grey lotka–volterra model and its application

Grey Lotka–Volterra model and its application

Lifeng Wu a,⁎, Sifeng Liu a, Yinao Wang b

a College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, Chinab School of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 June 2011Received in revised form 26 February 2012Accepted 20 April 2012Available online 30 June 2012

The aim of this study is to analyze the long-term relationship between the two variables and topredict the values of two variables in the social system or economic system. Based on the greymodeling method, a grey Lotka–Volterra model is proposed, and a linear programmingmethod is used to estimate the parameters of the grey Lotka–Volterra model under thecriterion of the minimization of mean absolute percentage error (MAPE). The obtainedsimulation results have been verified by three cases: the study of research and developmentinvestment (R&D) and gross domestic product (GDP), the study of fixed assets investment(FAI) and the (CPI), the study of energy consumption and the GDP. Comparisons of theobtained results with the traditional grey model demonstrate that the grey Lotka–Volterramodel is able to analyze the relationship between the two variables and predict the values ofthese variables effectively.

© 2012 Published by Elsevier Inc.

Keywords:Grey systemEnergy consumption and the gross domesticproductResearch and development investment andgross domestic productFixed assets investment and consumer priceindexForecasting

1. Introduction

The traditional GM(1,1) model proposed by Deng [1,2] has been widely applied in various fields, including wafer fabricationforecasting [3], opto-electronics industry output value forecasting [4], electricity costs forecasting [5], integrated circuit industry[6], product profit data [7] and vehicle fatality risk estimation [8]. However, GM(1,1), as a single variable forecasting model,cannot analyze the long-term relationship between the two variables and to predict the values of two variables in social system oreconomic system. Considering the complicated relationship that influences each other, it is necessary to analyze theserelationships. Therefore, in this work, the grey Lotka–Volterra model is used to analyze the relationship between two variables,and the discrete grey Lotka–Volterra model is used to forecast the values of two variables respectively.

The nonlinear least-squares method included in the software tool Eviews3.0 was applied to estimate the coefficients ofdiscrete Lotka–Volterra model [9–16], but the nonlinear least-squares method relies on the initial values and it may lead to largeerrors. Generally MAPE was employed to compare the performance of the forecasting model [1–8,17,18], but the objectivefunction is non-differentiable due to the presence of the absolute values when we estimate the parameters of the forecastingmodel under the criterion of the minimization of MAPE. So minimization of MAPE is computationally difficult.

So Zhou et al. adopted particle swarm optimization algorithm to estimate the parameters of the nonlinear grey Bernoullimodel under the criterion of the minimization of MAPE [19]. Wang and Hsu used a genetic algorithm to estimate the parametersof GM(1,1) model under the criterion of the minimization of MAPE [20,21]. Based on the grey direct modeling method, a linearprogramming method to estimate the parameters of the Lotka–Volterra model under the criterion of the minimization of MAPE isproposed [22]. In this paper, based on the grey modeling method, we present a linear programming method in order to estimateparameters of the grey Lotka–Volterra model under the criterion of the minimization of MAPE.

Technological Forecasting & Social Change 79 (2012) 1720–1730

⁎ Corresponding author.E-mail address: [email protected] (L. Wu).

0040-1625/$ – see front matter © 2012 Published by Elsevier Inc.doi:10.1016/j.techfore.2012.04.020

Contents lists available at SciVerse ScienceDirect

Technological Forecasting & Social Change

http://dx.doi.org/10.1016/j.techfore.2012.04.020

mailto:[email protected]

http://dx.doi.org/10.1016/j.techfore.2012.04.020

http://www.sciencedirect.com/science/journal/00401625

The rest of the paper proceeds as follows. The next section presents an overview of the relevant literature on the grey modeland the Lotka–Volterra model. The third section provides the grey Lotka–Volterra model. The fourth section proposes numericalexperiments. The final section discusses the conclusion.

2. Literature review

2.1. Original GM(1,1) [1,2,4]

Assume the original series to be

X 0ð Þ ¼ x 0ð Þ 1ð Þ; x 0ð Þ 2ð Þ;…; x 0ð Þ nð Þn o

;

and then the first-order accumulated generating operation(1-AGO) of X(0) is given by

x 1ð Þ kð Þ ¼Xki¼1

x 0ð Þ ið Þ; k ¼ 1;2;…;n


:

