The effect of vibropressing compaction process on the compressive strength based concrete paving blocks

Erno Widayanto; Agoes Soehardjono; Wisnumurti Wisnumurti; Achfas Zacoeb; Erno Widayanto; Agoes Soehardjono; Wisnumurti Wisnumurti; Achfas Zacoeb

doi:10.3934/matersci.2020.3.203

AIMS Materials Science

2020, Volume 7, Issue 3: 203-216. doi: 10.3934/matersci.2020.3.203

Previous Article Next Article

Research article Special Issues

The effect of vibropressing compaction process on the compressive strength based concrete paving blocks

1.
Department of Civil Engineering, Faculty of Engineering, University of Jember, Jember 68121, Indonesia
2.
Department of Civil, Faculty of Engineering, University of Brawijaya, Malang 68121, Indonesia

Received: 11 February 2020 Accepted: 13 May 2020 Published: 25 May 2020

Concrete paving blocks are a mixture of cement and aggregates, which using compressive energy blocks. Paving concrete is a dry concrete category, so it requires block energy in the compaction process. Block energy consists of manual blocks, block presses, and vibropressing blocks. They are pressing blocks to use hydraulics with a strong push of 75 kg/cm² with a 1, 2, and 3 s durations. Then the blocky vibropressing method, namely by making variations on the vibrating length between 4-8 seconds and a frequency of 25-50 Hz and pressing 75 kg/cm². All test specimens with the same volume ratio composition of 1 cement:4 sand:4 crushed stone ash 0-5 mm, and the water and cement ratio are 0.6. The results showed that the manual block and the pressing block did not have a sharp increase in compressive strength even though more blows in the manual bock and increased time in the bock pressing method. The vibropressing process shows a significant and linear growth in compressive strength with increase vibrations and frequencies. The conclusion is that the compressive strength of paving block concrete is very dependent on blocked in terms of the frequency and duration of vibrations.

Keywords:

Citation: Erno Widayanto, Agoes Soehardjono, Wisnumurti Wisnumurti, Achfas Zacoeb. The effect of vibropressing compaction process on the compressive strength based concrete paving blocks[J]. AIMS Materials Science, 2020, 7(3): 203-216. doi: 10.3934/matersci.2020.3.203

Related Papers:

[1]	Ekaterina Kldiashvili, Archil Burduli, Gocha Ghortlishvili . Application of Digital Imaging for Cytopathology under Conditions of Georgia. AIMS Medical Science, 2015, 2(3): 186-199. doi: 10.3934/medsci.2015.3.186
[2]	Anuj A. Shukla, Shreya Podder, Sana R. Chaudry, Bryan S. Benn, Jonathan S. Kurman . Non-small cell lung cancer: epidemiology, screening, diagnosis, and treatment. AIMS Medical Science, 2022, 9(2): 348-361. doi: 10.3934/medsci.2022016
[3]	Nicole Lavender, David W. Hein, Guy Brock, La Creis R. Kidd . Evaluation of Oxidative Stress Response Related Genetic Variants, Pro-oxidants, Antioxidants and Prostate Cancer. AIMS Medical Science, 2015, 2(4): 271-294. doi: 10.3934/medsci.2015.4.271
[4]	Masahiro Yasunaga, Shino Manabe, Masaru Furuta, Koretsugu Ogata, Yoshikatsu Koga, Hiroki Takashima, Toshirou Nishida, Yasuhiro Matsumura . Mass spectrometry imaging for early discovery and development of cancer drugs. AIMS Medical Science, 2018, 5(2): 162-180. doi: 10.3934/medsci.2018.2.162
[5]	Sherven Sharma, Pournima Kadam, Ram P Singh, Michael Davoodi, Maie St John, Jay M Lee . CCL21-DC tumor antigen vaccine augments anti-PD-1 therapy in lung cancer. AIMS Medical Science, 2021, 8(4): 269-275. doi: 10.3934/medsci.2021022
[6]	Ayomide Abe, Mpumelelo Nyathi, Akintunde Okunade . Lung cancer diagnosis from computed tomography scans using convolutional neural network architecture with Mavage pooling technique. AIMS Medical Science, 2025, 12(1): 13-27. doi: 10.3934/medsci.2025002
[7]	Timothy Hamerly, Margaret H. Butler, Steve T. Fisher, Jonathan K. Hilmer, Garth A. James, Brian Bothner . Mass Spectrometry Imaging of Chlorhexidine and Bacteria in a Model Wound. AIMS Medical Science, 2015, 2(3): 150-161. doi: 10.3934/medsci.2015.3.150
[8]	Prarthana Shrestha, Rik Kneepkens, Gijs van Elswijk, Jeroen Vrijnsen, Roxana Ion, Dirk Verhagen, Esther Abels, Dirk Vossen, and Bas Hulsken . Objective and Subjective Assessment of Digital Pathology Image Quality. AIMS Medical Science, 2015, 2(1): 65-78. doi: 10.3934/medsci.2015.1.65
[9]	Anne A. Adeyanju, Wonderful B. Adebagbo, Olorunfemi R. Molehin, Omolola R. Oyenihi . Exploring the multi-drug resistance (MDR) inhibition property of Sildenafil: phosphodiesterase 5 as a therapeutic target and a potential player in reversing MDR for a successful breast cancer treatment. AIMS Medical Science, 2025, 12(2): 145-170. doi: 10.3934/medsci.2025010
[10]	Salma M. AlDallal . Quick glance at Fanconi anemia and BRCA2/FANCD1. AIMS Medical Science, 2019, 6(4): 326-336. doi: 10.3934/medsci.2019.4.326

Abstract

1. Introduction

In mathematical modelling, the term diffusion is used to describe the motion of species from one region to another. Influenced by various natural factors, such as geographic, hydrological or climatic conditions and human activities, migrations occur between patches, which affects the population dynamics, for example the persistence and extinction of species ^{[1,2,3,4,5,6,7,8]}. The growth of species population is also affected by competition caused by disputing food, resources, territories and spouses, including intraspecific and interspecific competitions among populations. To see the effects of the diffusion and competition on population dynamics, we propose the following mathematical model with $n$ species, for $\;i, j = 1, 2, \ldots, n,$

$\begin{eqnarray} {\rm d}x_{i}(t)& = &x_{i}(t)\bigg[r_{i}-a_{ii}x_{i}(t)-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}x_{j}(t)+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}x_{j}(t)\\ &&-\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}x_{i}(t)\bigg] {\rm d}t, \end{eqnarray}$

(1)

where $x_{i}(t)$ is the population size at time $t$ of the $i$ th species, positive constants $r_{i}$ , $a_{ii}$ are the growth rate and the interspecific competition rate of the $i$ th species respectively, $a_{ij}>0\;(j\neq i)$ is the competition rate between species $i$ and $j$ , $D_{ij}\geq 0$ is the diffusion coefficient from species $j$ to species $i$ , $\alpha_{ij}\geq 0$ indicates the diffusion boundary condition.

Recently, time delays have been widely used in biological and ecological models in order to get more realistic mathematical models, for example ^{[9,10,11,12,13,14,15,16]}. In this paper, we also consider the time delay, which is accounted for the diffusion. For example, birds cannot migrate immediately after they were born, so the time delay here is the time it takes for them to learn to fly before they can migrate, and death can also occur in the process. Then, from (1) we have the model with time delays as follows

$\begin{eqnarray} {\rm d}x_{i}(t)& = &x_{i}(t)\bigg[r_{i}-a_{ii}x_{i}(t)-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}x_{j}(t)+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}x_{j}(t-\tau_{ij})\\ &&-\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}x_{i}(t)\bigg] {\rm d}t, \;i, j = 1, 2, \ldots, n, \end{eqnarray}$

(2)

where $\tau_{ij}\geq 0$ is the time delay and $d_{j}$ is the death rate of the $j$ th species. Let $\tau = \max_{i, j = 1, \ldots, n}\{\tau_{ij}\}$ and $C([-\tau, 0];R_{+}^{n})$ denote the family of all bounded and continuous functions from $[-\tau, 0]$ to $R_{+}^{n}$ . We assume model (2) is subject to the following initial condition

$\begin{equation} x(\theta) = \left(x_{1}(\theta), \ldots, x_{n}(\theta)\right)^{T} = \left(\phi_{1}(\theta), \ldots, \phi_{n}(\theta)\right)^{T} = \phi(\theta)\in C\left([-\tau, 0];R_{+}^{n}\right). \end{equation}$

