# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2022078
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## Stationary distribution, extinction, density function and periodicity of an n-species competition system with infinite distributed delays and nonlinear perturbations

 College of Science, China University of Petroleum (East China), Qingdao 266580, China

*Corresponding author: Baoquan Zhou

Received  July 2021 Revised  January 2022 Early access April 2022

Fund Project: 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.

Citation: Baoquan Zhou, Yucong Dai. Stationary distribution, extinction, density function and periodicity of an n-species competition system with infinite distributed delays and nonlinear perturbations. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2022078
##### References:
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##### References:
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Freedman, Stability conditions for two predator one prey systems, Acta. Appl. Math., 14 (1989), 3-10.  doi: 10.1007/BF00046669. [12] M. Gao and D. Jiang, Stationary distribution of a chemostat model with distributed delay and stochastic perturbations, Appl. Math. Lett., 123 (2022), Paper No. 107585, 7 pp. doi: 10.1016/j.aml.2021.107585. [13] C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Springer Berlin, 1983. [14] K. Golpalsamy, Globally asymptotic stability in a periodic Lotka-Volterra system, J. Austral. Math. Soc. Ser. B, 27 (1985), 66-72.  doi: 10.1017/S0334270000004768. [15] K. Gopalsamy, Time lags and global stability in two species competition, Bull. Math. Biol., 42 (1980), 729-737. [16] B. Han, D. Jiang, T. Hayat, A. Alsaedi and B. Ahmad, Stationary distribution and extinction of a stochastic staged progression AIDS model with staged treatment and second-order perturbation, Chaos. Soliton. Fract., 140 (2020), 110238, 19 pp. doi: 10.1016/j.chaos.2020.110238. [17] D. J. Higham, An algorithmic introduction to numerical simulation of stochastic differential equations, SIAM Rev., 43 (2001), 525-546.  doi: 10.1137/S0036144500378302. [18] D. J. Higham, X. Mao and C. Yuan, Almost sure and moment exponential stability in the numerical simulation of stochastic differential equations, SIAM J. Numer. Anal., 45 (2007), 592-609.  doi: 10.1137/060658138. [19] J. Hofbauer and K. Sigmund, The Theory of Evolution and Dynamical Systems, Cambridge University Press, Cambridge, 1988. [20] J. Hofbauer and K. Sigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.  doi: 10.1017/CBO9781139173179. [21] J. Hu, Z. Liu, L. Wang and R. Tan, Extinction and stationary distribution of a competition system with distributed delays and higher order coupled noises, Math. Biosci. Eng., 17 (2020), 3240-3251.  doi: 10.3934/mbe.2020184. [22] N. Ikeda and S. Watanade, A comparison theorem for solutions of stochastic differential equations and its applications, Osaka Math. J., 14 (1977), 619-633. [23] L. Imhof and S. Walcher, Exclusion and persistence in deterministic and stochastic chemostat models, J. Diff. Equ., 217 (2005), 26-53.  doi: 10.1016/j.jde.2005.06.017. [24] C. Ji, X. Yang and Y. Li, Permanence, extinction and periodicity to a stochastic competitive model with infinite distributed delays, J. Dyn. Differ. Equ., 33 (2021), 135-176.  doi: 10.1007/s10884-020-09850-7. [25] C. Ji, X. Yang and Y. Li, Periodic solutions for SDEs through upper and lower solutions, Discre. Contin. Dyn. Syst. B., 25 (2020), 4737-4754.  doi: 10.3934/dcdsb.2020122. [26] D. Jiang, C. Ji, X. Li and D. O'Regan, Analysis of autonomous Lotka-Volterra competition systems with random perturbation, J. Math. Anal. Appl., 390 (2012), 582-595.  doi: 10.1016/j.jmaa.2011.12.049. [27] X. Jiang and Y. 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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 $3000000$
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 $6000000$
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
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
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
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 $4000000$
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
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 $1000000$
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
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 $400000$
List of the initial value and biological parameters of two-species competition system (6)
 Parameter Value Source $(b_1, b_2, a_{11}, a_{12}, a_{21}, a_{22})$ $(1, 1.2, 0.3, 0.2, 0.3, 0.4)$ [26] $(x_1(0), x_2(0))$ $(1.5, 2)$ [26] $(y_{1, 2}(0), y_{2, 1}(0), \alpha_{12}, \alpha_{21})$ $(0.8, 0.2, 1, 1)$ Assumed
 Parameter Value Source $(b_1, b_2, a_{11}, a_{12}, a_{21}, a_{22})$ $(1, 1.2, 0.3, 0.2, 0.3, 0.4)$ [26] $(x_1(0), x_2(0))$ $(1.5, 2)$ [26] $(y_{1, 2}(0), y_{2, 1}(0), \alpha_{12}, \alpha_{21})$ $(0.8, 0.2, 1, 1)$ Assumed
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