The equation

x 0ð Þ kð Þ þ az 1ð Þ kð Þ ¼ b;

is called a GM(1,1) model, where z 1ð Þ kð Þ ¼ x 1ð Þ kð Þþx 1ð Þ kþ1ð Þ2 ; k ¼ 1;2;…;n−1. Use the ordinary least square method to estimate the

parameters

ab

� �¼ BTB

� �−1BTY

where

Y ¼x 0ð Þ 2ð Þx 0ð Þ 3ð Þ

⋮x 0ð Þ nð Þ

0BB@

1CCA;B ¼

−z 1ð Þ 2ð Þ 1−z 1ð Þ 3ð Þ 1

⋮ ⋮−z 1ð Þ nð Þ 1

0BB@

1CCA:

Solve the whitenization equation dx 1ð Þdt þ ax 1ð Þ ¼ b of the GM(1,1) model to obtain

x 1ð Þ kþ 1ð Þ ¼ x 0ð Þ 1ð Þ− ba

� �e−ak þ b

a

The inverse accumulated generating operation (IAGO) is

x 0ð Þ kþ 1ð Þ ¼ x 1ð Þ kþ 1ð Þ−x 1ð Þ kð Þ ¼ x 0ð Þ 1ð Þ− ba

� �e−ak 1−ea

� ; k ¼ 1;2;…;n−1:

2.2. Grey Verhulst model [2]

The equation

x 0ð Þ kð Þ þ az 1ð Þ kð Þ ¼ b z 1ð Þ kð Þh i2

is called the grey Verhulst model, where

x 1ð Þ kð Þ ¼Xki¼1

x 0ð Þ ið Þ; k ¼ 1;2;…;n;


;

1721L. Wu et al. / Technological Forecasting & Social Change 79 (2012) 1720–1730

z 1ð Þ kð Þ ¼ x 1ð Þ kð Þ þ x 1ð Þ kþ 1ð Þ2

; k ¼ 1;2;…;n−1:

The least squares estimate of the parameters sequence a and b of the grey Verhulst model is given by

ab

� �¼ BTB

� �−1BTY

where


⋮x 0ð Þ nð Þ

0BB@

1CCA; B ¼

−z 1ð Þ 2ð Þ z 1ð Þ 2ð Þh i2

−z 1ð Þ 3ð Þ z 1ð Þ 3ð Þh i2

⋮ ⋮−z 1ð Þ nð Þ z 1ð Þ nð Þ

h i2

0BBBBB@

1CCCCCA:

Solve the whitenization equation dx 1ð Þdt þ ax 1ð Þ ¼ b x 1ð Þ �2 of the grey Verhulst model to obtain

x 1ð Þ tð Þ ¼ ax 1ð Þ 0ð Þbx 1ð Þ 0ð Þ þ a−bx 1ð Þ 0ð Þ �

eat:

The time response sequence of the grey Verhulst model is given by

x 1ð Þ kþ 1ð Þ ¼ ax 1ð Þ 0ð Þbx 1ð Þ 0ð Þ þ a−bx 1ð Þ 0ð Þ �

eak; k ¼ 1;2;…;n−1:

The IAGO is

x 0ð Þ kþ 1ð Þ ¼ x 1ð Þ kþ 1ð Þ−x 1ð Þ kð Þ:

The Verhulst model is mainly used to study sigmoid processes, for instance, it is often used in the prediction of biologicalgrowth, economic life span of consumable products, etc.

2.3. The Lotka–Volterra model [9–16,23]

The Lotka–Volterra model has been developed to model the interaction between the two competing species based on thelogistic curve and to extended the analysis of technology diffusion in competitive or collaborative markets covered in relevantliterature [9–14,23]. The Lotka–Volterra model of two species, X and Y, is as follows

dxdt

¼ a1x tð Þ−b1x tð Þ2−c1x tð Þy tð Þ; ð1Þ

dydt

¼ a2y tð Þ−b2y tð Þ2−c2y tð Þx tð Þ: ð2Þ

where ai is the parameter of geometric growth for the species i when it is living alone, bi is the limitation parameter of nichecapacity for the species i, and ci is the interaction parameter with the other species, i=1,2. The multi-mode form could berevealed for the case of two species in Table 1.

Eqs. (1) and (2) could be transformed into the discrete Lotka–Volterra model in order to use discrete time data:

x kþ 1ð Þ ¼ α1x kð Þ1þ β1x kð Þ þ γ1y kð Þ ; k ¼ 1;2;…;n−1 ð3Þ

Table 1Multi-mode competitive relationship according to the sign of c1 and c2.

c1 c2 Type Explanation

+ + Pure competition Occurs when both species suffer from each other's existence+ − Predator–prey Occurs when one of them serves as direct food to the other− − Mutualism Occurs in the case of symbiosis or a win–win situation

