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

• *Corresponding author: Baoquan Zhou

This work is supported by the National Natural Science Foundation of China (No. 11871473) and Innovation project for Graduate Students of China University of Petroleum (East China) (No. YCX2021143)

• In this paper, we examine an n-species Lotka-Volterra competition system with general infinite distributed delays and nonlinear perturbations. The stochastic boundedness and extinction are first studied. Then we propose a new $p$-stochastic threshold method to establish sufficient conditions for the existence of stationary distribution $\ell(\cdot)$. By solving the corresponding Fokker–Planck equation, we derive the approximate expression of the distribution $\ell(\cdot)$ around its quasi-positive equilibrium. For the stochastic system with periodic coefficients, we use the $p$-stochastic threshold method again to obtain the existence of positive periodic solution. Besides, the related competition exclusion and moment estimate of species are shown. Finally, some numerical simulations are provided to substantiate our analytical results.

Mathematics Subject Classification: Primary: 37L55, 37H05; Secondary: 92B05.

 Citation: • • 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 $400000$

Table 1.  List of the initial value and biological parameters of two-species competition system (6)

 Parameter Value Source $(b_1, b_2, a_{11}, a_{12}, a_{21}, a_{22})$ $(1, 1.2, 0.3, 0.2, 0.3, 0.4)$  $(x_1(0), x_2(0))$ $(1.5, 2)$  $(y_{1, 2}(0), y_{2, 1}(0), \alpha_{12}, \alpha_{21})$ $(0.8, 0.2, 1, 1)$ Assumed
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