Stationary distribution, extinction, density function and periodicity of an n-species competition system with infinite distributed delays and nonlinear perturbations

Figure 1.  The diagrams track the number variations of species $ x_i\ (i = 1, 2) $ of deterministic system (1), and of stochastic system (6) with different cases of noise intensity, which include: (a) $ (\sigma_{13}, \sigma_{23}) = (0.2, 0.4) $, (b) $ (\sigma_{13}, \sigma_{23}) = (2, 0.4) $, (c) $ (\sigma_{13}, \sigma_{23}) = (0.2, 2) $ and (d) $ (\sigma_{13}, \sigma_{23}) = (2, 2) $. Other fixed parameters: $ (\sigma_{11}, \sigma_{12}, \sigma_{21}, \sigma_{22}) = (0.01, 0.02, 0.01, 0.02) $. All the iteration intervals are $ [0,600] $, and the number of iterations in the interval $ [0,600] $ is $ 3000000 $

Figure 2.  The left-hand column shows the number variations of species $ x_i\ (i = 1, 2) $ of deterministic system (1), and of stochastic system (6) on $ t\in[0, 2000] $. The right-hand column shows the corresponding frequency histograms of species $ x_i\ (i = 1, 2) $ on the iteration interval $ [0, 6000] $ with $ 6000000 $ iteration points, and the two marginal densities $ \Phi^{(i)}(x_i)\ (i = 1, 2) $ of the approximate density function $ \Phi(x_1, x_2, y_{1, 2}, y_{2, 1}) $. Fixed parameters: $ (\sigma_{k1}, \sigma_{k2}, \sigma_{k3}) = (0.005, 0.01, 0.03)\ (\forall\ k = 1, 2) $. The iteration interval is $ [0, 6000] $, and the number of iterations in the interval $ [0, 6000] $ is $ 6000000 $

Figure 3.  The blue, green and black lines separately denote the frequency histogram fitting curves of species $ x_i\ (i = 1, 2) $ of system (6) with the iteration time $ T $ equals to $ 1000 $, $ 2000 $ and $ 4000 $. The orange lines represent the marginal densities $ \Phi^{(i)}(x_i)\ (i = 1, 2) $ of the approximate density function $ \Phi(x_1, x_2, y_{1, 2}, y_{2, 1}) $. All of the parameter values are the same as in Figure 2

Figure 4.  The left-hand column presents the number variations of species $ x_i\ (i = 1, 2) $ of deterministic system (1), and of stochastic system (6) on $ t\in[0, 2000] $. The right-hand column shows the frequency histograms of species $ x_i\ (i = 1, 2) $ on the iteration interval $ [0, 6000] $ with $ 6000000 $ iteration points, and the two marginal densities $ \Phi^{(i)}(x_i)\ (i = 1, 2) $ of the approximate density function $ \Phi(x_1, x_2, y_{1, 2}, y_{2, 1}) $. Fixed parameters: $ \sigma_{k3} = 0.03 $ and $ \sigma_{kj} = 0\ (\forall\ k, j\in\mathbb{Z}_2) $. The iteration interval and the number of iterations are the same as Figure 2

Figure 5.  The blue, green and black lines separately represent the frequency histogram fitting curves of species $ x_i\ (i = 1, 2) $ of system (6) with the iteration time $ T $ equals to $ 1000 $, $ 2000 $ and $ 4000 $. The manganese purple lines denote the marginal densities $ \Phi^{(i)}(x_i)\ (i = 1, 2) $ of the approximate density function $ \Phi(x_1, x_2, y_{1, 2}, y_{2, 1}) $. All of the parameter values are the same as in Figure 4

Figure 6.  The variation trends of species $ x_i\ (i = 1, 2) $ of system (6) with different noise intensities $ (\sigma_{11}, \sigma_{12}, \sigma_{13}, \sigma_{21}, \sigma_{22}, \sigma_{23}) $, which include: (a) $ (0.04, 0.04, 0.04, 0.05, 0.05, 0.05) $, (b) $ (0.04, 0, 0, $ $ 0.05, 0, 0) $, (c) $ (0, 0.04, 0, 0, 0.05, 0) $ and (d) $ (0, 0, 0.04, 0, 0, 0.05) $. All the iteration interval are $ [0,400] $, and the number of iterations in the interval $ [0,400] $ is $ 4000000 $

Figure 7.  The diagrams track the number variations of species $ x_i\ (i = 1, 2) $ of system (7) with different cases of noise intensity, which include: (a) $ (\sigma_{13}, \sigma_{23}) = (0.2, 0.4) $, (b) $ (\sigma_{13}, \sigma_{23}) = (2, 0.4) $, (c) $ (\sigma_{13}, \sigma_{23}) = (0.2, 2) $ and (d) $ (\sigma_{13}, \sigma_{23}) = (2, 2) $. The other parameters: $ b_1(t) = 1+0.01\sin(\frac{t}{100\pi}) $, $ b_2(t) = 1.2+0.01\sin(\frac{t}{100\pi}) $, $ a_{11}(t) = 0.3+0.01\cos(\frac{t}{100\pi}) $, $ a_{12}(t) = 0.2+0.01\sin(\frac{t}{100\pi}) $, $ a_{21}(t) = 0.3+0.01\sin(\frac{t}{100\pi}) $ and $ a_{22}(t) = 0.4+0.01\cos(\frac{t}{100\pi}) $. The iteration interval and the number of iterations are the same as Figure 1

Figure 8.  The variation trends of species $ x_i\ (i = 1, 2) $ of deterministic system (1), and of stochastic system (7) with noise intensities $ \sigma_{kj} = 0.005\ (\forall\ k\in\mathbb{Z}_2, j\in\mathbb{Z}_3) $. The other parameter values are the same as in Figure 7. The iteration interval is $ [0, 1000] $, and the number of iterations in the interval $ [0, 1000] $ is $ 1000000 $

Figure 9.  The variation trends of species $ x_i\ (i = 1, 2) $ of deterministic system (1), and of stochastic system (7) with noise intensities $ \sigma_{kj} = 0.01\ (\forall\ k\in\mathbb{Z}_2, j\in\mathbb{Z}_3) $. The other parameter values, the iteration interval and the number of iterations are the same as in Figure 8

Figure 10.  The left-hand column shows the variation trends of species $ x_j\ (j\in\mathbb{Z}_3) $ of deterministic system (1), and of stochastic system (6) with noise intensities $ (\sigma_{i1}, \sigma_{i2}, \sigma_{i3}) = (0.01, 0.02, 0.05)\ (\forall\ i\in\mathbb{Z}_3) $. The right-hand column presents the corresponding frequency histograms of species $ x_j\ (j\in\mathbb{Z}_3) $ of system (6) on the iteration interval $ [0,400] $ with $ 400000 $ iteration points. The other parameters: $ (b_1, b_2, b_3) = (2, 2.2, 1.9) $, $ (a_{11}, a_{12}, a_{13}) = (0.4, 0.1, 0.05) $, $ (a_{21}, a_{22}, a_{23}) = (0.1, 0.5, 0.05) $, $ (a_{31}, a_{32}, a_{33}) = (0.1, 0.1, 0.8) $ and $ \alpha_{ik} = 1\ (\forall\ i\in\mathbb{Z}_3, k\in\mathbb{S}_3^i) $. The iteration interval is $ [0,400] $, and the number of iterations in the interval $ [0,400] $ is $ 400000 $