta + \frac{a_{011} ( 1 + a_{100} ) + a_{010}b_{011}}{a _{010} ( 1 + \lambda _{2} )}v\delta + \frac{a_{020} ( 1 + a_{100} ) + a_{010}b_{020}}{a_{010} ( 1 + \lambda _{2} )}v^{2}\\ &\qquad {} + \frac{a_{002} ( 1 + a_{100} )}{a_{010} ( 1 + \lambda _{2} )}\delta ^{2} + \frac{a_{300} ( 1 + a_{100} ) + a_{010}b_{300}}{a_{010} ( 1 + \lambda _{2} )}u ^{3} + \frac{a_{210} ( 1 + a_{100} ) + a_{010}b_{210}}{a _{010} ( 1 + \lambda _{2} )} u^{2}v \\ &\qquad {}+ \frac{a_{201} ( 1 + a_{100} )}{a_{010} ( 1 + \lambda _{2} )}u^{2} \delta + \frac{a_{111} ( 1 + a_{100} )}{a_{010} ( 1 + \lambda _{2} )}uv\delta + \frac{a_{102} ( 1 + a_{100} )}{a _{010} ( 1 + \lambda _{2} )}u\delta ^{2} \\ &\qquad {}+ \frac{a_{120} ( 1 + a_{100} ) + a_{010}b_{120}}{a _{010} ( 1 + \lambda _{2} )} uv^{2} + \frac{a_{021} ( 1 + a_{100} )}{a_{010} ( 1 + \lambda _{2} )}v^{2} \delta + \frac{a_{012} ( 1 + a_{100} )}{a_{010} ( 1 + \lambda _{2} )}v\delta ^{2}\\ &\qquad {} + \frac{a_{003} ( 1 + a_{100} )}{a _{010} ( 1 + \lambda _{2} )} \delta ^{3} + \frac{a_{030} ( 1 + a_{100} ) + a_{010}b_{030}}{a _{010} ( 1 + \lambda _{2} )} v^{3} + O \bigl( \bigl( \vert u \vert + \vert v \vert + \vert \delta \vert \bigr) ^{4} \bigr),\\ & u = a_{010} ( U + V ), v = ( - 1 - a_{100} )U + ( \lambda _{2} - a_{100} )V. \end{aligned}$$
Now we determine the center manifold of (4.3) at equilibrium point \(( 0,0 )\) in a small neighborhood of \(\delta = 0\). We can obtain that there exists a center manifold by the center manifold theorem, which can be written as follows:
$$ W^{c} ( 0,0 ) = \bigl\{ ( U,V ) \in R^{2}:Y = h ( U, \delta ) = c_{0}\delta + c_{1}U^{2} + c_{2}U \delta + c_{3}\delta ^{2} + O \bigl( \bigl( \vert U \vert + \vert \delta \vert \bigr)^{3} \bigr) \bigr\} , $$
where
$$\begin{aligned} &c_{0} = \frac{a_{001} ( 1 + a_{100} )}{a_{010} ( 1 - \lambda _{2}^{2} )},\qquad c_{1} = \frac{a_{200} ( 1 + a_{100} ) + a_{010}b_{200}}{a _{010} ( 1 - \lambda _{2}^{2} )}, \\ &c_{2} = - \frac{c_{0} [ a_{110} ( 1 + a_{100} ) + a _{010}b_{110} ] + a_{101} ( 1 + a_{100} ) + 2c_{1}a _{001} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )^{2}}, \\ &c_{3} = \frac{c_{0} [ a_{011} ( 1 + a_{100} ) + a _{010}b_{011} ] + c_{0}^{2} [ a_{020} ( 1 + a_{100} ) + a_{010}b_{020} ] + a_{002} ( 1 + a_{100} ) - c_{2}a_{001} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 - \lambda _{2}^{2} )} \\ &\phantom{c_{3} = }{} - \frac{c_{1} [ a_{001} ( \lambda _{2} - a_{100} ) ]^{2}}{a_{010}^{2} ( 1 + \lambda _{2} ) ( 1 - \lambda _{2}^{2} )}. \end{aligned}$$
We consider the following map originating from (4.3) restricted to the center manifold \(W^{c} ( 0,0 )\):
