Figure 2

(a) \(\alpha_{1} = 0.6\), \(\alpha_{2} = 0.5\) and \(\alpha_{3} = 0.74\) such that \(\rho_{1} > \tilde{\rho}_{1}\), \(\rho_{2} > \tilde{\rho}_{2}\) and \(G_{3} < \tilde{G}_{3}\) ((iv) of Theorem 2.4). Then the species \(z(t)\) goes to extinction a.s. and the species \(x(t)\) and \(y(t)\) are stochastically persistent with \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} x (s)\,ds = \frac{\rho_{1} - \tilde{\rho}_{1}}{a_{11}a_{22} - a_{12}a_{21}} = 0.4742\), \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} y (s)\,ds = \frac{\rho_{2} - \tilde{\rho}_{2}}{a_{11}a_{22} - a_{12}a_{21}} = 0.2970\). (b) Let \(\alpha_{1} = 0.53\), \(\alpha_{2} = 0.84\) and \(\alpha_{3} = 0.6\) such that \(\rho_{3} > \tilde{\rho}_{3}\), \(\rho_{4} > \tilde{\rho}_{4}\) and \(G_{2} < \tilde{G}_{2}\) ((v) of Theorem 2.4). Then the species \(y(t)\) goes to extinction a.s. and the species \(x(t)\) and \(z(t)\) are stochastically persistent with \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} x (s)\,ds = \frac{\rho_{3} - \tilde{\rho}_{3}}{a_{11}a_{33} - a_{13}a_{31}} = 0.3993\), \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} z (s)\,ds = \frac{\rho_{4} - \tilde{\rho}_{4}}{a_{11}a_{33} - a_{13}a_{31}} = 0.2003\). (c) Let \(\alpha_{1} = 0.98\), \(\alpha_{2} = 0.36\) and \(\alpha_{3} = 0.6\) such that \(\rho_{5} > \tilde{\rho}_{5}\), \(\rho_{6} > \tilde{\rho}_{6}\) and \(G_{1} > \tilde{G}_{1}\) ((vi) of Theorem 2.4). Then the species \(x(t)\) goes to extinction a.s. and the species \(y(t)\) and \(z(t)\) are stochastically persistent with \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} y (s)\,ds = \frac{\rho_{5} - \tilde{\rho}_{5}}{a_{22}a_{33} - a_{23}a_{32}} = 0.1605\), \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} z (s)\,ds = \frac{\rho_{6} - \tilde{\rho}_{6}}{a_{22}a_{33} - a_{23}a_{32}} = 0.4079\). (d) Let \(\alpha_{1} = 0.4\), \(\alpha_{2} = 0.45\) and \(\alpha_{3} = 0.48\) such that \(G_{1} > \tilde{G}_{1}\), \(G_{2} > \tilde{G}_{2}\) and \(G_{3} > \tilde{G}_{3}\) (Theorem 2.5). Then all the species are stochastically persistent with \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} x (s)\,ds =\frac{G_{1} - \tilde{G}_{1}}{G} = 0.5691\), \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} y (s)\,ds = \frac{G_{2} -\tilde{G}_{2}}{G} = 0.3371\), \(\lim_{t \to + \infty} \frac{1}{t}\int_{0}^{t} z (s)\,ds = \frac{G_{3} -\tilde{G}_{3}}{G} = 0.1605\).