Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

Kalabušić, S; Kulenović, MRS; Pilav, E

doi:10.1186/1687-1847-2011-29

Advances in Continuous and Discrete Models

Theory and Modern Applications

Table 1 The equilibrium points of System (1).

From: Dynamics of a two-dimensional system of rational difference equations of Leslie--Gower type

E ₁	$A_{1} > β_{1}, A_{2} < γ_{2} < A_{1} + A_{2} - β_{1}, \frac{(A_{1} - β_{1}) (γ_{2} - A_{2})}{B_{2}} < α_{1} \leq \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} > β_{1}, A_{2} > γ_{2}, α_{1} \leq \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}, A_{1} + γ_{2} \neq A_{2} + β_{1}$ or $A_{1} > β_{1}, A_{2} = γ_{2}, α_{1} \leq \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} > β_{1}, α_{1} > \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$
E ₁ ≡ E ₂ ≡ E ₃	$A_{1} > β_{1}, A_{1} + A_{2} = β_{1} + γ_{2}, α_{1} = \frac{(A_{1} - β_{1}) (γ_{2} - A_{2})}{B_{2}}$
E ₁ ≡ E ₃, E ₂	$A_{1} > β_{1}, A_{1} + A_{2} < β_{1} + γ_{2}, A_{2} < γ_{2}, α_{1} = \frac{(A_{1} - β_{1}) (γ_{2} - A_{2})}{B_{2}}$
E ₁, E ₂, E ₃	$A_{1} > β_{1}, A_{1} + A_{2} < β_{1} + γ_{2}, \frac{(A_{1} - β_{1}) (γ_{2} - A_{2})}{B_{2}} < α_{1} < \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$
E ₁, E ₂	$A_{1} > β_{1}, A_{2} < γ_{2}, α_{1} < \frac{(A_{1} - β_{1}) (γ_{2} - A_{2})}{B_{2}}$
E ₁ ≡ E ₂	$A_{1} > β_{1}, A_{2} < γ_{2} < A_{1} + A_{2} - β_{1}, α_{1} = \frac{(A_{1} - β_{1}) (γ_{2} - A_{2})}{B_{2}}$
E ₁, E ₂ ≡ E ₃	$A_{1} > β_{1}, A_{1} + A_{2} < β_{1} + γ_{2}, α_{1} = \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$
E ₂, E ₃	$A_{1} < β_{1}, A_{1} + γ_{2} > A_{2} + β_{1}, α_{1} < \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} = β_{1}, A_{1} + A_{2} < β_{1} + γ_{2}, α_{1} < \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$
E ₂ = E ₃	$A_{1} < β_{1} A_{1} + γ_{2} > A_{2} + β_{1}, α_{1} = \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} = β_{1}, A_{1} + A_{2} < β_{1} + γ_{2}, α_{1} = \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$
No equilibrium	$A_{1} < β_{1}, A_{2} < γ_{2} < - A_{1} + A_{2} + β_{1}, α_{1} \leq \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} < β_{1}, A_{2} \geq γ_{2}, α_{1} \leq \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} \leq β_{1}, α_{1} > \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$ or $A_{1} = β_{1}, A_{1} + A_{2} > γ_{2} + β_{1}, α_{1} \leq \frac{{(A_{1} - A_{2} - β_{1} + γ_{2})}^{2}}{4 B_{2}}$

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