(3)

Reference ^[17] suggests that the growth rate of organisms is generally affected by environmental fluctuations accounted for the disturbance of ecological environment in nature, consequently parameters in biologic models will exhibit random perturbations ^[18]. Thus, the deterministic models, like (2) are not applicable to capture the essential characters. In the past years, researchers have suggested the use of white noises to capture the main characters of these stochastic fluctuations, see ^{[18,19,20,21,22,23,24,25,26,27]} for example. Denote by $\{B_{i}(t)\}_{t\geq 0}, (i = 1, 2, \ldots, n)$ the independent standard Brownian motions defined on a complete probability space $(\Omega, \{F_{t}\}_{t\in R_{+}}, P)$ with $\sigma_{i}^{2}$ represents the intensity of the environment noises. Then, the growth rate subject to random perturbation can be described by

$r_{i}\rightarrow r_{i}+\sigma_{i} {\rm d}B_{i}(t),$

with which the model (2) reads

(4)

We further consider the optimal harvesting problem of model (4). The research on the optimal harvesting of the population is of great significance to the utilization and development of resources, and can also help mankind to get the optimal strategy of harvesting in order to obtain the most long-term benefits ^{[28,29,30,31,32,33,34,35]}. Then, we reach the following model accounted for harvesting:

(5)

where $h_{i}\geq 0$ denotes the harvesting effort of the species $i$ .

In the rest of the paper, we will devote ourselves to explore the dynamics and the optimal harvesting strategy of model (5). More precisely, in Section 2, we establish necessary conditions for persistence of species in mean and extinction of the species. In Section 3, we investigate conditions of stability, and prove asymptotic stability in distribution of the model, namely, there is a unique probability measure $\rho(\cdot)$ such that for each $\phi \in C([-\tau, 0];R_{+}^{n})$ , the transition probability $p(t, \phi, \cdot)$ of $x(t)$ converges weekly to $\rho(\cdot)$ when $t\rightarrow\infty$ . In Section 4, by the use of the Hessian matrix and theorems of optimal harvesting due to ^[36], we investigate the optimal harvesting effort and gain the maximum of expectation of sustainable yield (ESY). In Section 5, we numerically illustrate our theoretical results obtained in previous sections, and then conclude our study in Section 6.

2. Persistence and extinction

For the convenience of the following discussion, we define some notations as follows

$b_{i} = r_{i}-h_{i}-0.5\sigma_{i}^{2}, \;q_{i j} = a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}, \;c_{i} = b_{i}-\sum\limits_{j = 1, j\neq i}^{n}\frac{a_{ij}}{q_{ji}}b_{j}, \;i, j = 1, \ldots, n,$

and assume that $\sum_{j = 1, j\neq i}^{n}a_{ij}\geq \sum_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}$ holds in the rest of the paper.

Following the same argument as in ^[37], we can prove the existence of the positive solution.

Lemma 2.1. Given initial value (3), model (5) admits a unique global positive solution $x(t) = \left(x_{1}(t), \ldots, x_{n}(t)\right)^{T}$ almost surely. Furthermore, for each $p>1$ , there exists a positive constant $K = K(p)$ such that

$\begin{equation} \limsup\limits_{t\rightarrow+\infty}\mathbb{E}\bigg|x(t)\bigg|^{p}\leq K. \end{equation}$

(6)

To show our main result of this section, we consider the following auxiliary equations

$\begin{equation} {\rm d}\Phi_{i}(t) = \Phi_{i}(t)\bigg(r_{i}-h_{i}-a_{ii}\Phi_{i}(t)-\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{i j}\Phi_{i}(t)\bigg) {\rm d}t+\sigma_{i}\Phi_{i}(t) {\rm d}B_{i}(t), \end{equation}$

(7)

$\begin{eqnarray} {\rm d}\Psi_{i}(t)& = &\Psi_{i}(t)\bigg(r_{i}-h_{i}-a_{ii}\Psi_{i}(t)-\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\Phi_{j}(t)+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{i j}}\Phi_{j}(t-\tau_{i j})\\ &&-\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{ij}\Psi_{i}(t)\bigg) {\rm d}t+\sigma_{i}\Psi_{i}(t) {\rm d}B_{i}(t), \end{eqnarray}$

(8)

with initial value

$\Phi_{i}(\theta) = \Psi_{i}(\theta) = x_{i}(\theta), \;\theta\in[-\tau, 0], \;i = 1, 2, \ldots, n.$

By [38,Stochastic Comparison Theorem], we know that for $t\geq -\tau$ ,

$\begin{equation} \Psi_{i}(\theta)\leq x_{i}(\theta)\leq \Phi_{i}(\theta)\;a.s., \;i = 1, 2, \ldots, n. \end{equation}$

(9)

Remark 1. It is easy to see from ^[39] that the explicit solution of (7) is

$\begin{equation} \Phi_{i}(t) = \frac{\exp\{b_{i}t+\sigma_{i}B_{i}(t)\}}{\Phi_{i}^{-1}(0)+(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{i j})\int_{0}^{t}\exp\{b_{i}s+\sigma_{i}B_{i}(s)\} {\rm d}s}, \;i = 1, 2, \ldots, n. \end{equation}$

(10)

Similar calculation gives

$\begin{eqnarray} \Psi_{i}(t)& = &\exp\bigg\{b_{i}t-\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\int_{0}^{t}\Phi_{j}(s) {\rm d}s+\sum\limits_{j = 1, j\neq i} ^{n}D_{i j}e^{-d_{j}\tau_{i j}}\int_{0}^{t}\Phi_{j}(s-\tau_{i j}) {\rm d}s+\sigma_{i} {\rm d}B_{i}(t)\bigg\}\\ &&\times\bigg\{ \Psi_{i}^{-1}(0)+\bigg(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg)\int_{0}^{t}\exp\bigg\{b_{i}s-\sum\limits_{j = 1, j\neq i}^{n} a_{i j}\int_{0}^{s}\Phi_{j}(u) {\rm d}u\\ &&+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\int_{0}^{s}\Phi_{j}(u- \tau_{i j}) {\rm d}u+\sigma_{i}B_{i}(s)\bigg\} {\rm d}s\bigg\}^{-1}, \;i = 1, 2, \ldots, n. \end{eqnarray}$

(11)

Then, by using ^[40], we obtain the following.

Lemma 2.2. Let $b_{i}>0$ . Then, from (7) we have

$\begin{equation} \lim\limits_{t\rightarrow+\infty}t^{-1}\ln\Phi_{i}(t) = 0, \;\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}\Phi_{i}(s) {\rm d}s = \frac{b_{i}}{q_{ij}}, \;a.s., \;i = 1, 2, \ldots, n. \end{equation}$

(12)

Based on Lemma 2, we assume:

Assumption 2.1. $b_{i}>0, \;c_{i}>0, \;i = 1, 2, \ldots, n.$

Remark 2. A result due to Golpalsamy ^[10] and Assumption 2.1 imply that there exists a unique positive solution $(\det(A_{1})/\det (A), \ldots, \det(A_{n})/\det (A))^{T}$ for the following system

$\begin{equation} \left\{\begin{array}{l} {(a_{11}+\sum\limits_{j = 2}^{n}D_{1j}\alpha_{1j})x_{1}+(a_{12}-D_{12}e^{-d_{2}\tau_{12}})x_{2}+\ldots+(a_{1n}- D_{1n}e^{-d_{n}\tau_{1 n}})x_{n} = b_{1}\triangleq r_{1}-h_{1}-\frac{1}{2}\sigma_{1}^{2}, }\\ {(a_{21}-D_{21}e^{-d_{1}\tau_{21}})x_{1}+(a_{22}+\sum\limits_{j = 1, j\neq 2}^{n}D_{2j}\alpha_{2j})x_{2}+\ldots+(a_{2 n}-D_{2 n}e^{-d_{n}\tau_{2n}})x_{n} = b_{2}\triangleq r_{2}-h_{2}-\frac{1}{2}\sigma_{2}^{2}, }\\ {\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots}, \\ {(a_{n1}-D_{n1}e^{-d_{1}\tau_{n1}})x_{1}+(a_{n2}-D_{n2}e^{-d_{2}\tau_{n 2}})x_{2}+\ldots+ (a_{nn}+\sum\limits_{j = 1}^{n-1}D_{nj}\alpha_{nj})x_{n} = b_{n}\triangleq r_{n}-h_{n}-\frac{1}{2}\sigma_{n}^{2}}, \end{array} \right. \end{equation}$