1722 L. Wu et al. / Technological Forecasting & Social Change 79 (2012) 1720–1730

y kþ 1ð Þ ¼ α2y kð Þ1þ β2y kð Þ þ γ2x kð Þ ; k ¼ 1;2;…;n−1 ð4Þ

where αi and βi denote the logistic parameters for the species iwhen it is living alone, γi is the magnitude of the effect which eachspecies has on the rate of increase of the other, where

ai ¼ lnαi; bi ¼βiln αi

αi−1; ci ¼

γiln αi

αi−1; i ¼ 1;2: ð5Þ

3. The grey Lotka–Volterra model

Assume that


;Y 0ð Þ ¼ y 0ð Þ 1ð Þ; y 0ð Þ 2ð Þ;…; y 0ð Þ nð Þn o

;

are positive sequences of original data for the state variables of a system, where


;

y 1ð Þ kð Þ ¼Xki¼1

y 0ð Þ ið Þ; k ¼ 1;2;…;n;

Y 1ð Þ ¼ y 1ð Þ 1ð Þ; y 1ð Þ 2ð Þ;…; y 1ð Þ nð Þn o

:

We view x(1)(k) and y(1)(k) as research variables in order to explore the long-term relationship between X(0) and Y(0), becausethe mutual effects between X(0) and Y(0) have some lag period and lag length is difficult to ascertain. Actually, the data beforex(0)(k) often exert an influence on x(0)(k) and y(0)(k), that is to say the accumulation of x(0)(k) will exert an influence on x(0)(k)and y(0)(k). According the grey modeling method, at time k, the grey derivative of the X(1) is

dx kð Þ ¼ x 1ð Þ kþ 1ð Þ−x 1ð Þ kð Þ; k ¼ 1;2;…;n−1:

The mean generated sequence of consecutive neighbors of X(1) is

z 1ð Þx kð Þ ¼ x 1ð Þ kð Þ þ x 1ð Þ kþ 1ð Þ

2;

and the mean generated sequence of consecutive neighbors of Y(1) is

z 1ð Þy kð Þ ¼ y 1ð Þ kð Þ þ y 1ð Þ kþ 1ð Þ

2;

where k=1,2,…,n−1. The equation

x 0ð Þ kþ 1ð Þ≈a1z1ð Þx kð Þ−b1 z 1ð Þ

x kð Þ� �2−c1z

1ð Þx kð Þz 1ð Þ

y kð Þ

is called the grey Lotka–Volterra model. Given the error sequence

εk ¼ x 0ð Þ kþ 1ð Þ−a1z1ð Þx kð Þ þ b1 z 1ð Þ

x kð Þ� �2 þ c1z


y kð Þ

the least squares estimate of the parameter sequence a1, b1 and c1 of the grey Lotka–Volterra model is given by

a1

b1c1

0@

1A ¼ BTB

� �−1BTY


where


⋮x 0ð Þ nð Þ

0BB@

1CCA;B ¼

z 1ð Þx 2ð Þ − z 1ð Þ

x 2ð Þh i2 −z 1ð Þ

x 2ð Þz 1ð Þy 2ð Þ

z 1ð Þx 3ð Þ − z 1ð Þ

x 3ð Þh i2 −z 1ð Þ

x 3ð Þz 1ð Þy 3ð Þ

⋮ ⋮z 1ð Þx nð Þ − z 1ð Þ

x nð Þh i2 −z 1ð Þ

x nð Þz 1ð Þy nð Þ

0BBBBB@

1CCCCCA:

To estimate the unknown parameters a1, b1 and c1 under the criterion of the minimization of the MAPE, we set the objectivefunction as:

Minimize :1

n−1

Xn−1

k¼1

x 0ð Þ kþ 1ð Þ−a1z1ð Þx kð Þ þ b1 z 1ð Þ

x kð Þ� �2 þ c1z


y kð Þ��

x 1ð Þ kþ 1ð Þ :

The above objective function can be solved by the following linear programming model

Minimize :Xn−1

k¼1

eþi þ e−i

s.t.

a1z1ð Þx kð Þ−b1 z 1ð Þ

x kð Þ� �2−c1z


y kð Þ þ eþi −e−i �

x 1ð Þ kþ 1ð Þ ¼ x 0ð Þ kþ 1ð Þeþi ≥0e−i ≥0; k ¼ 1;2;…;n−1 ð6Þ

where

eþi ¼ εkj j þ εk2x 1ð Þ kþ 1ð Þ ; e

−i ¼ εkj j−εk

2x 1ð Þ kþ 1ð Þ :

We can get the estimated values a1; b1, and c1 of a1, b1 and c1 by solving problem (6). Then substituting a1; b1; and c1 intoEq. (5), we obtain the estimated values α1; β1; and γ1 of α1, β1 and γ1. Lastly, substituting α1; β1; and γ1 into Eq. (3), we canobtain the discrete grey Lotka–Volterra model

x 1ð Þ kþ 1ð Þ ¼ α1x1ð Þ kð Þ

1þ β1x1ð Þ kð Þ þ γ1y

1ð Þ kð Þ; k ¼ 1;2;…;n−1:

The IAGO is

x 0ð Þ kþ 1ð Þ ¼ x 1ð Þ kþ 1ð Þ−x 1ð Þ kð Þ:

Similarly, by using the same method, we can obtain

y 1ð Þ kþ 1ð Þ ¼ α2y1ð Þ kð Þ

1þ β2y1ð Þ kð Þ þ γ2x

1ð Þ kð Þ; k ¼ 1;2;…;n−1:

The IAGO is

y 0ð Þ kþ 1ð Þ ¼ y 1ð Þ kþ 1ð Þ−y 1ð Þ kð Þ:

We can judge the relationship between X(1) and Y(1) from the signs of c1 and c2.

4. Numerical experiment

Case 1. The research and development (R&D) investment and the gross domestic product (GDP) every year forecasting example.

While there is a growing literature that examines the relationship between the R&D invest and real GDP [24–28]. In this paper,we assume that the relationship between the R&D and the GDP every year in Zhejiang province in China satisfies the conditions ofthe grey Lotka–Volterra model. The sample data is from Ref. [27]. The accumulative total of R&D is designated as X(1) (a hundredmillion RMB), and the GDP every year as Y(0) (a hundred million RMB).


By the above linear programming method, we use the data from 1990 to 2004 to construct the grey Lotka–Volterramodel:

dx 1ð Þ

dt¼ 0:4744158x 1ð Þ tð Þ þ 0:001330321x 1ð Þ tð Þ2 þ 0:00001024117x 1ð Þ tð Þy 1ð Þ tð Þ ð7Þ

dy 1ð Þ

dt¼ 0:4865827y 1ð Þ tð Þ−0:00001884847y 1ð Þ tð Þ2−0:002627603y 1ð Þ tð Þx 1ð Þ tð Þ ð8Þ

In model (7) and model (8), c1b0,c2>0, it follows that, R&D investment and GDP are in the predator–prey system in the longrun, and R&D investment has a negative effect on GDP in the long-run. These results conform to the reality of China, and can beinterpreted as a sign of failure of the national innovation system. So this situation deserves government's reflection in the makingpolicies.

The discrete grey Lotka–Volterra model:

x 1ð Þ kþ 1ð Þ ¼ 1:60707507x 1ð Þ kð Þ1−0:001702314x 1ð Þ kð Þ þ 0:0000131049y 1ð Þ kð Þ ; k ¼ 1;2;…;n−1 ð9Þ

y 1ð Þ kþ 1ð Þ ¼ 1:626747626y 1ð Þ kð Þ1þ 0:000024278y 1ð Þ kð Þ−0:00338451x 1ð Þ kð Þ ; k ¼ 1;2;…;n−1: ð10Þ

Use model (9) and model (10) to predict respectively.For the accumulative total of R&D, we obtain the GM(1,1) model. Actual values and the forecasting values of the two compared

models are presented in Table 2. Model (9) reduces the MAPE of GM(1,1) model from 35.65% to 5.30%, which means that the greyLotka–Volterra model reaches the objective of minimizing forecast error. We can conclude that the grey Lotka–Volterra modelsignificantly enhances the precision of the grey forecasting model.

For the GDP every year, we obtain the GM(1,1) model. Actual values and the forecasting values of the two compared modelsare presented in Table 3. Grey Lotka–Volterra model reduces the MAPE of GM(1,1) model from 11.7% to 9.1%.

Case 2. The fixed assets investment (FAI) and the consumer price index (CPI) forecasting example

The FAI and the CPI are both very difficult to predict. So genetic programming is used to predict the CPI inflation in NewZealand [29]. A vector error-correction forecasting model is applied to predict the CPI of the US economy [30]. Black et al.predicted the CPI and used MAPE as a model evaluation criterion [31]. In this paper, we assume the relationship between the FAIand the CPI every year in China satisfying the conditions of the grey Lotka–Volterra model. The sample data is from Ref. [32] whichdiscussed the relationship between the FAI and the CPI. Due to the different measurement units of FAI and the CPI, the growth rateof FAI over preceding year is designated as X(0) (%), and that the growth rate of CPI over preceding year as Y(0) (%).

Table 2Forecasted values and MAPE.

