A Generalization of the Grey Lotka–Volterra Model and Application to GDP, Export, Import and Investment for the European Union

Abstract

This study proposes a generalized grey Lotka–Volterra model with a finite number of variables. The model is obtained by applying the grey modelling method to estimate the parameters of a finite dimensional quadratic Lotka–Volterra system. Subsequently, the model is used to analyze the competition and cooperation relationship between four macroeconomic indicators, namely Gross Domestic Product, Export, Import and Investment, and to obtain short-time forecasting for them. The data used in the empirical investigation cover the time periods 2005–2022 and 2011–2022, for the European Union. The empirical results are compared to the ones obtained by using the grey model $GM\left(\mathrm{1,1}\right)$ and the two-dimensional grey Lotka–Volterra model. Finally, economic interpretations of the empirical findings are formulated.

1. Introduction

Grey prediction models have been widely and successfully applied in various fields (industry, science and technology, economy and others). With more and more in-depth research in the field of economics, economic research approaches the grey theory by identifying the important role it can play in the economy. Grey prediction models have proven to be efficient and providing appropriate methodologies to handle economic problems that are very complex, due to their ability to adapt to model uncertainty and sometimes insufficient information for prediction and decision making [1,2,3,4,5,6,7,8,9].

In recent years, extended and modified grey prediction models have been developed based on the grey model $GM\left(\mathrm{1,1}\right),$ motivated by the practicality and accuracy of prediction.

However, the grey model $GM\left(\mathrm{1,1}\right)$ focuses mainly on linear relations and seasonal characteristics, ignoring nonlinear effects, such as competition and cooperation. Thus, due to the complexity of the economic system, research has permanently improved the basic grey models by adding multivariate prediction problems to them. The Lotka–Volterra model is viable in cyclical relationships. Since the economic system is characterized by a series of cycles, this model finds applicability in the economy. Lotka–Volterra equations can be useful in analyzing the long-term relationship between economic variables and predicting their values. Based on the grey modeling method, a grey Lotka–Volterra model was developed starting from the two-dimensional Lotka–Volterra Predator–Prey model [10,11,12,13].

This study has two main objectives. Firstly, it aims to obtain a generalized grey Lotka–Volterra model for a finite number of variables. The new model is derived by applying the grey direct modeling method in order to obtain a discrete model and to estimate the parameters of a finite dimensional quadratic Lotka–Volterra model. The coefficients of the model are estimated considering the criterion of minimizing the sum of squared errors. The accuracy of the estimations is established by using the mean absolute percentage error value.

The second objective of the study is to test the applicability and reliability of the theoretical model on a system of four macroeconomic indicators: Gross Domestic Product, Investment, Import and Export.

The impact of gross domestic product (GDP) elements (calculated by the expenditure method) on the economic growth of a country or group of countries has been studied by numerous authors. Different results were identified depending on the interval studied, the level of development of the states and the methods used. Investment and international trade are essential elements for the stable operation and long-term growth of the national economy. At the same time, due to the important role it plays in the process of economic development and social progress, many researchers have created different forecasting models to provide effective support to governments in the formulation of economic policies.

In general, studies have focused on analyzing the relationship between these variables and prediction on statistical methods (Linear regression model, ARIMA) and artificial intelligence, and less on grey prediction methods [14,15,16,17,18,19,20,21,22,23,24,25].

The generalized $s$-dimensional grey Lotka–Volterra model ($GLV\left(s\right)$) derived in this study is used to estimate a system of four economic variables (GDP, Investment, Import and Export), thus taking into account nonlinear effects and the mechanism of competition and cooperation between them. The purpose of the case study is to analyze the evolution and relationship between the economic variables using a $GLV\left(4\right)$ model. The empirical results are compared with the ones obtained using the $GM\left(\mathrm{1,1}\right)$ and $GLV\left(2\right)$ for pairs of two variables, aiming to verify in this context the validity of the new theoretically presented model.

The paper’s novelty consists in the development of a new model, obtained by generalizing the two-dimensional grey Lotka–Volterra model, and in analyzing the competition and cooperation relationship between four macroeconomic variables, by using this new model in the four-dimensional case.