$$\begin{aligned} F:U \to {}&{-} U + \frac{a_{001} ( \lambda _{2} - a_{100} )}{a _{010} ( 1 + \lambda _{2} )}\delta + \frac{a_{200} ( \lambda _{2} - a_{100} ) - a_{010}b_{200}}{a_{010} ( 1 + \lambda _{2} )}u^{2} \\ & {}+ \biggl[ c_{0}\frac{a_{110} ( \lambda _{2} - a_{100} ) - a_{010}b_{110}}{a_{010} ( 1 + \lambda _{2} )}+ \frac{a _{101} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )} \biggr]u\delta \\ &{}+ \frac{a_{300} ( \lambda _{2} - a_{100} ) - a_{010}b_{300}}{a_{010} ( 1 + \lambda _{2} )}u^{3} \\ & {} + \biggl[ c_{0}\frac{a_{011} ( \lambda _{2} - a_{100} )}{a _{010} ( 1 + \lambda _{2} )} + c_{0}^{2} \frac{a_{020} ( \lambda _{2} - a_{100} ) - a_{010}b_{020}}{a_{010} ( 1 + \lambda _{2} )} + \frac{a_{002} ( \lambda _{2} - a_{100} )}{a _{010} ( 1 + \lambda _{2} )} \biggr]\delta ^{2} \\ & {} + \biggl[ c_{0}\frac{a_{021} ( \lambda _{2} - a_{100} )}{a _{010} ( 1 + \lambda _{2} )} + c_{0} \frac{a_{012} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )} + \frac{a_{003} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )} \\ &{}+ c_{0} \frac{a_{030} ( \lambda _{2} - a _{100} ) - a_{010}b_{030}}{a_{010} ( 1 + \lambda _{2} )} \biggr]\delta ^{3} \\ & {} + \biggl[ c_{0}\frac{a_{111} ( \lambda _{2} - a_{100} )}{a _{010} ( 1 + \lambda _{2} )} + \frac{a_{102} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )} + c _{0}^{2}\frac{a_{120} ( \lambda _{2} - a_{100} ) - a_{010}b _{120}}{a_{010} ( 1 + \lambda _{2} )} \biggr]u\delta ^{2} \\ & {} + \biggl[ c_{0}\frac{a_{210} ( \lambda _{2} - a_{100} ) - a_{010}b_{210}}{a_{010} ( 1 + \lambda _{2} )} + \frac{a _{201} ( \lambda _{2} - a_{100} )}{a_{010} ( 1 + \lambda _{2} )} \biggr]u^{2}\delta + O ( 4 ). \end{aligned}$$
(4.4)
To enable Eq. (4.4) to undergo a flip bifurcation, it requires two discriminatory quantities \(\eta _{1}\) and \(\eta _{2}\) to be not zero, where
$$ \textstyle\begin{cases} \eta _{1} = ( \frac{2\partial ^{2}F}{\partial U\,\partial \delta } + \frac{\partial F}{\partial \delta } \frac{\partial ^{2}F}{\partial U ^{2}} )_{ ( 0,0 )} \ne 0, \\ \eta _{2} = ( \frac{1}{3}\frac{\partial ^{3}F}{\partial U^{3}} + \frac{1}{2} ( \frac{\partial ^{2}F}{\partial U^{2}} )^{2} ) _{ ( 0,0 )} \ne 0. \end{cases} $$
Therefore, based on the above analysis and the theorem in [24], we obtain the following theorem.
Theorem 4.1
If
\(\eta _{2} \ne 0\), the parameterδalters in the limited region of the point
\(( 0,0 )\), then system (3.1) undergoes a flip bifurcation at
\(E_{*}\). Moreover, the period-2 orbit that bifurcates from
\(E_{*}\)is stable (unstable) if
\(\eta _{2} > 0\ ( \eta _{2} < 0 )\).