Year Actual value (X(1)) Model (9) value GM(1,1) value

1990 2.041991 4.31 3.25 14.01992 7.77 6.80 18.61993 12.2 12.12 24.81994 20.08 18.71 33.01995 29.22 30.17 43.91996 39.72 42.68 58.51997 54.91 56.16 77.91998 74.61 75.32 103.81999 101.66 99.63 138.22000 138.26 133.07 184.12001 183 178.54 245.22002 240.65 234.003 326.62003 318.41 306.74 435.12004 413.41 408.85 579.5MAPE 5.30 35.65

MAPE ¼ 1n

Pnk¼1

Ak−P k

� �=Ak

�� 100% , Ak is the actual value, and P k is the fitted value.


By the above linear programming method, we use the data from 1995 to 2010 to construct the grey Lotka–Volterramodel:

dx 1ð Þ

dt¼ 0:153159x 1ð Þ tð Þ−0:0009786597x 1ð Þ tð Þ2 þ 0:004695725x 1ð Þ tð Þy 1ð Þ tð Þ ð11Þ

dy 1ð Þ

dt¼ 0:1148065y 1ð Þ tð Þ−0:005600275y 1ð Þ tð Þ2 þ 0:000727117y 1ð Þ tð Þx 1ð Þ tð Þ: ð12Þ

In model (11) and model (12), c1b0,c2b0 means that, the FAI and the CPI are in the mutualism system in the long run. The FAIin the future will increase in line with inflation as measured by the CPI, meanwhile, in spite of the slight shock, the FAI kept its fastgrowth and served as a major driver of China's economy. The FAI has become a major factor behind a higher CPI in China, and theCPI is a major gauge to measure inflation. So China's government needs to balance the growth rate of FAI and the growth rate ofCPI in making policies of economic development.


x 1ð Þ kþ 1ð Þ ¼ 1:16551028x 1ð Þ kð Þ1þ 0:001057582x 1ð Þ kð Þ−0:005074405y 1ð Þ kð Þ ; k ¼ 1;2;…;n−1 ð13Þ


Use model (13) and model (14) to predict respectively.For the growth rate of FAI, we obtain the grey Verhulst model as follows:

x 1ð Þ kþ 1ð Þ ¼ −1:472694−0:0467−0:0376eak

; k ¼ 1;2;…;n−1:

Actual values and the forecasting values of the two compared models are presented in Table 4. The grey Lotka–Volterra modelreduces the MAPE of the grey Verhulst model from 24.7623% to 12.6714%.

For the growth rate of CPI every year, we obtain the grey Verhulst model. Actual values and the forecasting values of the twocompared models are presented in Table 5. The grey Lotka–Volterra model reduces the MAPE of the grey Verhulst model from168.2882% to 38.6608%. The two models both yielded higher MAPE, mostly because the sample values are small. The grey Lotka–Volterra model yielded the higher MAPE, which means that the forecasting of the CPI is not fit to use the grey Lotka–Volterramodel. There are other factors except FAI, which affect the increase of CPI.

Case 3. Energy consumption and the gross domestic product (GDP) every year forecasting example

While there is a growing literature that examines the relationship between energy consumption and real GDP [33–38], thebulk of this literature assumes that the relationship between energy consumption and real GDP is positive in testing for causalitywithin a cointegrated and Granger causal framework [33,35–37], this assumption is only partially true [38]. So in this paper, weassume that the relationship between the energy consumption and the GDP every year in China satisfies the conditions of the


Year Actual value Y(0) Grey Lotka–Volterra model value GM(1,1) value

1990 898.01991 1081.8 541.4 1777.21992 1365.1 1676.9 2047.61993 1909.5 2041.8 2359.21994 2666.9 2710.6 2718.21995 3524.8 3592.1 3131.91996 4146.1 4332.3 3608.41997 4638.2 4594.6 4157.51998 4987.5 4824.6 4790.21999 5364.9 4958.1 5519.12000 6036.3 5406.0 6359.02001 6748.2 6342.2 7326.62002 7796.0 7295.0 8441.52003 9395.0 9084.7 9726.02004 11243.0 12443.8 11206.1MAPE 9.1 11.7

MAPE ¼ 1n

Pnk¼1

Ak−P k

� �=Ak



grey Lotka–Volterra model. The sample data is from the China Statistical Year Book (2007), and due to the different measurementunits of GDP and energy consumption, the growth rate of energy consumption over the preceding year is designated as X(0) (%),and the growth rate of GDP over the preceding year as Y(0) (%).

