From: Asymptotic dynamics of the Leslie-Gower competition system with Allee effects and stocking
Cases | Global dynamical behavior of ( 2.4 ) |
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1 | (2.4) has no interior steady state and \(E_{0}\) is GAS in \(\mathbb {R}_{+}^{2}\). |
2 | (2.4) has a unique interior steady state \(E^{*}\) which is GAS in Γ. |
3 | (2.4) has a unique interior steady state \(E^{*}\) which is non-hyperbolic. \(W^{s}\) separates \(\mathbb{R}_{+}^{2}\) into two positively invariant regions \(R_{1}\) and \(R_{2}\) such that \(E_{0}\) is GAS in \(R_{2}\) and solutions with initial conditions in \(R_{1}\) converge to \(E^{*}\). |
4 | (2.4) has two hyperbolic interior steady states \(E_{1}^{*}\) and \(E_{2}^{*}\). \(W_{2}^{s}\) separates \(\mathbb{R}_{+}^{2}\) into two positively invariant regions \(R_{1}\) and \(R_{2}\) such that \(E_{1}^{*}\) and \(E_{0}\) are GAS in \(R_{1}\) and \(R_{2}\), respectively. |
5 | (2.4) has two interior steady states, one is hyperbolic and the other is non-hyperbolic. The global stable manifold of the non-hyperbolic steady state separates \(\mathbb{R}_{+}^{2}\) into two positively invariant regions \(R_{1}\) and \(R_{2}\) such that the other interior steady state is GAS in \(R_{1}\) and \(E_{0}\) is GAS in \(R_{2}\). |
6 | (2.4) has three interior steady states \(E_{i}^{*}\), i = 1,2,3. \(W_{2}^{s}\) separates \(\mathbb{R}_{+}^{2}\) into two positively invariant regions \(R_{1}\) and \(R_{2}\) such that \(E_{1}^{*}\) and \(E_{3}^{*}\) are GAS in \(R_{1}\) and \(R_{2}\), respectively. |