Discrete competition models, including intra-specific competition models and interspecific competition models, play a important role in theoretical ecology and economics (see e.g. [1, 3, 7, 8]). Intra-specific competition refers to the competition among individuals of same species and interspecific competition to the competition between two or more species for some limiting resource. When one species is a better competitor, interspecific competition negatively influences the other species by reducing population sizes or growth rates, which in turn affects the population dynamics of the competitor. In 2011, based on the biological assumptions that each species is modeled by the logistic map, modeled species with non-overlapping generations, without interspecific competition and that one species will negatively affect the growth of the other species in the interspecific competition, Guzowska, Luís and Elaydi [5] developed the following logistic competition model

$ \displaystyle x_{n+1}=\frac{ax_{n}(1-x_{n})}{1+cy_{n}}, \\ \\ \displaystyle y_{n+1}=\frac{by_{n}(1-y_{n})}{1+dx_{n}}, $

(1.1)

where $x_{n}\in[0,1]$ and $y_{n}\in[0,1]$ represent the species $x$ density and species $y$ density at time $n$ respectively, the parameters $a\in(0,4]$ and $b\in(0,4]$ denote the intrinsic growth rates of species $x$ and $y$ respectively, and the parameters $c\in (0,1)$ and $d\in(0,1)$ are the competition parameters of species $y$ and $x$ respectively. As indicted in [5], system (1.1) has one extinction fixed point $E_{0}(0,0)$, two exclusion fixed points

$ E_{1}((a-1)/a,0),~~ E_{2}(0,(b-1)/b) $

and one coexistence fixed point $E_{3}(x_{0}, y_{0})$, where

$ x_{0}=\frac{ab-cb+c-b}{ab-cd}, ~~~y_{0}=\frac{ab-da-a+d}{ab-cd}. $

In [5] the stability of the four fixed points were investigated by center manifold theorem and Schwarzian derivative, the bifurcation scenario at $E_3$ is given in the parameter space, and, at fixed point $E_3$, fold and flip bifurcations route to chaos are exhibited via numerical simulations.

Up to now, what bifurcations happen at fixed points $E_0, E_1$ and $E_2$ is unknown. In this paper we discuss analytically these bifurcations. At first we give all topological types of the three fixed points and all non-hyperbolic cases in order to investigate bifurcations. Then, we show that system (1.1) undergoes transcritical bifurcation at $E_0$ as $(a, b)$ crossing two bifurcation curves $a=1$ or $b=1$. At last, we prove that, at fixed point $E_1$ (resp. $E_2$), a flip bifurcation occurs for $(a, b)$ crossing curves $a=3$ (resp. $b=3$) and a transcritical bifurcation happens for $(a, b)$ crossing $b=1+d(a-1)/a$ (resp. $a= 1+c(b-1)/b$).

System (1.1) can be described equivalently by the planar mapping $F:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$,

$ F(x, y)=(\frac{ax(1-x)}{1+cy}, \frac{by(1-y)}{1+dx}), $

(2.1)

whose Jacobian is given by

$ JF(x, y)=\begin{pmatrix} \frac{a(1-x)-ax}{1+cy}&\frac{-acx(1-x)}{(1+cy)^{2}} \\ \\ \frac{-bdy(1-y)}{(1+dx)^{2}}& \frac{b(1-y)-by}{1+dx} \\ \end{pmatrix}. $

(2.2)

We first give the topological types of fixed point $E_{0}$ and non-hyperbolic cases.

Lemma 1   Fixed point $E_{0}$ is non-hyperbolic if and only if $a=1$ or $b=1$. Otherwise, $E_0$ is one of the types in Table 1.

Proof   By (2.2) the Jacobian evaluated at the fixed point $E_{0}(0, 0)$ is given by

$ JF(0, 0)=\begin{pmatrix} a&0 \\ 0&b \\ \end{pmatrix}, $

(2.3)

which has eigenvalues $\lambda_{1}=a$ and $\lambda_{2}=b$. Hence it is easy to obtain the results in Table 1 by [4] (see p. 194-200).