By the above linear programming method, we use the data from 1999 to 2006 to construct the grey Lotka–Volterramodel

dx 1ð Þ

dt¼ 0:4130408x 1ð Þ tð Þ−0:01706505x 1ð Þ tð Þ2 þ 0:01128663x 1ð Þ tð Þy 1ð Þ tð Þ ð15Þ

dy 1ð Þ

dt¼ 0:6542780y 1ð Þ tð Þ−0:01428957y 1ð Þ tð Þ2 þ 0:007119915y 1ð Þ tð Þx 1ð Þ tð Þ: ð16Þ

In model (15) andmodel (16), c1b0,c2b0, it follows that, energy consumption and GDP are in the win–win situation system inthe long run, energy consumption has a significant positive effect on GDP in the long-run and GDP has a significant positive effecton energy consumption in the long-run, these results conform to the reality of China, and it implies that China is energy


Year Actual value X(0) Grey Lotka–Volterra model value Grey Verhulst model value

1995 17.469681996 14.45705 19.2691 18.12271997 8.848932 11.6807 18.76761998 13.89313 16.5463 19.40221999 5.099239 5.4512 20.02462000 10.25969 12.0179 20.63302001 13.05012 15.0684 21.22582002 16.89279 17.9527 21.80162003 27.7396 29.6778 22.35902004 26.83411 29.8685 22.89722005 25.96038 26.3789 23.41532006 23.90868 23.2950 23.91272007 24.8 27.3290 24.38882008 25.9 29.6701 24.84352009 30.0 23.1443 25.27672010 23.8 23.2051 25.6885MAPE 12.6714 24.76232011 22.8424 26.4485

MAPE ¼ 1n

Pnk¼1

Ak−P k

� �=Ak



Year Actual value Y(0) Grey Lotka–Volterra model value Grey Verhulst value

1995 17.11996 8.3 7.6703 8.75731997 2.8 2.5466 5.74881998 −0.8 −0.4168 4.20081999 −1.4 −1.1148 3.25942000 0.4 0.5389 2.62772001 0.7 0.8603 2.17552002 −0.8 −0.3754 1.83642003 1.2 1.6300 1.57342004 3.9 4.0318 1.36392005 1.8 2.1991 1.19342006 1.5 1.9244 1.05232007 4.8 4.7607 0.93392008 5.9 5.5805 0.83332009 −0.7 0.4870 0.74702010 3.3 3.4788 0.6723MAPE 38.6608 168.28822011 3.5575 0.5500

MAPE ¼ 1n

Pnk¼1

Ak−P k

� �=Ak



dependent, thus energy conservation policies maybe counter-productive, so the win–win situation is an important factor thatneeds to be considered in the making of government energy conservation policies.


x 1ð Þ kþ 1ð Þ ¼ 1:51140669x 1ð Þ kð Þ1þ 0:0211x 1ð Þ kð Þ−0:013974547y 1ð Þ kð Þ ; k ¼ 1;2;…;n−1 ð17Þ


Use model (17) and model (18) to predict respectively.For the growth rate of energy consumption over the preceding year, we obtain the grey Verhulst model, actual values and the

forecasting values of the two compared models are presented in Table 6. The grey Lotka–Volterra model reduces the MAPE of thegrey Verhulst model from 33.6% to 29.0%.

For the growth rate of GDP over the preceding year, we obtain the GM(1,1) model, actual values and the forecasting values ofthe two comparedmodels are presented in Table 7. TheMAPE of the grey Lotka–Volterra model is bigger and the forecasting valueis unsatisfactory, because there are other factors except energy consumption, which affect the increase of GDP, the Chinesegovernment must make full use of these factors to increase the GDP with lower growth of requirements for energy. Theforecasting of GDP is more fit to use the single variable forecasting model.

5. Conclusion

In order to analyze the relationship between the economic variables in social system or economic system and improve theprediction performance, the grey Lotka–Volterra model and a new approach to estimate the parameters of the grey Lotka–Volterra model are presented. Our results reveal that this approach is effective and feasible. It is very important to study the

Table 6The simulative values and MAPE of the grey Verhulst model and the grey Lotka–Volterra model.

Year X(0) Grey Lotka–Volterra model value Verhulst value

1999 1.2 1.2 1.22000 3.5 0.77 2.282001 3.4 6.14 4.032002 6.0 6.59 6.392003 15.3 10.9 8.892004 16.1 18.2 10.972005 10.6 12.91 12.352006 9.6 8.9 13.15MAPE 29.0 33.62007 7.8 8.49 13.79

MAPE ¼ 1n

Pnk¼1

Ak−P k

� �=Ak

�� 100% , Ak is the actual value, and P k is the predicted value.

Table 7The simulative values and MAPE of the GM(1,1) model and the grey Lotka–Volterra model value.