(13)

in which

$A = \left( \begin{array}{cccc} a_{11}+\sum_{j = 2}^{n}D_{1j}\alpha_{1j} & a_{12}-D_{12}e^{-d_{2}\tau_{12}} & \cdots & a_{1n}-D_{1n}e^{-d_{n}\tau_{1 n}} \\ a_{21}-D_{21}e^{-d_{1}\tau_{21}} & a_{22}+\sum\limits_{j = 1, j\neq 2}^{n}D_{2j}\alpha_{2j} & \cdots & a_{2 n}-D_{2 n}e^{-d_{n}\tau_{2n}} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1}-D_{n1}e^{-d_{1}\tau_{n1}} & a_{n2}-D_{n2}e^{-d_{2}\tau_{n 2}} & \cdots &a_{nn}+\sum_{j = 1}^{n-1}D_{nj}\alpha_{nj} \end{array} \right)$

and $A_{i}$ is the matrix given by using the $(b_{1}, b_{2}, \ldots, b_{n})^{T}$ to replace the $i$ th column of matrix $A$ .

Now we are in the position to show our main results.

Theorem 2.1. All species in system (5) are persistent in mean $a.s.$ , $i.e.$ ,

$\begin{equation} \lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}x_{i}(s) {\rm d}s = \det(A_{i})/\det(A) \gt 0\;\;a.s., \;i = 1, 2, \ldots, n. \end{equation}$

(14)

when Assumption 2.1 is satisfied.

Proof. Let $b_{i}>0$ , according to (12) that for $i, j = 1, 2, \ldots, n, \;j\neq i$ , one has

$\begin{equation} \lim\limits_{t\rightarrow+\infty}t^{-1}\int_{t-\tau_{ij}}^{t}\Phi_{j}(s) {\rm d}s = \lim\limits_{t\rightarrow+\infty}\bigg(t^{-1}\int_{0} ^{t}\Phi_{j}(s) {\rm d}s-t^{-1}\int_{0}^{t-\tau_{ij}}\Phi_{j}(s) {\rm d}s\bigg) = 0, \end{equation}$

(15)

which together with (9) yields

$\begin{equation} \lim\limits_{t\rightarrow+\infty}t^{-1}\int_{t-\tau_{i j}}^{t}x_{j}(s) {\rm d}s = 0, \;i, j = 1, 2, \ldots, n, \;j\neq i. \end{equation}$

(16)

By using Itô's formula to (5), one can see that

$\begin{align} &t^{-1}\ln x_{i}(t)-t^{-1}\ln x_{i}(0)\\ = &b_{i}- a_{ii}t^{-1}\int_{0}^{t}x_{i}(s) {\rm d}s- \sum\limits_{j = 1, j\neq i}^{n}a_{ij}t^{-1}\int_{0}^{t}x_{j}(s) {\rm d}s +\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}t^{-1}\int_{0}^{t}x_{j}(s-\tau_{ij}) {\rm d}s\\ &-\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}t^{-1}\int_{0}^{t}x_{j}(s) {\rm d}s+\sigma_{i}t^{-1}B_{i}(t)\\ = &b_{i}-\bigg[a_{ii}t^{-1}\int_{0}^{t}x_{i}(s) {\rm d}s+ \sum\limits_{j = 1, j\neq i}^{n}a_{ij}t^{-1}\int_{0}^{t}x_{j}(s) {\rm d}s- \sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}t^{-1}\int_{0}^{t}x_{j}(s) {\rm d}s\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}t^{-1}\int_{0}^{t}x_{i}(s) {\rm d}s\bigg]+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}t^{-1}\bigg[\int_{-\tau_{ij}}^{0}x_{j}(s) {\rm d}s\ -\int_{t-\tau_{i j}}^{t}x_{j}(s) {\rm d}s\bigg]\\ &+\sigma_{i}t^{-1}B_{i}(t), \;\;\;i, j = 1, 2\ldots, n, \;i\neq j. \end{align}$

(17)

According to (16) together with the property of Brownian motion, we obtain

$\lim\limits_{t\rightarrow+\infty}t^{-1}\bigg[\int_{-\tau_{i j}}^{0}x_{j}(s) {\rm d}s-\int_{t-\tau_{i j}}^{t}x_{j}(s) {\rm d}s\bigg] = 0,$

$\lim\limits_{t\rightarrow+\infty}t^{-1}B_{i}(t) = 0, \;\;\lim\limits_{t\rightarrow+\infty}t^{-1}\ln x_{i}(0) = 0, \;a.s.$

We next to show that

$\lim\limits_{t\rightarrow+\infty}t^{-1}\ln x_{i}(t) = 0, \;i = 1, 2, \ldots, n.$

In view of (9) and (12), we have

$\liminf\limits_{t\rightarrow+\infty}t^{-1}\ln \Psi_{i}(t)\leq \liminf\limits_{t\rightarrow+\infty}t^{-1}\ln x_{i}(t)\leq \limsup\limits_{t\rightarrow+\infty}t^{-1}\ln x_{i}(t)\leq \limsup\limits_{t\rightarrow+\infty}t^{-1}\ln \Phi_{i}(t) = 0.$

Therefore we obtain

$\begin{equation} \liminf\limits_{t\rightarrow+\infty}t^{-1}\ln \Psi_{i}(t)\geq 0\;a.s., \;i = 1, 2, \ldots, n. \end{equation}$

(18)

From (15) and (12), we get

$\begin{align} &\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}\Phi_{j}(s-\tau_{i j}) {\rm d}s\\ & = \lim\limits_{t\rightarrow+\infty}t^{-1}\bigg(\int_{0}^{t}\Phi_{j}(s) {\rm d}s-\int_{t-\tau_{i j}}^{t}\Phi_{j}(s) {\rm d}s +\int_{\tau_{i j}}^{0}\Phi_{j}(s) {\rm d}s\bigg)\\ & = \frac{b_{j}}{q_{ji}}, \;\;a.s., \;\;i, j = 1, 2\ldots, n, \;i\neq j. \end{align}$

By using $\lim_{t\rightarrow+\infty}t^{-1}B_{i}(t) = 0$ together with what we have just obtained, yields that for any given $\varepsilon>0$ , there exists a $T = T(\omega)$ thus for $t\geq T, \;i, j = 1, 2\ldots, n, \;i\neq j,$

$b_{j}/q_{ji}-\varepsilon\leq t^{-1}\int_{0}^{t}\Phi_{j}(s-\tau_{ij}) {\rm d}s\leq b_{j}/q_{ji}+\varepsilon, \;\;-\varepsilon\leq t^{-1}\sigma_{i}B_{i}(t)\leq \varepsilon.$

Applying these inequalities to (11), we have

$\begin{align} &\frac{1}{\Psi_{i}(t)}\\ = &\exp\bigg\{-b_{i}t+\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\int_{0}^{t}\Phi_{j}(s) {\rm d}s- \sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\int_{0}^{t}\Phi_{j}(s-\tau_{ij}) {\rm d}s- \sigma_{i}B_{i}(t)\bigg\}\\ &\times \bigg\{\Psi_{i}^{-1}(0)+\left(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}\right) \int_{0}^{t}\exp\bigg\{b_{i}s-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\int_{0}^{s}\Phi_{j}(u) {\rm d}u\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\int_{0}^{s}\Phi_{j}(u-\tau_{ij}) {\rm d}u+ \sigma_{i}B_{i}(s)\bigg\} {\rm d}s\bigg\} \\ = &\exp\bigg\{-b_{i}t+\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\int_{0}^{t}\Phi_{j}(s) {\rm d}s- \sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\int_{0}^{t}\Phi_{j}(s-\tau_{ij}) {\rm d}s- \sigma_{i}B_{i}(t)\bigg\}\\ &\times \bigg\{\Psi_{i}^{-1}(0)+\left(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}\right) \int_{0}^{T}\exp\bigg\{b_{i}s-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\int_{0}^{s}\Phi_{j}(u) {\rm d}u\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\int_{0}^{s}\Phi_{j}(u-\tau_{ij}) {\rm d}u+ \sigma_{i}B_{i}(s)\bigg\} {\rm d}s\\ &+\bigg(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}\bigg) \int_{T}^{t}\exp\bigg\{b_{i}s-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\int_{0}^{s}\Phi_{j}(u) {\rm d}u\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\int_{0}^{s}\Phi_{j}(u-\tau_{ij}) {\rm d}u+ \sigma_{i}B_{i}(s)\bigg\} {\rm d}s\bigg\} \\ \leq& \exp\bigg\{t\bigg[-b_{i}+\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg)- \sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \varepsilon\bigg]\bigg\}\\ &\times \bigg\{\Psi_{i}^{-1}(0)+M_{ij}+\bigg(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}\bigg) \int_{T}^{t}\exp\bigg\{s\bigg[b_{i}-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg) +\varepsilon\bigg]\bigg\} {\rm d}s\bigg\}, \;i, j = 1, \ldots, n, \end{align}$

in which $M_{ij}>0$ is a constant. Note that $c_{i} = b_{i}-\sum_{j = 1, j\neq i}^{n}\frac{a_{ij}}{q_{ji}}b_{j}>0$ , thereby for large enough $t$ , one has that