From the lemma it is obvious that the bifurcations occur at the fixed point $E_{0}$ if $a=1$ or $b=1$.

Theorem 1   If $a$ (resp. $b$) crosses 1 and $b\neq 1$ (resp. $a\neq 1$), then the map $F$ undergoes a transcritical bifurcation at fixed point $E_{0}(0, 0)$.

Proof   We prove one case that $a$ crosses 1 and $b\neq 1$. The proof of the other case is similar. From $0<c<1, 0<d<1, 0<x<1$ and $0<y<1$, map (2.1) can expand the following form at $a=1:$

$ \left( \begin{array}{c} x \\ y \end{array}\right)\rightarrow\left(\begin{array}{c} x \\ by \end{array}\right)- \left( \begin{array}{c} x^{2}+cxy+\mathcal{O}(3) \\ by^{2}+bdxy+\mathcal{O}(3) \\ \end{array} \right), $

(2.4)

where $\mathcal{O}(3)$ is a function with order at least $3$ in the variables. We choose $\epsilon=a-1$ as a bifurcation parameter to study the bifurcation of the mapping $F$ at the fixed point $E_{0}(0, 0)$, where $|\epsilon|\ll 1$. We consider a perturbation of (2.4) as follows:

$ \left( \begin{array}{c} x \\ y \end{array}\right)\rightarrow\left(\begin{array}{c} x \\ by \end{array}\right)- \left( \begin{array}{c} x^{2}-x\epsilon+cxy+\mathcal{O}(3) \\ by^{2}+bdxy+\mathcal{O}(3) \\ \end{array} \right). $

(2.5)

System (2.4) can be rewritten in the following suspended form

$ \left( \begin{array}{c} x \\ \epsilon\\ y \end{array}\right)\rightarrow\left(\begin{array}{c} x \\ \epsilon\\ by \end{array}\right)- \left( \begin{array}{c} x^{2}-x\epsilon+cxy+\mathcal{O}(3), \\ 0\\ by^{2}+bdxy+\mathcal{O}(3), \end{array} \right). $

(2.6)

By the center manifold theory (see p. 33--35 in [2]) the center manifold of system (2.6) can expressed locally as follows:

$ W^{c}(O)=\{(x, y, \varepsilon)\in \mathbb{R}^{3} | y=h(x, \epsilon), h(0, 0)=D h(0, 0)=0, |x|<\varepsilon, |\epsilon|<\delta\}, $

where $\varepsilon$ and $\delta$ are sufficient small positives. Assume that $h(x, \epsilon)$ has the following form

$ y=h(x, \epsilon)=c_{1}x^{2}+c_{2}x \epsilon+c_{3}\epsilon^{2}+\mathcal{O}(3), $

(2.7)

which must satisfy

$ h\left(x-x^{2}+x\epsilon-cxh(x, \epsilon)+\mathcal{O}(3), \epsilon\right)=bh(x, \epsilon)-bh^2(x, \epsilon)-bdxh(x, \epsilon)+\mathcal{O}(3) $

(2.8)

by the center manifold theorem. Comparing coefficients of $x^{2}, x\epsilon$ and $\epsilon^{2}$ in (2.8) we obtain that

$ c_{1}=c_{2}= c_{3}=0, $

and (2.7) has the determinative form

$ y=h(x, \epsilon)=\mathcal{O}(3). $

(2.9)

Substituting (2.9) into the first two equations in (2.6) yields

$ \left( \begin{array}{c} x\\ \epsilon \end{array}\right)\rightarrow\left(\begin{array}{c} x-x^{2}+x\epsilon+\mathcal{O}(3)\\ \epsilon \end{array}\right), $

which defines a one-dimensional mapping $(x, \epsilon)\rightarrow f_{1}(x, \epsilon)$ by

$ f_{1}(x, \epsilon)=x-x^{2}+x\epsilon+\mathcal{O}(3). $

From

$ \left.\frac{\partial^{2}f_{1}}{\partial x \partial \epsilon}\right|_{(x, \epsilon)=(0, 0)}=1 $

(2.10)

and

$ \left.\frac{\partial^{2}f_{1}}{\partial x^{2}}\right|_{(x, \epsilon)=(0, 0)}=-2, $

we get that the map $F$ undergoes a transcritical bifurcation on the center manifold at $E_{0}$ (see p. 504--507 in [10]). This completes the proof.