4.2 N-S bifurcation
From Theorem 3.1(4)(b), by using the bifurcation theorem [26,27,28] and selecting arbitrary parameters \(( r_{2},m,K,c,b,T,T_{n},n )\), we write system (3.1) in the form
$$ \textstyle\begin{cases} x \to x\exp [ \frac{r_{2} ( 1 - \frac{x}{K} ) ( x - n )}{x + m} - \frac{bTxy}{1 + cx + bT_{n}x^{2}} ], \\ y \to x [ 1 - \exp ( - \frac{bTxy}{1 + cx + bT_{n}x^{2}} ) ], \end{cases} $$
(4.5)
\(E_{*}\) is the only positive equilibrium of system (4.5). We consider the following perturbation of (4.5), with \(\delta _{*}\) used as the bifurcation parameter:
$$\textstyle\begin{cases} x \to x\exp [ \frac{ ( \delta _{*} + r_{2} ) ( 1 - \frac{x}{K} ) ( x - n )}{x + m} - \frac{bTxy}{1 + cx + bT_{n}x^{2}} ], \\ y \to x [ 1 - \exp ( - \frac{bTxy}{1 + cx + bT_{n}x^{2}} ) ], \end{cases} $$
(4.6)
where \(\vert \delta _{*} \vert \ll 1\).
Let \(u = x - x_{*}\), \(v = y - y_{*}\), the equilibrium \(E_{*}\) is transformed to the origin point \(( 0,0 )\), we obtain
$$ \textstyle\begin{cases} u \to a_{100}u + a_{010}v + a_{200}u^{2} + a_{110}uv + a_{020}v^{2} + a_{300}u^{3} + a_{210}u^{2}v \\ \phantom{u \to}{}+ a_{120}uv^{2} + a_{030}v^{3} + O ( 4 ), \\ v \to b_{100}u + b_{010}v + b_{200}u^{2} + b_{110}uv + b_{300}u^{3} + b_{210}u^{2}v + b_{020}v^{2}\\ \phantom{u \to}{} + b_{120}uv^{2} + b_{030}v^{3} + O ( 4 ), \end{cases} $$
(4.7)
where the coefficient is given in (4.2) and \(r = \delta _{*} + r_{2}\). The characteristic equation associated with the linearization of system (4.7) at \(( 0,0 )\) is given by
$$ \lambda ^{2} + p ( \delta _{*} )\lambda + q ( \delta _{*} ) = 0, $$
(4.8)
where
$$\begin{aligned} &p ( \delta _{*} ) = - \bigl[ 1 + ( \delta _{*} + r _{2} )x_{*}G - H + L \bigr], \\ &q ( \delta _{*} ) = ( \delta _{*} + r_{2} )x _{*}GL + \frac{bTx_{*}^{2}}{M}, \end{aligned}$$
we obtain
$$ \lambda _{1,2} = - \frac{p ( \delta _{*} )}{2} \pm \frac{i}{2} \sqrt{4q ( \delta _{*} ) - p^{2} ( \delta _{*} )}, $$
and
$$ \vert \lambda \vert = \sqrt{q ( \delta _{*} )},\qquad d = \frac{d \vert \lambda \vert }{d\delta _{*}} \bigg| _{\delta _{*} = 0} = \frac{x_{*}GL}{2} \ne 0. $$
Moreover, if \(\delta _{*} = 0\), we have \(\lambda _{1,2}^{k} \ne 1\ ( k = 1,2,3,4 )\), which is equivalent to \(p ( 0 ) \ne - 2,0,1,2\). Based on Theorem 3.1(b), we have \(p ( 0 ) \ne - 2,2\), then we only need to require \(p ( 0 ) \ne 0,1\), which leads to
$$ ( H - L )L, ( H - L - 1 )L \ne 1 - \frac{bTx ^{2}}{M}. $$
(4.9)
Therefore, the eigenvalues \(\lambda _{1,2}\) do not lie in the intersection of the unit circle with the coordinate axes when \(\delta _{*} = 0\) and condition (4.9) holds.
