0,\\ \dot y\big|_{y=0}= D^u x>0,\\ \dot x\big|_{x=p}= p(r^u-Dl-d^lC_0^l-a^lp)+D^uy<0,\\ \dot y\big|_{y=q}= q(r^u-D^l-b^lq)+D^ux<0. \end{gather*} This completes the proof. \end{proof} \begin{theorem} \label{thm3.4} For system \eqref{e3.1}, if $r^u(D^l-r^u)(d^lC_0^l+D^l-r^u)$, or $r^u\geq D^l$ then $(x^*,y^*)$ is the unique positive equilibrium, it is globally asymptotically stable. \end{theorem} \begin{proof} We construct the Liapunov function $$ V(x,y)=\alpha(x-x^*-x^*\ln\frac{x}{x^*})+\beta(y-y^*-y^*\ln\frac{y}{y^*}), $$ where $\alpha,\beta$ are positive constants. Calculating the derivative of $V(x,y)$ along \eqref{e3.1}, we have \begin{align*} V'_{\eqref{e3.1}}(x,y) &= \alpha(x-x^*)\frac{\dot x}{x}+\beta(y-y^*)\frac{\dot y}{y} \\ &= -\alpha a^l(x-x^*)^2-\beta b^l(y-y^*)^2\\ &\quad +\alpha D^u(x-x^*)(\frac{y}{x}-\frac{y^*}{x^*})+\beta D^u(y-y^*)(\frac{x}{y}-\frac{x^*}{y^*})\\ &= -x^* a^l(x-x^*)^2-y^* b^l(y-y^*)^2\\ &\quad - D^u[\sqrt{\frac{y}{x}}(x-x^*)-\sqrt{\frac{x}{y}}(y-y^*)]^2 \leq 0, \end{align*} In fact, we choose that $\alpha=x^*$, $\beta=y^*$. We can see that in the domain $OABCO$, $V'_{\eqref{e3.1}}=0$ if and only if $x=x^*$, $y=y^*$. Hence $(x^*,y^*)$ is globally asymptotically stable. This completes the proof. \end{proof} The specific computation is similar to above-proved theorems, for the system \eqref{e3.2} has two equilibria $O(0,0)$ and $(x^{**},y^{**})$. \begin{theorem} \label{thm3.6} The point $(0,0)$ is always an equilibrium of system \eqref{e3.2}. If $r^l (D^u-r^l)(d^uC_0^u+D^u-r^l)~$, or $r^l\geq D^u$, then there exists a unique positive equilibrium $(x^{*},y^{*})$ of system \eqref{e3.2} which is a stable node and globally asymptotically stable. \end{theorem} In other words, for the systems \eqref{e3.1}, \eqref{e3.2}, when the only nonnegative equilibrium $(0,0)$ exists, it is stable node and is globally asymptotically stable. If the $(0,0)$ is unstable, then there exists a unique positive equilibrium which is globally asymptotically stable. \section{Permanence and Extinction} In this section, we study the permanence and extinction of population of system \eqref{e2.3}. \begin{theorem} \label{thm4.1} (1) If $r^u (D^u-r^l)(d^uC_0^u+D^u-r^l)$, or $r^l\geq D^u$, then the system \eqref{e2.3} is permanent; (2) If $r^u (D^ul-r^u)(d^lC_0^l+D^l-r^u)$, or $r^u\geq D^l$ holds. By the above discussion of the theorem 3.5 and 3.7, we know that the system \eqref{e3.1} and \eqref{e3.2} have the globally asymptotically stable positive equilibria $(x^*,y^*)$ and $(x^{**},y^{**})$, the trivial equilibrium $(0,0)$ is unstable. We construct a positively invariant region for system \eqref{e2.3}. \begin{figure}[ht] \begin{center} \includegraphics[width=0.3\textwidth]{fig3} %{3} \end{center} \caption{ The rectangle $ABCD$ with $A(p_1,q_1),$ $B(p_2,q_1),$ $C(p_2,q_2),$ $D(p_1,q_2)$ and $P(x^{**},y^{**}),$ $Q(x^*,y^*)$} \label{fig:figure3} \end{figure} where $p_1,p_2,q_1,q_2$ are positive constants satisfying $$ p_1<\min\{x^*,x^{**}\},\quad p_2>\max\{x^*,x^{**}\}, $$ \begin{align*} & \frac{p_1}{D^l}(a^up_1+d^uC_0^u+D^u-r^l)\\ & 0,\\ \dot {x}(t)\big|_{x=p_2}\leq p_2(r^u-D^l-d^lC_0^l+a^lp_2)+D^uy\big|_{q_1\leq y\leq q_2}<0,\\ \dot {y}(t)\big|_{y=q_1}\geq q_1(r^l-D^u-b^uq_1)+D^lx\big|_{p_1\leq x\leq p_2}>0,\\ \dot {y}(t)\big|_{y=q_2} \leq q_2(r^u-D^l-b^lq_2)+D^ux\big|_{p_1\leq x\leq p_2}<0, \end{gather*} So the compact confined set $ABCD$ in $R^2_+$, the phase trajectories of the system \eqref{e2.3} starting from the boundary always point into the enclosed domain. According to the Kamke theorem and definition \ref{def2.1}, for any positive solution $(x(t),y(t))$ of \eqref{e2.3} with positive initial value, there exists a time T, when $(x(t),y(t))$ goes in the $ABCD$ and never leaves for all $t>T$. Hence the system \eqref{e2.3} is permanent. Let $(x(t),y(t))$ be an arbitrary positive solution of system \eqref{e2.3} with the positive initial value; $(x^*(t),y^*(t))$ and $(x^{**}(t),y^{**}(t))$ are the same of systems \eqref{e3.1} and \eqref{e3.2} respectively. Choose initial value $x^{**}(0)=x(0)=x^*(0)$, $y^{**}(0)=y(0)=y^*(0)$, If the condition (2) of the theorem exists, then the conditions $r^l