In order that $E_{1}$ has biological significance, we have $a>1$. By (2.2) the Jacobian evaluated at the fixed point $E_{1}$ is given by

$ JF((a-1)/a, 0)=\begin{pmatrix} 2-a&-\frac{c}{a}(a-1) \\ 0&\frac{ab}{ad+a-d} \\ \end{pmatrix}, $

(3.1)

whose eigenvalues are $\lambda_{1}=2-a$ and $\lambda_{2}=ab/(ad+a-d)$. Hence we have the following results.

Lemma 2  The fixed point $E_{1}$ is not hyperbolic if and only if $a=3$ or $b=1+d(a-1)/a$. Otherwise, $E_1$ is one of the types in Table 2.

Proof  Solving $|\lambda_1|=|2-a|<1$ yields $1<a<3$. Obviously $\lambda_2>0$, from $\lambda_2=ab/(ad+a-d)<1$ we get $0<b<1+d(a-1)/a$. Hence $E_1$ is stable node for $1<a\leq3$ and $0<b<1+d(a-1)/a$ (refers to case $E_{1}$-Ⅰ). Similarly, we can obtain the other three cases in Table 2. This completes the proof.

From the lemma, it is obvious that the bifurcation occurs at the fixed point $E_{1}$ if $a=3$ or $b=1+d(a-1)/a$. Let $u=x-(a-1)/a$ and $v=y$. Then we get map $\tilde{F}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$,

$ \left( \begin{array}{c} u \\ v \\ \end{array} \right)\rightarrow \left( \begin{array}{c} (2-a)u-\frac{c}{a}(a-1)v-au^{2}-c(2-a)uv+\frac{c^{2}}{a}(a-1)v^{2}+\mathcal{O}_{1}(3) \\ \frac{ab}{a+d(a-1)}v-\frac{a^{2}bd}{(a+d(a-1))^{2}}uv-\frac{ab}{a+d(a-1)}v^{2}+\mathcal{O}_{2}(3)\\ \end{array} \right). $

(3.2)

Note that there isn't the term $u^{3}$ in $\mathcal{O}_{1}(3)$ and the term $v^{3}$ in $\mathcal{O}_{2}(3)$. Its Jacobian evaluated at the $O$, $J\tilde{F}(0, 0)$, is equal to $JF((a-1)/a, 0)$. One can easily see that the matrix $J\tilde{F}(0, 0)$ has eigenvectors $(1, 0)^{T}$ and

$ \left(\frac{ad+a-b}{(2-a)(ad+a-d)-ab}, \frac{c}{a}(a-1)\right)^{T} $

corresponding to $\lambda_{1}=2-a$ and $\lambda_{2}=ab/(ad+a-d)$, respectively, where $T$ denotes the transpose of matrices. One can check that $\lambda_{1}\neq \lambda_{2}$ if $a=3$ or $b=1+d(a-1)/a$. Hence the matrix $J\tilde{F}(0, 0)$ can be diagonalized by the change of variables $(u, v)^{T}=H_{1}(\xi, \eta)^{T}$, where

$ H_{1}=\left( \begin{array}{cc} 1&\frac{ad+a-b}{(2-a)(ad+a-d)-ab} \\ \\ 0&\frac{c}{a}(a-1) \\ \end{array} \right), $

and therefore the map $\tilde{F}$ can be changed into the mapping $G:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$,