$\begin{align} &\Psi_{i}^{-1}(0)+M_{ij}\\ &\leq \bigg(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}\bigg) \int_{T}^{t}\exp\bigg\{s\bigg[b_{i}-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg) +\varepsilon\bigg]\bigg\} {\rm d}s. \end{align}$

Hence for sufficiently large $t$ , we obtain

$\begin{align} &\frac{1}{\Psi_{i}(t)}\\ \leq&\exp\bigg\{t\bigg[-b_{i}+\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg)- \sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \varepsilon\bigg]\bigg\}\\ &\times 2\bigg(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg) \int_{T}^{t}\exp\bigg\{s\bigg[b_{i}-\sum\limits_{j = 1, j\neq i}^{n}a_{ij}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg) +\varepsilon\bigg]\bigg\} {\rm d}s \\ = &\frac{2\bigg(a_{ii}+\sum_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg)} {b_{i}-\sum_{j = 1, j\neq i}^{n}a_{i j}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \sum_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg) +\varepsilon}\\ &\times \exp\bigg\{t\bigg[-b_{i}+\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg)- \sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \varepsilon\bigg]\bigg\}\\ &\times \exp\bigg\{\bigg[b_{i}-\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg)+ \varepsilon\bigg](t-T)\bigg\}. \end{align}$

Rearranging this inequality shows that

$\begin{eqnarray} t^{-1}\ln \Psi_{i}(t)&\geq& t^{-1}\ln \frac{b_{i}-\sum_{j = 1, j\neq i}^{n}a_{i j}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)+ \sum_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg)+ \varepsilon}{2\bigg(a_{ii}+\sum_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg)}\\ &&-2\varepsilon \bigg(\sum\limits_{j = 1, j\neq i}^{n}a_{i j}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}+1\bigg)+ \bigg[b_{i}-\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\bigg(\frac{b_{j}}{q_{ji}}-\varepsilon\bigg)\\ &&+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(\frac{b_{j}}{q_{ji}}+\varepsilon\bigg)+ \varepsilon\bigg]\frac{T}{t}. \end{eqnarray}$

Since $t$ is large enough and $\varepsilon$ is arbitrary, we get (14). This completes the proof of Theorem 2.1.

Corollary 2.1. If there is a $b_{i}<0$ , then according to (17), one has $\limsup_{t\rightarrow+\infty}t^{-1}\ln x_{i}(t)\leq b_{i}<0, \;a.s.$ It is to say $\lim_{t\rightarrow+\infty}x_{i}(t) = 0, \;a.s.$ , which means that the $i$ th species in system (5) will die out.

3. Stability in distribution

In this section, we study the stability of the model. To this end, we suppose the following holds:

Assumption 3.1. $a_{ii}+\sum_{j = 1, j\neq i}^{n}D_{ij}\alpha_{ij}\geq \sum_{j = 1, j\neq i}^{n}a_{ji}+\sum_{j = 1, j\neq i}^{n}D_{ji}e^{-d_{i}\tau_{ji}}, \;i = 1, 2, \ldots, n.$

Then, we can prove the following.

Theorem 3.1. The system (5) is asymptotically stable in distribution if Assumption 3.1 holds.

Proof. Given two initial values $\phi(\theta), \psi(\theta) \in C\left([-\tau, 0];R_{+}^{n}\right)$ of model (5), the corresponding solutions are $x^{\phi}(t) = (x_{1}^{\phi_{1}}(t), \ldots, x_{n}^{\phi_{n}}(t))^{T}$ and $x^{\psi}(t) = (x_{1}^{\psi_{1}}(t), \ldots, x_{n}^{\psi_{n}}(t))^{T}$ respectively. Let

$V(t) = \sum\limits_{i = 1}^{n}\bigg|\ln x_{i}^{\phi_{i}}(t)-\ln x_{i}^{\psi_{i}}(t)\bigg|+ \sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}} \int_{t-\tau_{i j}}^{t}\bigg|x_{j}^{\phi_{j}}(s)-x_{j}^{\psi_{j}}(s)\bigg| {\rm d}s.$

Applying Itô's formula yields

$\begin{align} & {\rm d}^{+}V(t)\\ = &\sum\limits_{i = 1}^{n} {\rm sgn}\bigg(x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg) {\rm d}\bigg(\ln x_{i}^{\phi_{i}}(t)-\ln x_{i}^{\psi_{i}}(t)\bigg) +\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{ij}e^{-d_{j}\tau_{ij}}\bigg|x_{j}^{\phi_{j}}(t)-x_{j}^{\psi_{j}}(t)\bigg| {\rm d}t\\ &-\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg|x_{j}^{\phi_{j}}(t-\tau_{i j})- x_{j}^{\psi_{j}}(t-\tau_{i j})\bigg| {\rm d}t\\ = &\sum\limits_{i = 1}^{n} {\rm sgn}\bigg(x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg) \bigg[-a_{ii}\bigg(x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg) -\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\bigg(x_{j}^{\phi_{j}}(t)-x_{j}^{\psi_{j}}(t)\bigg)\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg(x_{j}^{\phi_{j}}(t-\tau_{i j})-x_{j}^{\psi_{j}}(t-\tau_{i j})\bigg) -\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg(x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg)\bigg] {\rm d}t\\ &+\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg|x_{j}^{\phi_{j}}(t)-x_{j}^{\psi_{j}}(t)\bigg| {\rm d}t -\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg|x_{j}^{\phi_{j}}(t-\tau_{i j})- x_{j}^{\psi_{j}}(t-\tau_{i j})\bigg| {\rm d}t\\ \leq& -\sum\limits_{i = 1}^{n}a_{ii}\bigg|x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg| {\rm d}t +\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\bigg|x_{j}^{\phi_{j}}(t)-x_{j}^{\psi_{j}}(t)\bigg| {\rm d}t\\ &+\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg|x_{j}^{\phi_{j}}(t-\tau_{i j}) -x_{j}^{\psi_{j}}(t-\tau_{i j})\bigg| {\rm d}t +\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg|x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg| {\rm d}t\\ &+\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg|x_{j}^{\phi_{j}}(t)-x_{j}^{\psi_{j}}(t)\bigg| {\rm d}t -\sum\limits_{i = 1}^{n}\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg|x_{j}^{\phi_{j}}(t-\tau_{i j}) -x_{j}^{\psi_{j}}(t-\tau_{i j})\bigg| {\rm d}t\\ = &-\sum\limits_{i = 1}^{n}\bigg(a_{ii}-\sum\limits_{j = 1, j\neq i}^{n}a_{ji}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j} -\sum\limits_{j = 1, j\neq i}^{n}D_{ji}e^{-d_{i}\tau_{ji}}\bigg)\bigg|x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)\bigg| {\rm d}t. \end{align}$

Therefore

$\mathbb{E}(V(t))\leq V(0)-\sum\limits_{i = 1}^{n}\bigg(a_{ii}-\sum\limits_{j = 1, j\neq i}^{n}a_{ji}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j} -\sum\limits_{j = 1, j\neq i}^{n}D_{ji}e^{-d_{i}\tau_{ji}}\bigg)\int_{0}^{t}\mathbb{E} \bigg|x_{i}^{\phi_{i}}(s)-x_{i}^{\psi_{i}}(s)\bigg| {\rm d}s.$

