Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 9786107, 9 pages

http://dx.doi.org/10.1155/2016/9786107

## Research on the Concentration Prediction of Nitrogen in Red Tide Based on an Optimal Grey Verhulst Model

^{1}The Key Laboratory of Intelligent Manufacturing and Robotics, School of Mechatronic Engineering and Automation, Shanghai University, Mailbox 232, No. 149 Yanchang Road, Shanghai 200072, China^{2}Department of Mechanical Engineering, College of Engineering, University of Michigan, Ann Arbor, MI 48105, USA

Received 21 March 2016; Revised 2 August 2016; Accepted 15 August 2016

Academic Editor: Rosana Rodriguez-Lopez

Copyright © 2016 Xiaomei Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In order to reduce the harm of red tide to marine ecological balance, marine fisheries, aquatic resources, and human health, an optimal Grey Verhulst model is proposed to predict the concentration of nitrogen in seawater, which is the key factor in red tide. The Grey Verhulst model is established according to the existing concentration data series of nitrogen in seawater, which is then optimized based on background value and time response formula to predict the future changes in the nitrogen concentration in seawater. Finally, the accuracy of the model is tested by the posterior test. The results show that the prediction value based on the optimal Grey Verhulst model is in good agreement with the measured nitrogen concentration in seawater, which proves the effectiveness of the optimal Grey Verhulst model in the forecast of red tide.

#### 1. Introduction

With the population expansion, land resources are becoming more and more precious, which leads to the shortage of material resources and the crisis of energy. The development of marine resources has become an effective way to relieve the pressure of resources and environment in the 21st century. With rapid development of marine resources, a variety of marine disasters follow as a result. In particular, the occurrence of red tide as well as the harm caused by it is frequently increasing [1]. Many researches have shown that the eutrophication of the seawater is the primary condition of the occurrence of red tide. The increase of nitrogen, phosphorus, and other nutrient salts in seawater greatly promotes the eutrophication of seawater [2]. Moreover, the nitrogen concentration in seawater is regarded as a key factor to predict the occurrence of red tide. Measured results have shown that the change of the nitrogen concentration in seawater is not monotonous.

Through the analysis of Grey system model and traditional Verhulst model, it is found that Grey system model is suitable to describe the monotonous change process, but it can be used in small sample data as well [3, 4]. In contrast, traditional Verhulst model is suitable for nonmonotonous data, but large samples are required [5]. In light of the characteristics of the change of the nitrogen concentration in seawater, Grey Verhulst model is applied to predict the nitrogen concentration in seawater. In Grey Verhulst model, an accumulation result of the original data is used to expand the scope of the application of the traditional Verhulst model [6, 7]. Therefore, Grey Verhulst model has been widely used in recent years [8–11].

In order to improve the accuracy of the prediction, an optimal Grey Verhulst model is proposed to predict the nitrogen concentration in seawater. The experimental results show its high precision and small error compared with other models [12], a testament to the effectiveness of the optimal Grey Verhulst model. So the optimal Grey Verhulst model can be applied to forecast red tide.

#### 2. Related Work

##### 2.1. Research on Red Tide Disaster

Red tide is an abnormal ecological phenomenon which is caused by fulminating proliferation or accumulation of plankton in seawater [13]. According to statistics, the frequency and the cumulative occurrence area of red tide are both increasing year by year.

Although the mechanism of the occurrence of red tide has not been determined yet, the main reasons that increase the frequency of red tide are widely recognized as follows [14]:(1)More and more eutrophic seawater(2)The increase of the utilization and the development of coastal water, such as the development of aquaculture, which leads to marine pollution(3)The increasing marine traffic, which is considered to expand the distribution of some harmful algae(4)Abnormal climate events, such as Nino and Southern Oscillation phenomenon(5)Decreasing efforts in the marine environmental protection and careless attitude towards the red tide

A large number of studies have shown that the occurrence of red tide is most strongly associated with seawater eutrophication [2]. Therefore, the research on the forecast of the concentration of nitrogen in seawater has great significance in the prediction of red tide disaster.

##### 2.2. Grey Verhulst Model

Grey system theory was established and developed by Professor Julong Deng at the beginning of 1980s, which has been successfully applied in industrial, agricultural, economic, and other fields, solving many practical problems in production and scientific research. Specifically, Grey system theory is mainly used in small sample monotonous data. Grey system theory can effectively deal with incomplete and uncertain information. The Grey model (GM) is the core of Grey system theory, which collects available data to obtain the internal regularity without using any assumptions. The forecasting accuracy is related to the sample number in GM. However, Gray model always needs to be combined with other methods to optimize the model, which can increase the accuracy of the prediction. For example, the combination of Grey model GM() with three-point moving average proposed by Professor Mao and Chirwa has been proven to be a more powerful forecasting tool and yields far much better predictions for vehicle fatality risk rates [15]. Its application to the UK and US data sets yields exact predictions that are of high repeatability with characteristics depicting high reliability and efficiency [16]. The paper is based on the Grey theory combined with the Verhulst model to predict nitrogen concentration which is the key factor of red tide. Traditional Verhulst model was put forward by Verhulst in the study of biological reproduction rules. The model is mainly used in large amount of data. Grey Verhulst model extends traditional Verhulst model so that it can be used in the unimodal type data.

In order to improve the accuracy of prediction, some researchers have optimized Grey Verhulst model. Evans proposed a Generalized Grey Verhulst model in which a new parameter estimation method was proposed on the basis of the relationship of background value and simulative function. The amount of British steel input was predicted by Generalized Grey Verhulst model to prove its effectiveness [17]. Chunguang et al. established an unbiased Grey Verhulst model according to the objective function which is the minimum value of the square of subtraction between reciprocal accumulating generating sequence and its inversely simulative value [16]. Wang et al. established a new Grey Verhulst model and its application is put forward [18]. Julong improved the simulative accuracy by using Fourier transform to correct simulation residual, and the trend of the euro against the dollar was predicted by this model to prove the good forecasting effect [19]. According to the analysis of the existing Grey Verhulst models, there are few researches on the model from the perspective of the initial value and the simulative value.

In order to predict the nitrogen concentration in seawater and avoid the error accumulation problem, a new method to optimize the time response function of Grey Verhulst model is proposed according to the criterion of minimum sum-square of difference between the raw data vector and the simulated data vector. The Logistic curve is used to fit the raw data, which optimizes background value and improves the prediction accuracy.

#### 3. The Optimization of the Grey Verhulst Model

##### 3.1. The Optimization of the Background Value

The Grey Verhulst model GM is constituted by a first-order differential equation containing only one variable [20–22]. Assuming that is a nonnegative raw data sequence and that is an accumulative sequence of , can be defined as follows [23–26]:

In (1), is the number of data in the sequence.

The generated mean sequence of is defined as

At this time, the power model of GM is defined as follows:

The whitening equation of GM is defined as follows [27–30]:

When , according to (4), is calculated as [31–33]According to (5), has S-type growth, which is shown in Figure 1 (the sequence of is shown in Figure 1).