$ \left( \begin{array}{c} \xi \\ \eta \\ \end{array} \right)\rightarrow \left( \begin{array}{c} h_{10}\xi \\ g_{01}\eta \\ \end{array} \right)+\left( \begin{array}{c} h_{20}\xi^{2}+h_{11}\xi\eta+h_{02}\eta^{2}+\mathcal{O}_{1}(3) \\ g_{11}\xi\eta+g_{02}\eta^{2}+\mathcal{O}_{2}(3) \\ \end{array} \right), $

(3.3)

where

$ h_{10}\quad:=\quad2-a,~~g_{01}:=\frac{ab}{ad+a-d},~~h_{20}:=-a,~~g_{11}=-a^2bd/(ad+a-d)^2,\\ $

$ h_{11}\quad:=\quad(-11a^4bd-17a^2bd^2+6d^2ab+19a^3bd+2a^5bd+17a^3bd^2-a^3b^2d+a^2b^2d\\ \quad\quad\quad\quad\quad-7a^4bd^2-10a^2bd-4a^4d^2b+7a^3d^2b-6a^4db+8a^3db-6a^{2}d^2b-4a^{2}db\\ \quad\quad\quad\quad\quad-2a^3b^{2}+2a^5db+a^5b+12a^2d+a^6d^3+a^5b-4a^4b+4a^3b+3a^6d^2\\ \quad\quad\quad\quad\quad+3a^6d-2a^4b+2a^3b+2d^2ab+42a^2d^2-12ad^2-60a^3d^2-30a^3d\\ \quad\quad\quad\quad\quad+21a^4d^3+45a^4d^2-35a^3d^3-7a^5d^3-18a^5d^2+34a^2d^3+30a^4d\\ \quad\quad\quad\quad\quad-15a^5d-18ad^3-4a^3-4a^5+6a^4+4d^3+a^6+a^5bd^2+a^4b^{2}d+a^4b^{2}\\ \quad\quad\quad\quad\quad+a^5d^2b-3a^3b^{2}d+2da^{2}b^{2})a/((ad+a-d)(-3ad-2a+2d\\ \quad\quad\quad\quad\quad+a^2d+a^2+ab)(-3ad-2a+2d+a^2d+a^2+ab)(a-1)), $

$ h_{02}\quad:=\quad a(-ca^{4}d^{2}b+3ca^{3}d^{2}b-2ca^{4}db+4ca^{3}db-3ca^{2}d^{2}b-2ca^{2}db-6ca^{2}bd+ca^{5}bd^{2} \\ \quad\quad\quad\quad\quad+2ca^{5}bd+ca^{4}b^{2}+9ca^{3}bd^{2}-7ca^{2}bd^{2}-5ca^{4}bd^{2}-9ca^{4}bd-2ca^{3}b^{2}-2ca^{3}b^{2}d\\ \quad\quad\quad\quad\quad+ca^{2}b^{2}d+ca^{4}b^{2}d+ca^{3}b^{3}+2cd^{2}ab+13ca^{3}bd-8a^{2}bd+a^{3}b^{3}+a^{5}b+3ca^{4}\\ \quad\quad\quad\quad\quad-ca^{5}+2cd^{3}-2ca^{3}-4a^{4}b-4a^{3}b^{2}+2a^{4}b^{2}+a^{5}bd^{2}+2a^{5}bd+ca^{5}b-4ca^{4}b\\ \quad\quad\quad\quad\quad-2ca^{3}b^{2}+ca^{4}b^{2}+4ca^{3}b-ca^{4}b+ca^{3}b+cd^{2}ab+6ca^{2}d+21ca^{2}d^{2}-6cad^{2}\\ \quad\quad\quad\quad\quad-27ca^{3}d^{2}-15ca^{3}d+6ca^{4}d^{3}+15ca^{4}d^{2}-14ca^{3}d^{3}-ca^{5}d^{3}-3ca^{5}d^{2}\\ \quad\quad\quad\quad\quad+16ca^{2}d^{3}+12ca^{4}d-3ca^{5}d-9cad^{3}+13a^{3}bd^{2}-12a^{2}bd^{2}-6a^{4}bd^{2}\\ \quad\quad\quad\quad\quad-10a^{4}bd-6a^{3}b^{2}d+4a^{2}b^{2}d+2a^{4}b^{2}d+4d^{2}ab+16a^{3}bd-3ca^{3}b^{2}d+2cda^{2}b^{2}\\ \quad\quad\quad\quad\quad+ca^{4}b^{2}d+4a^{3}b)/(c(-3ad-2a+2d+a^{2}d+a^{2}+ab)(a-1)(-3ad-2a\\ \quad\quad\quad\quad\quad+2d+a^{2}d+a^{2}+ab)^{2}), $