Together with $\mathbb{E}(V(t))\geq 0$ , one has

$\sum\limits_{i = 1}^{n}\bigg(a_{ii}-\sum\limits_{j = 1, j\neq i}^{n}a_{ji}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j} -\sum\limits_{j = 1, j\neq i}^{n}D_{ji}e^{-d_{i}\tau_{ji}}\bigg)\int_{0}^{t}\mathbb{E} \bigg|x_{i}^{\phi_{i}}(s)-x_{i}^{\psi_{i}}(s)\bigg| {\rm d}s\leq V(0) \lt \infty.$

Hence we have $\mathbb{E}\bigg|x_{i}^{\phi_{i}}(s)-x_{i}^{\psi_{i}}(s)\bigg|\in L^{1}[0, \infty), \;i = 1, 2, \ldots, n.$ At the same time, by using (5) we obtain that

$\begin{align} &\mathbb{E}(x_{i}(t))\\ = &x_{i}(0)+\int_{0}^{t}\bigg[\mathbb{E}(x_{i}(s))(r_{i}-h_{i})-a_{ii}\mathbb{E}(x_{i}(s))^{2}- \sum\limits_{j = 1, j\neq i}^{n}a_{i j}\mathbb{E}(x_{i}(s)x_{j}(s))\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\mathbb{E}(x_{i}(s)x_{j}(s-\tau_{i j})) -\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\mathbb{E}(x_{i}(s))^{2}\bigg] {\rm d}s \\ = &x_{i}(0)+\int_{0}^{t}\bigg[\mathbb{E}(x_{i}(s))(r_{i}-h_{i})-a_{ii}\mathbb{E}(x_{i}(s))^{2}- \sum\limits_{j = 1, j\neq i}^{n}a_{i j}\mathbb{E}(x_{i}(s)x_{j}(s))\\ &-\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\mathbb{E}(x_{i}(s))^{2}\bigg] {\rm d}s +\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg[\int_{-\tau_{ij}}^{0}\mathbb{E}(x_{i}(s)x_{j}(s)) {\rm d}s\\ &+\int_{0}^{t}\mathbb{E}(x_{i}(s)x_{j}(s)) {\rm d}s- \int_{t-\tau_{ij}}^{t}\mathbb{E}(x_{i}(s)x_{j}(s)) {\rm d}s\bigg]\\ \leq& x_{i}(0)+\int_{0}^{t}\bigg[\mathbb{E}x_{i}(s)(r_{i}-h_{i})-a_{ii}\mathbb{E}(x_{i}(s))^{2}- \sum\limits_{j = 1, j\neq i}^{n}a_{i j}\mathbb{E}(x_{i}(s)x_{j}(s))\\ &-\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\mathbb{E}(x_{i}(s))^{2}\bigg] {\rm d}s +\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\bigg[\int_{-\tau_{ij}}^{0}\mathbb{E}(x_{i}(s)x_{j}(s)) {\rm d}s\\ &+\int_{0}^{t}\mathbb{E}(x_{i}(s)x_{j}(s)) {\rm d}s\bigg]. \end{align}$

That is to say $\mathbb{E}(x_{i}(t))$ is continuously differentiable with respect of $t$ . Computing by (5) leads to

$\begin{align} &\frac{ {\rm d}\mathbb{E}(x_{i}(t))}{ {\rm d}t}\\ \leq& \mathbb{E}(x_{i}(t))(r_{i}-h_{i})-\bigg(a_{ii}+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}\alpha_{i j}\bigg)\mathbb{E}(x_{i}(t))^{2} -\sum\limits_{j = 1, j\neq i}^{n}a_{i j}\mathbb{E}(x_{i}(t)x_{j}(t))\\ &+\sum\limits_{j = 1, j\neq i}^{n}D_{i j}e^{-d_{j}\tau_{i j}}\mathbb{E}(x_{i}(t)x_{j}(t))\\ \leq& \mathbb{E}(x_{i}(t))r_{i}\leq r_{i}K, \end{align}$

in which $K>0$ is a constant. It implies that $\mathbb{E}(x_{i}(t))$ is uniformly continuous. Using ^[41], we get

$\begin{equation} \lim\limits_{t\rightarrow+\infty}\mathbb{E}|x_{i}^{\phi_{i}}(t)-x_{i}^{\psi_{i}}(t)| = 0, \;a.s., \;i = 1, 2, \ldots, n. \end{equation}$

(19)

Denote $p(t, \phi, {\rm d} y)$ as the transition probability density of the process $x(t)$ and $P(t, \phi, A)$ represents the probability of event $x(t)\in A$ . By (6) and [42,Chebyshev's inequality], we can obtain that the family of $p(t, \phi, {\rm d} y)$ is tight. Now define $\Gamma\left(C\left([-\tau, 0];R^n_+\right)\right)$ as the probability measures on $C\left([-\tau, 0];R^n_+\right)$ . For arbitrary two measures $P_1, P_2\in \Gamma$ , we define the metric

$\begin{equation*} d_\mathbb{L}(P_1, P_2) = \sup\limits_{v\in \mathbb{L}}\bigg|\int_{R^n_+}v(x)P_1(\mathrm{d}x)-\int_{R^n_+}v(x)P_2(\mathrm{d}x)\bigg|, \end{equation*}$

where

$\begin{equation*} \mathbb{L} = \left\{v:C\left([-\tau, 0];R^3_+\right)\rightarrow R:||v(x)-v(y)||\leq\parallel x-y\parallel, | v(\cdot)|\leq1\right\}. \end{equation*}$

Since $\{p(t, \phi, {\rm d} y)\}$ is tight, then according to (19) we know that for any $\varepsilon>0$ , there is a $T>0$ satisfies that for $t\geq T, s>0$ ,

$\begin{equation*} \sup\limits_{v\in \mathbb{L}}\bigg|\mathbb{E}v(x(t+s))-\mathbb{E}v(x(t))\bigg|\leq\varepsilon. \end{equation*}$

Therefore $\{p(t, \xi, \cdot)\}$ is Cauchy in $\Gamma$ with metric $d_\mathbb{L}$ , in which $\xi\in C\left([-\tau, 0];R^n_+\right)$ is arbitrary given. Hence there exists a unique $\kappa(\cdot)\in \Gamma(C([-\tau, 0];R^n_+))$ such that $\lim\limits_{t\rightarrow\infty}d_\mathbb{L}(p(t, \xi, \cdot), \kappa(\cdot)) = 0$ . At the same time, it follows from (19) that

$\begin{equation*} \lim\limits_{t\rightarrow\infty}d_\mathbb{L}(p(t, \phi, \cdot), p(t, \xi, \cdot)) = 0. \end{equation*}$

Consequently,

$\begin{equation*} \lim\limits_{t\rightarrow\infty}d_\mathbb{L}(p(t, \phi, \cdot), \kappa(\cdot))\leq\lim\limits_{t\rightarrow \infty}d_\mathbb{L}(p(t, \phi, \cdot), p(t, \xi, \cdot))+\lim\limits_{t\rightarrow\infty}d_\mathbb{L} (p(t, \xi, \cdot), \kappa(\cdot)) = 0. \end{equation*}$

This completes the proof of Theorem 3.1.

4. Optimal harvesting

In this section, we consider the optimal harvesting problem of system (5). Our purpose is to find the optimal harvesting effort $H^{*} = (h_{1}^{*}, \ldots, h_{n}^{*})$ such that:

(ⅰ) $Y(H) = \lim_{t\rightarrow+\infty}\sum_{i = 1}^{n}\mathbb{E}(h_{i}x_{i}(t))$ is maximum;

(ⅱ) Every $x_{i}(i = 1, 2, \ldots, n)$ is persistent in the mean.