Year Y(0) Grey Lotka–Volterra model value GM(1,1) model value

1999 7.6 7.6 7.62000 8.4 5.21 8.272001 8.3 11.32 8.682002 9.1 9.05 9.122003 10 8.76 9.582004 10.1 10.90 10.062005 10.4 10.61 10.572006 11.1 7.80 11.10MAPE 15.84 1.812007 11.4 6.62 11.66

MAPE ¼ 1n

Pnk¼1

Ak−P k

� �=Ak

�� 100% , Akis the actual value, and P k is the predicted value.


relationship between the economic variables, which is helpful to the government in making policies of social development. Futurestudies could investigate the relationship between other economic variables in other countries or areas in order to confirm thegenerality of the grey Lotka–Volterra model.

Acknowledgments

The authors are grateful to the editors for their valuable comments. Thanks to the anonymous reviewers for their veryhelpful comments. This work was supported by Funding of Jiangsu Innovation Program for Graduate Education(No.CXLX12_0176), Funding for Outstanding Doctoral Dissertation in NUAA (No.BCXJ12-13) and the Fundamental ResearchFunds for the Central Universities. At the same time, the authors would like to acknowledge the support of the NationalNatural Science Foundation of China (No.90924022, 70971064 and 70901041),the Science Foundation for the Excellent andCreative group in Science and Technology of Jiangsu Province (No.Y0553-091), and the Foundation for outstanding teachinggroup of China (No.10td128).

References

[1] J.L. Deng, Introduction Grey system theory, J. Grey Syst. 1 (1) (1989) 191–243.[2] S.F. Liu, Y. Lin, Grey Systems: Theory and Practical Applications, Springer-Verlag London Ltd., London, 2010.[3] Yu-Shan Chen, Bi-Yu Chen, Applying DEA, MPI, and grey model to explore the operation performance of the Taiwanese wafer fabrication industry, Technol.

Forecast. Soc. Change 78 (3) (2011) 536–546.[4] C.T. Lin, S.Y. Yang, Forecast of the output value of Taiwan's opto-electronics industry using the Grey forecasting model, Technol. Forecast. Soc. Change 70 (2)

(2003) 177–186.[5] Shun-Chung Lee, Li-Hsing Shih, Forecasting of electricity costs based on an enhanced gray-based learning model: a case study of renewable energy in

Taiwan, Technol. Forecast. Soc. Change 78 (7) (2011) 1242–1253.[6] Li-Chang Hsu, Applying the Grey prediction model to the global integrated circuit industry, Technol. Forecast. Soc. Change 70 (6) (2003) 563–574.[7] Yuan-Yeuan Tai, Jenn-Yang Lin, Ming-Shi Chen, Ming-Chyuan Lin, A grey decision and prediction model for investment in the core competitiveness of

product development, Technol. Forecast. Soc. Change 78 (7) (2011) 1254–1267.[8] Mingzhi Mao, E.C. Chirwa, Application of grey model GM(1,1) to vehicle fatality risk estimation, Technol. Forecast. Soc. Change 73 (5) (2006) 588–605.[9] Bi-Huei Tsai, Yiming Li, Cluster evolution of IC industry from Taiwan to China, Technol. Forecast. Soc. Change 76 (8) (2009) 1092–1104.

[10] S.Y. Chiang, G.G. Wong, Competitive diffusion of personal computer shipments in Taiwan, Technol. Forecast. Soc. Change 78 (3) (2011) 526–535.[11] S. Lee, D. Lee, H. Oh, Technological forecasting at the Korean stock market: a dynamic competition analysis using Lotka–Volterra model, Technol. Forecast.

Soc. Change 72 (8) (2005) 1044–1057.[12] J. Kim, D.J. Lee, J. Ahn, A dynamic competition analysis on the Korean mobile phone market using competitive diffusion model, Comput. Ind. Eng. 51 (1)

(2006) 174–182.[13] S.A. Morris, D. Pratt, Analysis of the Lotka–Volterra competition equations as a technological substitution model, Technol. Forecast. Soc. Change 70 (2)

(2003) 103–133.[14] S.C. Bhargava, Generalized Lotka–Volterra equations and the mechanism of technological substitution, Technol. Forecast. Soc. Change 35 (4) (1989)

319–326.[15] S.Y. Chiang, An application of Lotka-Volterra model to Taiwan's transition from 200 mm to 300 mm silicon wafers, Technol. Forecast. Soc. Change 79 (2)

(2012) 383–392.[16] V.B. Kreng, Hsi Tse Wang, The competition and equilibrium analysis of LCD TV and PDP TV, Technol. Forecast. Soc. Change 78 (3) (2011) 448–457.[17] F.M. Tseng, H.C. Yub, G.H. Tzeng, Combining neural network model with seasonal time series ARIMA model, Technol. Forecast. Soc. Change 69 (1) (2002)

71–87.[18] F.M. Tseng, H.C. Yu, G.H. Tzeng, Applied hybrid grey model to forecast seasonal time series, Technol. Forecast. Soc. Change 67 (7) (2001) 291–302.[19] J.Z. Zhou, R.C. Fang, Y.H. Li, Y.C. Zhang, B. Peng, Parameter optimization of nonlinear grey Bernoulli model using particle swarm optimization, Appl. Math.