$ g_{02}\quad:=\quad a^2b(dca-dc-3ad-2a+2d+a^2d+a^2+ab)/(c(a-1)(-3ad-2a\\ \quad\quad\quad\quad\quad+2d+a^2d+a^2+ab)(ad+a-d)). $

Theorem 2   If $1<a<4$, then the map $F$ undergoes flip bifurcation at the fixed point $E_{1}$ as $a$ crossing 3 and $b\neq 1+d(a-1)/a$. More concretely, for the restriction of mapping $F$ to a one-dimensional center manifold, a stable 2-periodic orbit emerges near the fixed point $E_{1}$ for $a-3>0$ small.

Proof  We choose $a$ as bifurcation parameter. Rewrite system (3.3) in the suspended form

$ \left( \begin{array}{c} \xi \\ a\\ \eta \\ \end{array} \right)\rightarrow \left( \begin{array}{c} h_{10}\xi \\ -a\\ g_{01}\eta \\ \end{array} \right)+\left( \begin{array}{c} h_{20}\xi^{2}+h_{11}\xi\eta+h_{02}\eta^{2}+\mathcal{O}_{1}(3) \\ 0\\ g_{11}\xi\eta+g_{02}\eta^{2}+\mathcal{O}_{2}(3) \\ \end{array} \right), $

(3.4)

so as to involve the parameter $a$ explicitly in the discussion. The suspended system (3.4) has a two-dimensional center manifold

$ W_{1}^{c}(O)= \{(\xi, \eta, a)\in \mathbb{R}^{3} : \eta=h_{1}(\xi, a), h_{1}(0, 3)=D h_{1}(0, 3)=0, |\xi|<\varepsilon_{1}, |a-3|<\delta_{1}\}, $

(3.5)

where $\varepsilon_{1}$ and $\delta_{1}$ are sufficient small positives. Assume that $h_{1}(\xi, a)$ has the following form

$ \eta=h_{1}(\xi, a)=b_{1}\xi^{2}+b_{2}\xi a+b_{3}a^{2}+\mathcal{O}_{1}(3), $

(3.6)

which must satisfy

$ \begin{split} &h_{1}(h_{10}\xi+h_{20}\xi^{2}+h_{11}\xi h_{1}(\xi, a)+h_{02}h_{1}^{2}(\xi, a)+\mathcal{O}(3), a)\\ =&g_{01}h_{1}(\xi, a)+g_{11}\xi h_{1}(\xi, a)+g_{02}h_{1}^{2}(\xi, a) \end{split} $

(3.7)

by the center manifold theorem (see p. 33--35 in [2]). Comparing coefficients of $\xi^{2}, a\xi$ and $a^{2}$ in (3.7) we obtain that $\label{all c} b_{1}=b_{2}= b_{3}=0, $ and (3.6) has the determinative form

$ y=h_{1}(\xi, a)=\mathcal{O}_{1}(3). $

(3.8)

Substituting (3.14) into the first two equations in (3.4) yields

$ \left( \begin{array}{c} \xi\\ a \end{array}\right)\rightarrow\left(\begin{array}{c} (2-a)\xi-a\xi^{2}+\mathcal{O}_{1}(4)\\ -a \end{array}\right), $

which defines a two-dimensional mapping $(\xi, a)\rightarrow f_{2}(\xi, a)$ by $\label{f2} f_{2}(\xi, a)= (2-a)\xi-a\xi^{2}+\mathcal{O}_{1}(4).$ From that there isn't the term $u^3$ in $\mathcal{O}_{1}(3)$ in system (3.2), it is not difficult to follow that there isn't the term $\xi^3$ in $\mathcal{O}_{1}(4)$. One can check that