Before we give our main results, we define

$\begin{equation} \Theta = (\theta_{1}, \theta_{2}, \ldots, \theta_{n})^{T} = [A(A^{-1})^{T}+I]^{-1}G, \end{equation}$

(20)

in which $G = (r_{1}-0.5\sigma_{1}^{2}, r_{2}-0.5\sigma_{2}^{2}, \ldots, r_{n}-0.5\sigma_{n}^{2})^{T}$ and $I$ is the unit matrix, and make an assumption:

Assumption 4.1. $A^{-1}+(A^{-1})^{T}$ is positive definite,

Theorem 4.1. Suppose Assumptions 3.1 and 4.1 hold, and If these following inequalities

$\begin{equation} \theta_{i}\geq 0, \;b_{i}\mid_{h_{i} = \theta_{i}} \gt 0, \;c_{i}\mid_{h_{m} = \theta_{m}, \;m = 1, 2, \ldots, n} \gt 0, i = 1, \cdots, n \end{equation}$

(21)

are satisfied. Then, for system (5) the optimal harvesting effort is

$\begin{equation*} H^{*} = \Theta = [A(A^{-1})^{T}+I]^{-1}G \end{equation*}$

and the maximum of ESY is

$\begin{equation} Y^{*} = \Theta^{T}A^{-1}(G-\Theta). \end{equation}$

(22)

Proof. Denote $W = \{H = (h_{1}, \ldots, h_{n})^{T}\in R^{n}\mid b_{i}>0, \;c_{i}>0, \;h_{i}>0, \;i = 1, \ldots, n\}$ . Easily we can see that for any $H\in W$ , (14) is satisfied. Note that $\Theta\in W$ , then $W$ is not empty. According to (14), we have that for every $H\in W$ ,

$\begin{equation} \lim\limits_{t\to+\infty} t^{-1}\int_0^tH^\mathrm{T}x(s)\mathrm{d}s = \sum\limits_{i = 1}^n h_i\lim\limits_{t\to+\infty} t^{-1}\int_0^tx_{i}(s)\mathrm{d}s = H^\mathrm{T}A^{-1}(G-H). \end{equation}$

(23)

Applying Theorem 4.1, there is a unique invariant measure $\rho(\cdot)$ for model (5). By [43,Corollary 3.4.3], we obtain that $\rho(\cdot)$ is strong mixing. Meanwhile, it is ergodic according to [43,Theorem 3.2.6]. It means

$\begin{equation} \lim\limits_{t\to+\infty} t^{-1}\int_0^tH^\mathrm{T}x(s)\mathrm{d}s = \int_{R^n_+}H^\mathrm{T}x\rho(\mathrm{d}x). \end{equation}$

(24)

Let $\mu(x)$ represent the stationary probability density of system (5), then we have

$\begin{equation} Y(H) = \lim\limits_{t\to+\infty}\sum\limits_{i = 1}^n\mathbb{E}(h_ix_{i}(t)) = \lim\limits_{t\to+\infty}\mathbb{E}(H^\mathrm{T}x(t)) = \int_{R^n_+}H^\mathrm{T}x\mu(x)\mathrm{d}x. \end{equation}$

(25)

Since the invariant measure of model (9) is unique, one has

$\begin{equation} \int_{R^n_+}H^\mathrm{T}x\mu(x)\mathrm{d}x = \int_{R^n_+}H^\mathrm{T}x\rho(\mathrm{d}x). \end{equation}$

(26)

In other words,

$\begin{equation} Y(H) = H^\mathrm{T}A^{-1}(G-H). \end{equation}$

(27)

Assume that $\Theta = (\theta_{1}, \theta_{2}, \ldots, \theta_{n})^{T}$ is the solution of the following equation

$\begin{equation} \begin{aligned} \frac{\mathrm{d}Y(H)}{\mathrm{d}H}& = \frac{\mathrm{d}H^\mathrm{T}}{\mathrm{d}H}A^{-1}(G-H)+ \frac{\mathrm{d}}{\mathrm{d}H}\left[(G-H)^\mathrm{T}(A^{-1})^\mathrm{T}\right]H\\ & = A^{-1}G-\left[A^{-1}+(A^{-1})^\mathrm{T}\right]H\\ & = 0. \end{aligned} \end{equation}$

(28)

Thus, $\Theta = [A(A^{-1})^\mathrm{T}+I]^{-1}G$ . By using of the Hessian matrix (see ^[44,45]),

$\begin{equation*} \begin{aligned} \frac{\mathrm{d}}{\mathrm{d}H^\mathrm{T}}\left[\frac{\mathrm{d}Y(H)}{\mathrm{d}H}\right] & = \left(\frac{\mathrm{d}}{\mathrm{d}H}\left[\left(\frac{\mathrm{d}Y(H)}{\mathrm{d}H}\right)^\mathrm{T}\right]\right) ^\mathrm{T} \\ & = \left(\frac{\mathrm{d}}{\mathrm{d}H}\left[G^\mathrm{T}(A^{-1})^\mathrm{T}-H^\mathrm{T}[A^{-1}+(A^{-1})^\mathrm {\mathrm{T}}]\right]\right)^\mathrm{T}\\ & = -A^{-1}-(A^{-1})^\mathrm{T} \end{aligned} \end{equation*}$

is negative defined, then $\Theta$ is the unique extreme point of $Y(H)$ . That is to say, if $\Theta\in W$ and under the condition of (21), the optimal harvesting effort is $H^* = \Theta$ and $Y^*$ is the maximum value of ESY. This completes the proof of Theorem 4.1.

5. Numerical simulations

To see our analytical results more clearly, we shall give some numerical simulations in this section. Without loss of generality, we consider the following system

$\begin{equation} \left\{\begin{aligned} {\rm d}x_{1}(t) = &x_{1}(t)\bigg[r_{1}-h_{1}-a_{11}x_{1}(t)-a_{12}x_{2}(t)+ D_{12}e^{-d_{2}\tau_{12}}x_{2}(t-\tau_{12})-D_{12}\alpha_{12}x_{1}(t)\bigg] {\rm d}t\\ &+\sigma_{1}x_{1}(t) {\rm d}B_{1}(t), \\ {\rm d}x_{2}(t)& = x_{2}(t)\bigg[r_{2}-h_{2}-a_{22}x_{2}(t)-a_{21}x_{1}(t)+ D_{21}e^{-d_{1}\tau_{21}}x_{1}(t-\tau_{21})-D_{21}\alpha_{21}x_{2}(t)\bigg] {\rm d}t\\ &+\sigma_{2}x_{2}(t) {\rm d}B_{2}(t), \end{aligned} \right. \end{equation}$

(29)

which is the case when $n = 2$ in (5), with initial value

$x(\theta) = \phi(\theta)\in C\left([-\tau, 0];R_{+}^{2}\right), \;\tau = \max\{\tau_{1}, \tau_{2}\},$

where $r_{i}>0, \;a_{ij}>0, \;\tau_{i}\geq 0, \;i, j = 1, 2$ .

Firstly, we discuss the persistence in mean of $x_{1}$ and $x_{2}$ . For that, we take the parameter values as follows:

Table 1. Parameter Values for Figure 1–3.

Parameter	Value	Parameter	Value	Parameter	Value
$r_{1}$	0.9	$h_{1}$	0.1	$\alpha_{12}$	0.6
$r_{2}$	0.8	$h_{2}$	0.05	$\alpha_{21}$	0.3
$a_{11}$	0.6	$a_{12}$	0.2	$a_{21}$	0.35
$a_{22}$	0.23	$\sigma_{1}$	0.05	$\sigma_{2}$	0.05
$d_{1}$	1	$d_{2}$	1	$D_{12}$	4
$D_{21}$	3	$\tau_{12}$	8	$\tau_{21}$	5

| Show Table

DownLoad: CSV

The initial values are $x_{1}(\theta) = 0.5+0.01\sin\theta$ , $x_{2}(\theta) = 0.5+0.02\sin\theta$ , $\theta\in[-\tau, 0]$ . Simple calculations show that $b_{1} = 0.7988>0, \;b_{2} = 0.7488>0, \;c_{1} = 0.6662>0, \;c_{2} = 0.6556>0$ implying Assumption 2.1 is satisfied. Then by Theorem 2.1, we can obtain that in (29)

Figure 1. Time series of species

$x_{1}$ and

$x_{2}$ of system (29) with initial values

$x_{1}(\theta) = 0.5+0.01\sin\theta$ ,

$x_{2}(\theta) = 0.5+0.02\sin\theta$ ,

$\theta\in[-\tau, 0]$ and parameter values in Table 1.