The paper is organized as follows: After a review of the pursued objectives and the novelty of the study described in Section 1, in Section 2 the $GM\left(\mathrm{1,1}\right)$ and the two-dimensional grey Lotka–Volterra models are reviewed. In Section 3 the new model based on the generalization of the $GLV\left(2\right)$ model is presented. In Section 4, the $GLV\left(4\right)$ model is used to estimate the evolution and the relationships between for four economic indicators (GDP, Investment, Export and Import). The results of the estimates and forecasts are compared to the ones obtained using $GM\left(\mathrm{1,1}\right)$ model for each indicator. The estimated relationships are compared to the ones obtained using $GLV\left(2\right)$ models for pairs of the four considered economic indicators. The performance of the model and the results are discussed. Economic interpretations of the empirical results are formulated. Section 5 contains conclusions and further research directions.

2. About the $\mathit{G}\mathit{M}\mathbf{\left(}\mathbf{1,1}\mathbf{\right)}$ and Grey Lotka–Volterra Models

Together with many statistical methods, the grey model $GM\left(\mathrm{1,1}\right)$ predicts the future values of a single variable, starting with a sequence of existing values. However, as in many fields (such as economy or sociology) where two or more variables interact, the models concerning a single variable cannot suggest the relationship between these variables. This is why more complicated grey-type models were developed [6,7].

2.1. The Grey Model $GM\left(\mathrm{1,1}\right)$

In order to forecast the future behavior of a variable, the $GM\left(\mathrm{1,1}\right)$ model starts with a data sequence [2,3,8]:

$$X=\left\{x\right(1), x(2),\dots ,x(k),\dots ,x(n\left)\right\}$$

(1)

Considering the accumulated sequence (also called Accumulating Generation Operator [7]):

$$Y=\left\{y\right(1), y(2),\dots ,y(k),\dots ,y(n\left)\right\}, \mathrm{where} y\left(k\right)=x\left(1\right)+x\left(2\right)+\dots +x\left(k\right),$$

(2)

and the mean generation of consecutive neighbors of $Y$ sequence:

$$Z=\left\{z\right(2), z(3),\dots ,z(k),\dots ,z(n\left)\right\}, \mathrm{where} z\left(k\right)=\frac{y(k-1)+y\left(k\right)}{2}, k=\mathrm{2,3},\dots ,n,$$

(3)

then the $GM\left(\mathrm{1,1}\right)$ model reads [9]:

$$x\left(k\right)+az\left(k\right)=b, k=\mathrm{2,3},\dots ,n.$$

(4)

Theorem 1 

([9]). The least square estimates $a^{*},b^{*}$ of the grey difference Equation (4) satisfy

$$\left(\begin{array}{c}{a}^{*}\\ b^{*}\end{array}\right)=(B^{T}B{)}^{-1}{B}^{T}M$$

(5)

where $B$ and $M$ are given by

$$M=\left(\begin{array}{c}x\left(2\right)\\ ⋮\\ x\left(n\right)\end{array}\right) , B=\left(\begin{array}{cc}-z\left(2\right)& 1\\ ⋮& ⋮\\ -z\left(n\right)& 1\end{array}\right).$$

Consider the linear differential equation:

$$\stackrel{.}{y}\left(t\right)=-ay\left(t\right)+b,$$

(6)

with the initial condition $y\left(0\right)=y_0$, possessing the unique solution:

$$y\left(t\right)=\left(y_0-\frac{b}{a}\right)e^{-at}+\frac{b}{a}.$$

(7)

If the coefficients a and b are given by (5) and $y_0=y\left(1\right)=x\left(1\right),$ then the values of the sequence $Y$ are discrete values of the solution of Cauchy problem $y_0=y\left(1\right)$ for Equation (6). Consequently, the estimated values of the sequence $X$ are:

$$x^{*}(k+1)=y^{*}(k+1)-y^{*}\left(k\right)=(y_0-\frac{b^{*}}{a^{*}})e^{-a^{*}k}(e^{-a^{*}}-1), k=1, 2,\dots$$

(8)

The mean absolute percentage error ($MAPE$) of the estimation is computed as follows:

$$MAPE=\frac{100}{n}\sum _{k=2}^n\left|\frac{x\left(k\right)-x^{*}\left(k\right)}{x\left(k\right)}\right|.$$

(9)

The accuracy of the model is given by the values of $MAPE$ as in

Table 1. The accuracy of the grey model.