$ \left.\frac{\partial^{2} f_{2}}{\partial\xi\partial a} \right|_{(\xi, a)=(0, 3)}=-1 $

and

$ \left.\left[\frac{1}{2}\left(\frac{\partial^{2}f_{2}}{\partial \xi^{2}}\right)^{2}+ \frac{1}{3}\left(\frac{\partial^{3}f_{2}}{\partial \xi^{3}}\right)\right] \right|_{(\xi, a)=(0, 3)}=18>0. $

Hence the transversality condition and non-degeneracy condition of Theorem 4.3 in [9] are satisfied, which implies that a flip bifurcation occurs at $\xi=0$ as $a$ crossing 3 and a stable cycle of period two arises in system (3.3). So the map $F$ undergoes flip bifurcation at the fixed point $E_{1}$ on the center manifold if $a$ crosses 3 and $b\neq 1+d(a-1)/a$.

Theorem 3  If $a\neq3$ and $(a, b)$ crosses $b= 1+d(a-1)/a$, then system (2.1) undergoes a transcritical bifurcation at the fixed point $E_{1}$.

Proof  We choose $b$ as bifurcation parameter. Rewrite system (3.3) in the suspended form

$ \left( \begin{array}{c} \xi \\ b\\ \eta \\ \end{array} \right)\rightarrow \left( \begin{array}{c} h_{10}\xi \\ b\\ g_{01}\eta \\ \end{array} \right)+\left( \begin{array}{c} h_{20}\xi^{2}+h_{11}\xi\eta+h_{02}\eta^{2}+\mathcal{O}_{1}(3) \\ 0\\ g_{11}\xi\eta+g_{02}\eta^{2}+\mathcal{O}_{2}(3) \\ \end{array} \right). $

(3.9)

The suspended system (3.9) has a two-dimensional center manifold

$ \begin{split} W_{2}^{c}(O)=& \left\{(\xi, \eta, b)\in \mathbb{R}^{3} : \xi=h_{2}(\eta, b), h_{2}(0, b_{0})= D h_{2}(0, b_{0})=0, \right.\\ &\left. |\eta|<\varepsilon_{2}, |b-b_{0})|<\delta_{2}\right\}, \end{split} $

(3.10)

where $\varepsilon_{2}$ and $\delta_{2}$ are sufficient small positives and $b_{0}=1+d(a-1)/a$. Assume that $h_{2}(\eta, b)$ has the following form

$ \xi=h_{2}(\eta, b)=a_{1}\eta^{2}+a_{2}\eta b+a_{3}b^{2}+\mathcal{O}(3), $

(3.11)

which must satisfy

$ \quad\quad h_{2}(g_{01}\eta+g_{11}\eta h_{2}(\eta, b)+g_{02}\eta^{2}+\mathcal{O}_{2}(3), b) $

(3.12)

$ =\quad h_{10}h_{2}(\eta, b)+h_{20}h_{2}^{2}(\eta, b)+h_{11}\eta h_{2}(\eta, b)+h_{02}\eta^{2}+\mathcal{O}_{1}(3) $

(3.13)

by the center manifold theorem (see p. 33-35 in [2]). Comparing coefficients of $\eta^{2}, b\eta$ and $b^{2}$ in (3.12) we obtain that ${a_1}g_{01}^2 = {h_{02}} + {a_1}{h_{01}},{a_2}{g_{01}} = {a_2}{h_{10}},{a_3} = {a_3}{h_{10}},$ from which we find $a_{2}=a_{3}=0$ and