DownLoad: Full-Size Img PowerPoint

$\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}x_{1}(s) {\rm d}s = \det(A_{1})/\det(A) = 0.2268 \gt 0\;\;a.s.,$

$\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}x_{2}(s) {\rm d}s = \det(A_{2})/\det(A) = 0.5964 \gt 0\;\;a.s..$

Applying the Milstein numerical method in ^[47], we then obtained the numerical solution of system (29), see Figure 1. It shows that $x_{1}$ and $x_{2}$ respectively asymptotical approach to 0.2268 and 0.5964 time averagely. And this agrees well with our results in Theorem 2.1. Then we research the distributions of $x_{1}$ and $x_{2}$ under the same conditions. Obviously, we have $a_{11}+D_{12}\alpha_{12}\geq a_{21}+D_{21}e^{-d_{1}\tau_{21}}, \;a_{22}+D_{21}\alpha_{21}\geq a_{12}+D_{12}e^{-d_{2}\tau_{12}}$ , it is to say Assumption 3.1 is satisfied. Thus by Theorem 3.1, system (29) is asymptotically stable in distribution as suggested by Figure 2.

Figure 2. Distributions of species

$x_{1}$ and

$x_{2}$ of system (29) with initial values

$x_{1}(\theta) = 0.5+0.01\sin\theta$ ,

$x_{2}(\theta) = 0.5+0.02\sin\theta$ ,

$\theta\in[-\tau, 0]$ and parameter values in Table 1.

DownLoad: Full-Size Img PowerPoint

Lastly, we consider the optimal harvesting strategy of system (29). It is easy to see that the Assumption 2.1 and Assumption 3.1 are satisfied. Furthermore, we have

$\Theta = (\theta_{1}, \theta_{2})^{T} = [A(A^{-1})^{T}+I]^{-1}(r_{1}-0.5\sigma_{1}^{2}, r_{2} -0.5\sigma_{2}^{2})^{T} = (0.4817, 0.3820)^{T},$

in which $I = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)$ . Since condition (21) is satisfied, by Theorem 4.1, the optimal harvesting effort is

$H^{*} = \Theta = (\theta_{1}, \theta_{2})^{T} = [A(A^{-1})^{T}+I]^{-1}(r_{1}-0.5\sigma_{1}^{2}, r_{2} -0.5\sigma_{2}^{2})^{T} = (0.4817, 0.3820)^{T},$

on the other hand, the maximum of ESY is

$Y^{*} = \Theta^{T}A^{-1}(r_{1}-0.5\sigma_{1}^{2}-\theta_{1}, r_{2}- 0.5\sigma_{2}^{2}-\theta_{2})^{T} = 0.1789.$

By using the Monte Carlo method (see ^[48]) and the parameters in Table 1, we can obtain Figure 3, showing our results in Theorem 4.1.

Figure 3. The harvesting policies

$E[h_{1}x_{1}(t)+h_{2}x_{2}(t)]$ of system (29) with initial values

$x_{1}(\theta) = 0.5+0.01\sin\theta$ ,

$x_{2}(\theta) = 0.5+0.02\sin\theta$ ,

$\theta\in[-\tau, 0]$ and parameter values in Table 1. The red line is with

$h_{1} = h_{1}^{*} = 0.4817, \;h_{2} = h_{2}^{*} = 0.3820$ , the green line is with

$h_{1} = 0.53, \;h_{2} = 0.2$ , the blue line is with

$h_{1} = 0.1, \;h_{2} = 0.2$ .

DownLoad: Full-Size Img PowerPoint

Table 2. Parameter Values for Figure 4-6.

Parameter	Value	Parameter	Value	Parameter	Value
$r_{1}$	2	$h_{1}$	0.4452	$\alpha_{12}$	0.8
$r_{2}$	1.12	$h_{2}$	0.3307	$\alpha_{13}$	0.67
$r_{3}$	0.6	$h_{3}$	0.3307	$\alpha_{21}$	0.56
$\alpha_{23}$	0.8	$\alpha_{31}$	0.6	$\alpha_{32}$	0.77
$a_{11}$	0.18	$a_{12}$	0.35	$a_{13}$	0.3
$a_{21}$	0.45	$a_{22}$	0.22	$a_{23}$	0.6
$a_{31}$	0.4	$a_{32}$	0.3	$a_{33}$	0.2
$\sigma_{1}$	0.05	$\sigma_{2}$	0.05	$\sigma_{3}$	0.05
$d_{1}$	0.39	$d_{2}$	0.57	$d_{3}$	0.37
$\tau_{12}$	3	$\tau_{13}$	3	$\tau_{21}$	5
$\tau_{22}$	5	$\tau_{31}$	4	$\tau_{32}$	5.5
$D_{12}$	4	$D_{13}$	5	$D_{21}$	2.4
$D_{23}$	4	$D_{31}$	2	$D_{32}$	2.5

| Show Table

DownLoad: CSV

Next, we consider a case of three species.

$\begin{equation} \left\{\begin{aligned} {\rm d}x_{1}(t) = &x_{1}(t)\bigg[r_{1}-h_{1}-a_{11}x_{1}(t)-\bigg(a_{12}x_{2}(t)+a_{13}x_{3}(t)\bigg)+ \bigg(D_{12}e^{-d_{2}\tau_{12}}x_{2}(t-\tau_{12})\\ &+D_{13}e^{-d_{3}\tau_{13}}x_{3}(t-\tau_{13})\bigg) -\bigg(D_{12}\alpha_{12}x_{1}(t)+D_{13}\alpha_{13}x_{1}(t)\bigg)\bigg] {\rm d}t\\ &+\sigma_{1}x_{1}(t) {\rm d}B_{1}(t), \\ {\rm d}x_{2}(t) = &x_{2}(t)\bigg[r_{2}-h_{2}-a_{22}x_{2}(t)-\bigg(a_{21}x_{1}(t)+a_{23}x_{3}(t)\bigg)+ \bigg(D_{21}e^{-d_{1}\tau_{21}}x_{1}(t-\tau_{21})\\ &+D_{23}e^{-d_{3}\tau_{23}}x_{3}(t-\tau_{23})\bigg) -\bigg(D_{21}\alpha_{21}x_{2}(t)+D_{23}\alpha_{23}x_{2}(t)\bigg)\bigg] {\rm d}t\\ &+\sigma_{2}x_{2}(t) {\rm d}B_{2}(t), \\ {\rm d}x_{3}(t) = &x_{3}(t)\bigg[r_{3}-h_{3}-a_{33}x_{3}(t)-\bigg(a_{31}x_{1}(t)+a_{32}x_{2}(t)\bigg)+ \bigg(D_{31}e^{-d_{1}\tau_{31}}x_{1}(t-\tau_{31})\\ &+D_{32}e^{-d_{2}\tau_{32}}x_{2}(t-\tau_{32})\bigg) -\bigg(D_{31}\alpha_{31}x_{3}(t)+D_{32}\alpha_{32}x_{3}(t)\bigg)\bigg] {\rm d}t\\ &+\sigma_{3}x_{3}(t) {\rm d}B_{3}(t). \end{aligned} \right. \end{equation}$

(30)

We use the following parameter values:

The initial values are $x_{1}(\theta) = 0.5+0.01\sin\theta$ , $x_{2}(\theta) = 0.5+0.02\sin\theta$ , $x_{3}(\theta) = 0.5+0.001\sin\theta$ , $\theta\in[-\tau, 0]$ . Easily we get that $b_{1} = 0.1.5536>0, \;b_{2} = 0.7881>0, \;b_{3} = 0.2681>0, \;c_{1} = 1.4502>0, \;c_{2} = 0.0552>0, \;c_{3} = 0.0229>0.$ Thus Assumption 2.1 is hold. By Theorem 2.1, we have for (30)

$\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}x_{1}(s) {\rm d}s = \det(A_{1})/\det(A) = 0.2543 \gt 0\;\;a.s.,$

$\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}x_{2}(s) {\rm d}s = \det(A_{2})/\det(A) = 0.1601 \gt 0\;\;a.s.,$

$\lim\limits_{t\rightarrow+\infty}t^{-1}\int_{0}^{t}x_{3}(s) {\rm d}s = \det(A_{3})/\det(A) = 0.0730 \gt 0\;\;a.s..$

The numerical results of Theorem 2.1 when $n = 3$ are shown in Figure 4.

Figure 4. Time series of species

$x_{1}$ ,

$x_{2}$ and

$x_{3}$ of system (30) with initial values

$x_{1}(\theta) = 0.5+0.01\sin\theta$ ,

$x_{2}(\theta) = 0.5+0.02\sin\theta$ ,

$x_{3}(\theta) = 0.5+0.001\sin\theta$ ,

$\theta\in[-\tau, 0]$ and parameter values in Table 2.

