Table 2 Global behavior of System (1).

$\begin{array}{cc}\hfill {ℛ}_1& A_1>{\beta }_1,A_2<{\gamma }_2<A_1+A_2-{\beta }_1,\hfill \\ \frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}<{\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\hfill \end{array}$

or

$\begin{array}{cc}\hfill {ℛ}_2& A_1>{\beta }_1,A_2>{\gamma }_2,{\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2},\hfill \\ A_1+{\gamma }_2\ne A_2+{\beta }_1,\hfill \end{array}$

or

${ℛ}_3{A}_1>{\beta }_1,A_2={\gamma }_2,{\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}$

or

${ℛ}_4{A}_1>{\beta }_1,{\alpha }_1>\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}$

There exists a unique equilibrium E ₁, and it is globally asymptotically stable (G.A.S.). The basin of attraction of E ₁ is given by $ℬ$ ₁(E ₁) = [0, ∞)²

${ℛ}_5{A}_1>{\beta }_1,{\gamma }_2+{\beta }_1\le A_1+A_2,{\alpha }_1=\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}$

There exists a unique equilibrium E ₁ = E ₂ which is non-hyperbolic. Furthermore, this equilibrium is the global attractor. Its basin of attraction is given by $ℬ$(E ₁) = [0, +∞)². This is an example of globally attractive non-hyperbolic equilibrium point

${ℛ}_6{A}_1>{\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,{\alpha }_1=\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}$

There exist two equilibrium points E = E ₁ = E ₃ which is non-hyperbolic, and E ₂, which is locally asymptotically stable. Furthermore, the x-axis is the basin of attraction of E ₁. The equilibrium point E ₂ is globally asymptotically stable with the basin of attraction $ℬ$(E ₂) = [0, +∞) × [0, +∞)

${ℛ}_7{A}_2>{\beta }_1,A_2<{\gamma }_2,{\alpha }_1<\frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}$

There exist two equilibrium points E ₁, which is a saddle, and E ₂, which is a locally asymptotically stable equilibrium point. Furthermore, the x-axis is the global stable manifold of $\mathcal{W}$ ^s(E ₁). The equilibrium point E ₂ is globally asymptotically stable with the basin of attraction $ℬ$(E ₂) = [0, +∞) × [0, +∞)

${ℛ}_8{A}_1<{\beta }_1,A_1+{\gamma }_2>A_2+{\beta }_1,{\alpha }_1<\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}$

There exist two equilibrium points E ₃, which is a saddle, and E ₂, which is locally asymptotically stable. Furthermore, there exists the global stable manifold $ℬ$ ^s(E ₃) that separates the positive quadrant so that all orbits below this manifold are asymptotic to (+∞, 0), and all orbits above this manifold are asymptotic to the equilibrium point E ₂. All orbits that starts on $ℬ$ ^s(E ₃) are attracted to E ₃

or

$\begin{array}{cc}\hfill {ℛ}_9& A_1={\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,\hfill \\ {\alpha }_1<\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\hfill \end{array}$

$\begin{array}{cc}\hfill {ℛ}_{10}& A_1>{\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,\hfill \\ \frac{\left(A_1-{\beta }_1\right)\left({\gamma }_2-A_2\right)}{B_2}<{\alpha }_1<\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2},\hfill \end{array}$

There exist three equilibrium points E ₁, E ₂, and E ₃, where E ₁ and E ₂ are locally asymptotically stable and E ₃ is a saddle. There exists the global stable manifold $\mathcal{W}$ ^s(E ₃) that separates the positive quadrant so that all orbits below this manifold are attracted to the equilibrium point E ₁, and all orbits above this manifold are attracted to the equilibrium point E ₂. All orbits that starts on $\mathcal{W}$ ^s(E ₃) are attracted to E ₃. The global unstable manifold $\mathcal{W}$ ^s(E ₃) is the graph of a continuous strictly decreasing function, and $\mathcal{W}$ ^u(E ₃) has endpoints E ₂ and E ₁

${ℛ}_{11}{A}_1>{\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,{\alpha }_1=\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}$

There exist two equilibrium points E = E ₂ = E ₃ and E ₁. E ₁ is locally asymptotically stable and E is non-hyperbolic. There exists a continuous increasing curve $\mathcal{W}$ _E which is a subset of the basin of attraction of E. All orbits that start below this curve are attracted to E ₁. All orbits that start above this curve are attracted to E

${ℛ}_{12}{A}_1<{\beta }_1,A_1+{\gamma }_2>A_2+{\beta }_1,{\alpha }_1=\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}$

There exists a unique equilibrium point E = E ₂ = E ₃ which is non-hyperbolic. There exists a continuous increasing curve $\mathcal{W}$ _E which is a subset of basin of attraction of E. All orbits that start below this curve are attracted to (+∞, 0). All orbits that start above this curve are attracted to E. This is an example of semi-stable non-hyperbolic equilibrium point

or

$\begin{array}{cc}\hfill {ℛ}_{13}& A_1={\beta }_1,A_1+A_2<{\beta }_1+{\gamma }_2,\hfill \\ {\alpha }_1=\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\hfill \end{array}$

$\begin{array}{cc}\hfill {ℛ}_{14}& A_1<{\beta }_1,A_2<{\gamma }_2<-A_1+A_2+{\beta }_1,\hfill \\ {\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\hfill \end{array}$

System (1) does not posses an equilibrium point. Its behavior is as follows x _n → ∞, y _n → ∞, n → ∞

or

$\begin{array}{cc}\hfill {ℛ}_{15}& A_1<{\beta }_1,A_2\ge {\gamma }_2,\hfill \\ {\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\hfill \end{array}$

or

${ℛ}_{16}{A}_1\le {\beta }_1,{\alpha }_1>\frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}$

or

$\begin{array}{cc}\hfill {ℛ}_{17}& A_1={\beta }_1,A_1+A_2>{\gamma }_2+{\beta }_1,\hfill \\ {\alpha }_1\le \frac{{\left(A_1-A_2-{\beta }_1+{\gamma }_2\right)}^2}{4B_2}\hfill \end{array}$