$ a_{1}=\frac{h_{02}}{g_{01}^{2}-h_{10}}. $

Hence (3.11) has the determinative form

$ \xi=h_{2}(\eta, b)=\frac{h_{02}}{g_{01}^{2}-h_{10}}\eta^{2}+\mathcal{O}(3). $

(3.14)

Substituting (3.14) into the last two equations in (3.9) yields

$ \left( \begin{array}{c} b\\ \\ \eta \end{array}\right)\rightarrow\left(\begin{array}{c} b\\ \\ g_{01}\eta+g_{02}\eta^{2}+\frac{g_{11}h_{02}}{g_{01}^{2}-h_{01}}\eta^{3}+\mathcal{O}(4) \end{array}\right), $

which defines a two-dimensional mapping $(\eta, b)\rightarrow f_{3}(\eta, b)$ by

$ f_{3}(\eta, b)= g_{01}\eta+g_{02}\eta^{2}+\frac{g_{11}h_{02}}{g_{01}^{2}-h_{01}}\eta^{3}+\mathcal{O}(4). $

By $a>1$ and $d>0$ we obtain that

$ \frac{\partial^{2} f_{3}}{\partial \eta \partial b}|_{(\eta, b)=(0, b_{0})}=\frac{a}{(a-1)d+a}>0 $

(3.15)

and

$ \begin{split} &\left.\frac{\partial^{2} f_{3}}{\partial \eta^{2}}\right|_{(\eta, b)=(0, b_{0})} = \frac{2a^2\left[1+\frac{d}{a}(a-1)\right]g_{1}(a)}{c(a-1)[(a-1)d+a]g_{2}(a)}>0, \end{split} $

(3.16)

where

$ g_{1}(a)=(1+d)a^2+(-2d+dc-1)a-d(-1+c) $

and

$ g_{2}(a)=(1+d)a^2+(-2d-1)a+d. $

In fact, from $a>1, 0<d<1$ and $0<c<1$, one can check that $g_{1}(1)=g_{2}(1)=0$,

$ g_{1}'(a)=(2a-1)+2d(a-1)+dc>0 $

and

$ g_{2}'(a)=(2a-1)+2d(a-1)>0, $

which imply that $g_{1}(a)>0$ and $g_{2}(a)>0$. Hence (3.16) is true, and the mapping $G$ undergoes a transcritical bifurcation on the center manifold at $E_1$ if $a>1$, $a\neq 3$ and $b=1+(a-1)b/a$. The proof is completed.

Using the same arguments we have the following results.

Theorem 4  If $1<b<4$, then the map $F$ undergoes flip bifurcation at the fixed point $E_{2}$ when $b$ crosses 3 and $a\neq 1+c(b-1)/b$. More concretely, the bifurcation is supercritical and a stable 2-periodic orbit emerges near the fixed point $E_{2}$ when $b>3$. If $b\neq3$ and $(a, b)$ crosses $a= 1+c(b-1)/b$, the transcritical bifurcation occurs at the fixed point $E_{2}$ in system (2.1).

Sometimes flip bifurcation is also called period-doubling bifurcation (see p.114 in [9]). Theorem 2 (resp. Theorem 4) shows that a 2-periodic oscillation of the population sizes in species $x$ (resp. $y$) emerges near the equilibrium $(a-1)/a$ (resp. $(b-1)/b$).

In this paper we only discuss the codimension 1 local bifurcations at fixed points $E_0, E_1$ and $E_2$. In fact, if $a=1$ and $b=1$ in Theorem 1, the map $F$ has a double multiplier $1$, which implies that 1:1 resonance may occur at the fixed point $E_{0}$ (see p. 410--415 in [9]). If $a=3$ (resp. $b=3$) and $b=1+d(a-1)/a$ (resp. $a=1+c(b-1)/a$), the map $F$ has eigenvalues -1 and 1. A fold-flip bifurcation may occur at the fixed point $E_{1}$ (resp. $E_{2}$) in the system (2.1) (see e.g. [6]). All of these codimension 2 bifurcations will involve more complicated computation. We leave these to our next work.