DownLoad: Full-Size Img PowerPoint

The stable distribution for $n = 3$ are shown in Figure 5.

Figure 5. Distributions of species

$x_{1}, \;x_{2}$ and

$x_{3}$ of system (30) with initial values

$x_{1}(\theta) = 0.5+0.01\sin\theta$ ,

$x_{2}(\theta) = 0.5+0.02\sin\theta$ ,

$x_{3}(\theta) = 0.5+0.001\sin\theta$ and parameter values in Table 2.

DownLoad: Full-Size Img PowerPoint

To numerical illustrate the optimal harvesting effort of (30), we set

$\Theta = (\theta_{1}, \theta_{2}, \theta_{3})^{T} = [A(A^{-1})^{T}+I]^{-1}(r_{1}-0.5\sigma_{1}^{2}, r_{2} -0.5\sigma_{2}^{2}, r_{3}-0.5\sigma_{3}^{2})^{T} = (1.1052, 0.5537, 0.1663)^{T},$

which yield $H^{*} = \Theta = (1.1052, 0.5537, 0.1663)^{T}$ , and the maximum of ESY is $Y^{*} = 0.2263$ , see Figure-6.

Figure 6. The harvesting policies

$E[h_{1}x_{1}(t)+h_{2}x_{2}(t)+h_{3}x_{3}(t)]$ of system (29) with initial values

$x_{1}(\theta) = 0.5+0.01\sin\theta$ ,

$x_{2}(\theta) = 0.5+0.02\sin\theta$ ,

$x_{3}(\theta) = 0.5+0.001\sin\theta$ and parameter values in Table 2. The red line is with

$h_{1} = h_{1}^{*} = 1.1052, \;h_{2} = h_{2}^{*} = 0.5537, \;h_{3} = h_{3}^{*} = 0.1663$ , the green line is with

$h_{1} = 0.1, \;h_{2} = 0.6, \;h_{3} = 0.6$ , the blue line is with

$h_{1} = 0.35, \;h_{2} = 0.4, \;h_{3} = 0.1$ .

DownLoad: Full-Size Img PowerPoint

6. Conclusions and discussions

In this paper, a stochastic n-species competitive model with delayed diffusions and harvesting has been considered. We studied the persistence in mean of every population, which is biologically significant because it shows that all populations can coexist in the community. Since the model (5) does not have a positive equilibrium point and its solution can not approach a positive value, we considered its asymptotically stable distribution. By using ergodic method, we obtained the optimal harvesting policy and the maximum harvesting yield of system (5). We have also done some numerical simulations of the situations for $n = 2$ and $n = 3$ in model (5) to illustrate our theoretical results as it is very useful whether in terms of mathematics or biology to visualize our conclusions.

Our studies showed some interesting results

(a) Both environmental disturbance and diffused time delay can effect the persistence and optimal harvesting effort of system (5)..

(b) Environmental noises have no effect on asymptotic stability in distribution of system (5), but the time delays have.

There are other meaningful aspects that can be studied further since our paper only consider the effects of white noises on population growth rate. In future, for example, we can consider the situation when white noises also have influences over harvesting (see ^[45]) and non-autonomous system (see ^[46]); the time delay will also be reflected in competition (see ^[49]). Furthermore, we can consider something more complex models such as the ones with regime-switching (see ^[50,51]) or Lévy jumps (see ^[14,42]).

Acknowledgments

This work was supported by the Research Fund for the Taishan Scholar Project of Shandong Province of China, and the SDUST Research Fund (2014TDJH102).

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this paper.

References

[1]	Shackel B (2003) The challenges of concrete block paving as a mature technology. Proceedings of 7th International Conference Concrete Block Paving, 1-9.
[2]	Nevill AM, Brooks JJ (2010) Concrete Technology, 2nd Eds., England: Pearson Education Limited.
[3]	Penteado CSG, de Carvalho EV, Lintz RCC (2016) Reusing ceramic tile polishing waste in paving block manufacturing. J Clean Prod 112: 514-520. doi: 10.1016/j.jclepro.2015.06.142
[4]	Wattanasiriwech D, Saiton A, Wattanasiriwech S (2009) Paving blocks from ceramic tile production waste. J Clean Prod 17: 1663-1668. doi: 10.1016/j.jclepro.2009.08.008
[5]	Uygunolu T, Topcu IB, Gencel O, et al. (2012) The effect of fly ash content and types of aggregates on the properties of pre-fabricated concrete interlocking blocks (PCIBs). Constr Build Mater 30: 180-187. doi: 10.1016/j.conbuildmat.2011.12.020
[6]	Gencel O, Ozel C, Koksal F, et al. (2012) Properties of concrete paving blocks made with waste marble. J Clean Prod 21: 62-70. doi: 10.1016/j.jclepro.2011.08.023
[7]	Agyeman S, Obeng-ahenkora NK, Assiamah S, et al. (2019) Exploiting recycled plastic waste as an alternative binder for paving blocks production. Case Stud Constr Mater 11: e00246.
[8]	Udawattha C, Galabada H, Halwatura R (2017) Mud concrete paving block for pedestrian pavements. Case Stud Constr Mater 7: 249-262
[9]	De Silva P, Sagoe-Crenstil K, Sirivivatnanon V (2007) Kinetics of geopolymerization: role of Al2O₃ and SiO₂. Cement Concrete Res 37: 512-518 doi: 10.1016/j.cemconres.2007.01.003
[10]	Arslan B, Kamas T (2017) Investigation of aggregate size effects on the compressive behavior of concrete by electromechanical and mechanical impedance spectroscopy. Procedia Struct Integr 5: 171-178. doi: 10.1016/j.prostr.2017.07.093
[11]	Ling T, Nor H, Mudiyono R (2006) The effect of cement and water cement ratio on concrete paving block. Constr Build Environ 3: 26-27.
[12]	Baskaran K, Gopinath K (2013) Study on applicability of ACI and DOE mix design methods for paving blocks. Annual Transactions of Institution of Engineers, Sri Lanka, 127-134.
[13]	Xuan D, Zhan B, Poon CS (2016) Development of a new generation of eco-friendly concrete blocks by accelerated mineral carbonation. J Clean Prod 133: 1235-1241. doi: 10.1016/j.jclepro.2016.06.062
[14]	Djamaluddin AR, Caronge MA, Tjaronge MW, et al. (2020) Evaluation of sustainable concrete paving blocks incorporating processed waste tea ash. Case Stud Constr Mater 12: e00325.
[15]	Sulistyana P, Widoanindyawati V, Pratamab MMD (2014) The influence of compression applied during production to the compression strength of dry concrete: An experimental study. Procedia Eng 95: 465-472. doi: 10.1016/j.proeng.2014.12.206
[16]	ACI Committee 309 (2011) Behavior of fresh concrete during vibration.
[17]	Xiao YJ, Liu R, Song HP, et al. (2015) The characteristics of perlite sound absorption board formed by vibration molding. Open Mater Sci J 9: 39-42. doi: 10.2174/1874088X01509010039
[18]	Boral limited (2006) DS2006 compaction of concrete. Available from: https://www.boral.com/news-announcements/management-presentations.
[19]	Badan Standardisasi Nasional (1996) Bata beton (paving block). SNI 03-0691-1996.
[20]	Iffat S (2015) Relation between density and compressive strength of hardened concrete. Concrete Res Lett 6: 182-189.
[21]	Wersall C (2016) Frequency optimization of vibratory rollers and plates for compaction of granular soil. Available from: http://www.diva-portal.org/smash/record.jsf?pid=diva2%3A929931&dswid=1941.
[22]	Koh HB, Yeoh D, Shahidan S (2017) Effect of re-vibration on the compressive strength and surface hardness of concrete. IOP Conference Series: Materials Science and Engineering, 271: 012057. doi: 10.1088/1757-899X/271/1/012057
[23]	Arslan ME, Yozgat E, Pul S, et al. (2011) Effects of vibration time on strength of ordinary and high performance concrete. Proceedings of the 4th WSEAS international conference on Energy and development-environment-biomedicine, 270-274.
[24]	Kovalska A, Auzins J (2011) Investigation of vibropressing process technology. Proceedings of the 10th International Scientific Conference, 26: 408-412.

Reader Comments

Your name:*

Email:*
© 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)