Let \(\delta _{*} = 0\), \(\mu = - \frac{p ( 0 )}{2}\), \(\omega = \frac{ \sqrt{4q ( 0 ) - p^{2} ( 0 )}}{2}\), we make an invertible matrix:
using translation
system (4.7) becomes
where
$$\begin{aligned} &\overline{f} ( U,V ) = \frac{1}{a_{010}} \bigl( a_{200}u ^{2} + a_{110}uv + a_{020}v^{2} + a_{300}u^{3} + a_{210}u^{2}v + a _{120}uv^{2} + a_{030}v^{3} \bigr) + O ( 4 ), \\ &\overline{g} ( U,V ) = \biggl( \frac{a_{200} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{200}}{\omega } \biggr)u ^{2} + \biggl( \frac{a_{110} ( \mu - a_{100} )}{\omega a _{010}} - \frac{b_{110}}{\omega } \biggr)uv\\ &\phantom{\overline{g} ( U,V ) =}{} + \biggl( \frac{a_{020} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{020}}{ \omega } \biggr)v^{2} \\ & \phantom{\overline{g} ( U,V ) =}{} + \biggl( \frac{a_{300} ( \mu - a_{100} )}{\omega a _{010}} - \frac{b_{300}}{\omega } \biggr)u^{3} + \biggl( \frac{a_{210} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{210}}{ \omega } \biggr)u^{2}v \\ &\phantom{\overline{g} ( U,V ) =}{}+ \biggl( \frac{a_{120} ( \mu - a_{100} )}{ \omega a_{010}} - \frac{b_{120}}{\omega } \biggr)uv^{2} \\ & \phantom{\overline{g} ( U,V ) =}{} + \biggl( \frac{a_{030} ( \mu - a_{100} )}{\omega a _{010}} - \frac{b_{030}}{\omega } \biggr)v^{3} + O \bigl( \bigl( \vert u \vert + \vert v \vert \bigr)^{4} \bigr), \end{aligned}$$
and
$$\begin{aligned} &u^{2} = a_{010}^{2}U^{2}, \qquad uv = a_{010} ( \mu - a_{100} )U^{2} - a_{010}\omega UV, \\ &v^{2} = ( \mu - a_{100} )^{2}U^{2} - 2\omega ( \mu - a_{100} )UV + \omega ^{2}V^{2},\qquad u^{3} = a_{010}^{3}U^{3}, \\ &u^{2}v = a_{010}^{2} ( \mu - a_{100} )U^{3} - a_{010}^{2} \omega U^{2}V, \\ &uv^{2} = a_{010} ( \mu - a_{100} )^{2}U^{3} - 2a_{010} \omega U^{2}V + a_{010}\omega ^{2}UV^{2}, \\ &v^{3} = ( \mu - a_{100} )U^{3} - \omega ^{3}V^{3} - 3 \omega ( \mu - a_{100} )^{2}U^{2}V + 3 ( \mu - a _{100} )\omega ^{2}UV^{2}. \end{aligned}$$
Therefore
$$\begin{aligned} &\overline{f}_{UU} = 2a_{200}a_{010} + 2a_{110} ( \mu - a_{100} ) + \frac{2a_{020} ( \mu - a_{100} )^{2}}{a_{010}}, \\ &\overline{f}_{UV} = - a_{110}\omega - \frac{2a_{020}\omega ( \mu - a_{100} )}{a_{010}},\qquad \overline{f}_{VV} = \frac{2a_{020}\omega ^{2}}{a_{010}}, \\ &\overline{f}_{UUU} = 6a_{300}a_{010}^{2} + 6a_{210}a_{010} ( \mu - a_{100} ) + 6a_{120} ( \mu - a_{100} )^{2} + \frac{6a_{030} ( \mu - a_{100} )}{a_{010}}, \\ &\overline{f}_{UUV} = - 2a_{010}a_{210}\omega - 4a_{120}\omega - \frac{6a _{030}\omega ( \mu - a_{100} )^{2}}{a_{010}},\qquad \overline{f}_{VVV} = - \frac{6a_{030}\omega ^{3}}{a_{010}}, \\ &\overline{f}_{UVV} = 2a_{120}\omega ^{2} + \frac{6a_{030} ( \mu - a_{100} )\omega ^{2}}{a_{010}}, \\ &\overline{g}_{UU} = 2a_{010}^{2} \biggl( \frac{a_{200} ( \mu - a _{100} )}{\omega a_{010}} - \frac{b_{200}}{\omega } \biggr) + 2a_{010} ( \mu - a_{100} ) \biggl( \frac{a_{110} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{110}}{\omega } \biggr) \\ & \phantom{\overline{g}_{UU} =}{}+ 2 ( \mu - a_{100} )^{2} \biggl( \frac{a_{020} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{020}}{\omega } \biggr), \\ &\overline{g}_{UV} = - a_{010}\omega \biggl( \frac{a_{110} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{110}}{\omega } \biggr) - 2\omega ( \mu - a_{100} ) \biggl( \frac{a_{020} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{020}}{ \omega } \biggr), \\ &\overline{g}_{VV} = 2\omega ^{2} \biggl( \frac{a_{020} ( \mu - a _{100} )}{\omega a_{010}} - \frac{b_{020}}{\omega } \biggr), \\ &\overline{g}_{UUU} = 6a_{010}^{3} \biggl( \frac{a_{300} ( \mu - a _{100} )}{\omega a_{010}} - \frac{b_{300}}{\omega } \biggr) + 6a_{010}^{2} ( \mu - a_{100} ) \biggl( \frac{a_{210} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{210}}{\omega } \biggr) \\ &\phantom{\overline{g}_{UUU} = }{} + 6a_{010} ( \mu - a_{100} )^{2} \biggl( \frac{a_{120} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{120}}{ \omega } \biggr) \\ &\phantom{\overline{g}_{UUU} = }{}+ 6 ( \mu - a_{100} ) \biggl( \frac{a _{030} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{030}}{ \omega } \biggr), \\ &\overline{g}_{UVV} = 2a_{010}\omega ^{2} \biggl( \frac{a_{120} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{120}}{\omega } \biggr) + 6 ( \mu - a_{100} )\omega ^{2} \biggl( \frac{a _{030} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{030}}{ \omega } \biggr), \\ &\overline{g}_{UUV} = - 2a_{010}^{2}\omega \biggl( \frac{a_{210} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{210}}{\omega } \biggr) - 4a_{010}\omega \biggl( \frac{a_{120} ( \mu - a_{100} )}{ \omega a_{010}} - \frac{b_{120}}{\omega } \biggr) \\ &\phantom{\overline{g}_{UUV} =}{} - 6\omega ( \mu - a_{100} )^{2} \biggl( \frac{a_{030} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{030}}{ \omega } \biggr), \\ &\overline{g}_{VVV} = - 6\omega ^{3} \biggl( \frac{a_{030} ( \mu - a_{100} )}{\omega a_{010}} - \frac{b_{030}}{\omega } \biggr). \end{aligned}$$
To enable system (4.7) to undergo an N-S bifurcation, we require the following discriminatory quantity θ to be not zero:
$$ \theta = - \biggl[ \operatorname{Re} \biggl( \frac{ ( 1 - 2 \lambda )\overline{\lambda }^{2}}{1 - \lambda } \xi _{20}\xi _{11} \biggr) - \frac{1}{2} \vert \xi _{11} \vert ^{2} - \vert \xi _{02} \vert ^{2} + \operatorname{Re} ( \overline{\lambda } \xi _{21} ) \biggr]_{\delta = 0}, $$
where
$$\begin{aligned} &\xi _{20} = \frac{1}{8} \bigl[ \overline{f}_{UU} - \overline{f}_{VV} + 2\overline{g}_{UV} + i ( \overline{g}_{UU} - \overline{g}_{VV} - 2 \overline{f}_{UV} ) \bigr], \\ &\xi _{11} = \frac{1}{4} \bigl[ \overline{f}_{UU} + \overline{f}_{VV} + i ( \overline{g}_{UU} + \overline{g}_{VV} ) \bigr], \\ &\xi _{02} = \frac{1}{8} \bigl[ \overline{f}_{UU} - \overline{f}_{VV} - 2\overline{g}_{UV} + i ( \overline{g}_{UU} - \overline{g}_{VV} + \overline{f}_{UV} ) \bigr], \\ &\xi _{21} = \frac{1}{16} \bigl[ \overline{f}_{UUU} + \overline{f}_{VVV} + 2\overline{g}_{UUV} + i ( \overline{g}_{UUU} + \overline{g} _{UVV} - 2 \overline{f}_{VVV} ) \bigr]. \end{aligned}$$
Therefore, according to the above analysis and the theorem in [24], we obtain the following theorem.
Theorem 4.2
System (3.1) undergoes an N-S bifurcation at equilibrium
\(E_{*}\)if conditions in Theorem
3.1(4)(b) and
\(\theta \ne 0\)hold and
\(\delta _{*}\)varies in a small vicinity of the origin. Moreover, if
\(\theta < 0\) (or
\(\theta > 0\)), then an attracting (or repelling) invariant closed curve bifurcates from
\(E_{*}\)for
\(\delta _{*} > 0\) (or
\(\delta _{*} < 0\)).