Comput. 207 (2) (2009) 292–299 (15).[20] L.C. Hsu, Forecasting the output of integrated circuit industry using genetic algorithm based multivariable grey optimization models, Expert Syst. Appl. 36

(4) (2009) 7898–7903.[21] C.H. Wang, L.C. Hsu, Using genetic algorithms grey theory to forecast high technology industrial output, Appl. Math. Comput. 195 (2008) 256–263.[22] Wu. Lifeng, Yinao Wang, Estimation the parameters of Lotka–Volterra model based on grey direct modelling method and its application, Expert Syst. Appl.

38 (6) (2011) 6412–6416.[23] Theodore Modis, US Nobel laureates: logistic growth versus Volterra–Lotka, Technol. Forecast. Soc. Change 78 (4) (2011) 559–564.[24] Xia Gao, Xiaochuan Guo, Katz J. Sylvan, Jiancheng Guan, The Chinese innovation system during economic transition: a scale-independent view, J. Informetr. 4

(4) (2010) 618–628.[25] Mario Coccia, What is the optimal rate of R&D investment to maximize productivity growth? Technol. Forecast. Soc. Change 76 (3) (2009) 433–446.[26] Olof Ejermo, Astrid Kander, Martin Svensson Henning, The R&D-growth paradox arises in fast-growing sectors, Res. Policy 40 (5) (2011) 664–672.[27] Cheng Hua, Wu Xiaohui, R&D investment, R&D stock and elasticity production of R&D stock-Empirical study based on age effectiveness function, Stud. Sci.

Sci. 24 (8) (2006) 108–113.[28] Alessandro Sterlacchini, R&D, higher education and regional growth: Uneven linkages among European regions, Res. Policy 37 (6–7) (2008) 1096–1107.[29] H.Y. Xie, M.J. Zhang, P. Andreae, Genetic programming for New Zealand CPI inflation prediction, 2007 IEEE Congress on Evolutionary Computation, 2007,

pp. 2538–2545.[30] Richard G. Anderson, Dennis L. Hoffman, Robert H. Rasche, A vector error-correction forecasting model of the US economy, J. Macroecon. 24 (4) (2002)

569–598.[31] David C. Black, Paul R. Corrigan, Michael R. Dowd, New dogs and old tricks: do money and interest rates still provide information content for forecasts of

output and prices? Int. J. Forecast. 16 (2) (2000) 191–205.[32] ZhuZhaolin , LiWanfeng , The analysis of the relationship between fixed assets investment and CPI in China, Price Theory Pract. 28 (3) (2008) 62–63.[33] Ramakrishnan Ramanathan, A multi-factor efficiency perspective to the relationships among world GDP, energy consumption and carbon dioxide emissions,

Technol. Forecast. Soc. Chang. 73 (2006) 483–494.[34] Li Feia, Suocheng Dong, Li Xue, Quanxi Liang, Wangzhou Yang, Energy consumption–economic growth relationship and carbon dioxide emissions in China,

Energy Policy 39 (2011) 568–574.[35] Ugur Soytas, Ramazan Sari, Energy consumption and GDP: causality relationship in G-7 countries and emerging markets, Energy Econ. 25 (2003) 33–37.[36] A. Talha Yalta, Analyzing energy consumption and GDP nexus using maximum entropy bootstrap: the case of Turkey, Energy Econ. 33 (2011) 453–460.[37] Bernard C. Beaudreau, On the methodology of energy-GDP Granger causality tests, Energy 35 (9) (2010) 3535–3539.[38] Paresh Kumar Narayan, Seema Narayan, Stephan Popp, A note on the long-run elasticities from the energy consumption-GDP relationship, Appl. Energy 87

(2010) 1054–1057.


Lifeng Wu is a Ph.D. candidate in the College of Economics and Management, Nanjing University of Aeronautics and Astronautics, his research interests focus onthe grey system modeling and operation research. He has authored 9 papers which appeared in journals and conferences.

Sifeng Liu is a professor and tutor of Ph.D of the College of Economics andManagement, Nanjing University of Aeronautics and Astronautics. He is the chairman ofthe IEEE Grey System Society and vice chairman of IEEE SMC China (Beijing) Branch. His main research direction is grey system theory.

Yinao Wang is a professor of the College of Mathematics and information Science at Wenzhou University, His research interests focus on the grey modeling andits application, he has published more than 30 papers in journals and conferences.


grey lotka–volterra model and its application

Documents