$MAPE<10$	High accuracy
$10\le MAPE<20$	Good accuracy
$20\le MAPE<50$	Reasonable accuracy
$50<MAPE$	Lack of accuracy

.

2.2. The Grey Lotka–Volterra Model

In order to estimate the evolution of two coexisting variables, the grey model $GM\left(\mathrm{1,1}\right)$ can be applied for each of them, to obtain predicted values. However, the relationship between the variables which interact may be emphasized using the two-dimensional grey Lotka–Volterra model, denoted $GLV\left(2\right)$.

Consider the quadratic Lotka–Volterra system of differential equations:

$$\left\{\begin{array}{c}{\dot{y}}_1=y_1\left(a_1-b_1{y}_1-c_1{y}_2\right),\\ {\dot{y}}_2=y_2\left(a_2-b_2{y}_1-c_2{y}_2\right),\end{array}\right.$$

(10)

where $y_1,y_2$ are the variables, functions of time $t$, $a_i,b_i,c_i,i=\mathrm{1,2}$, are real coefficients, while the dot over quantities signifies differentiation with respect to the time $t$.

Theorem 2 

([10,11]). The functions $y_1\left(t\right)$ and $y_2\left(t\right)$that verify

$$\begin{gathered} y_1\left(t+h\right)=\frac{{\alpha }_1{y}_1\left(t\right)}{1+{\beta }_1{y}_1\left(t\right)+{\gamma }_1{y}_2\left(t\right)},\mathrm{with} {\alpha }_1=e^{a_{1}h},{\beta }_1=\frac{b_1}{a_1}\left({\alpha }_1-1\right),{\gamma }_1=\frac{c_1}{a_1}\left({\alpha }_1-1\right), \\ {y}_2(t+h)=\frac{{\alpha }_2{y}_2\left(t\right)}{1+{\beta }_2{y}_1\left(t\right)+{\gamma }_2{y}_2\left(t\right)}, \mathrm{with} {\alpha }_2=e^{a2h},{\beta }_2=\frac{b_2}{a_2}\left({\alpha }_2-1\right),{\gamma }_2=\frac{c_2}{a_2}\left({\alpha }_2-1\right), \end{gathered}$$

(11)

are solutions of system (10).

As a consequence, the continuous model (10) can be converted to the following Lotka–Volterra difference equations [10]:

$$y_1\left(k+1\right)=\frac{{\alpha }_1{y}_1\left(k\right)}{1+{\beta }_1{y}_1\left(k\right)+{\gamma }_1{y}_2\left(k\right)}, y_2\left(k+1\right)=\frac{{\alpha }_2{y}_2\left(t\right)}{1+{\beta }_2{y}_1\left(k\right)+{\gamma }_2{y}_2\left(k\right)}, \mathrm{for} k=1,\dots ,n-1,$$

where the parameters ${\alpha }_{\mathrm{i}}$, ${\beta }_{\mathrm{i}},$ ${\gamma }_{\mathrm{i}}$, $i=\mathrm{1,2}$, are giving by (11).

Consider with two data sequences:

$$X_1=\left\{x_1\right(1), x_1(2),\dots ,x_1(k),\dots ,x_1(n\left)\right\}; X_2=\left\{x_2\right(1), x_2(2),\dots ,x_2(k),\dots ,x_2(n\left)\right\},$$

(12)

the accumulating sequences

$$\begin{gathered} Y_1=\left\{y_1\right(1), y_1(2),\dots ,y_1(k),\dots ,y_1(n\left)\right\}, \mathrm{where} y_1\left(k\right)=x_1\left(1\right)+x_1\left(2\right)+\dots +x_1\left(k\right), \\ {Y}_2=\left\{y_2\right(1), y_2(2),\dots ,y_2(k),\dots ,y_2(n\left)\right\}, \mathrm{where} y_2\left(k\right)=x_2\left(1\right)+x_2\left(2\right)+\dots +x_2\left(k\right), \end{gathered}$$

(13)

and the mean generation of consecutive neighbors sequences:

$$\begin{gathered} Z_1=\left\{z_1\right(2), z_1(3),\dots ,z_1(k),\dots ,z_1(n\left)\right\}, \mathrm{where} z_1\left(k\right)=\frac{y_1(k-1)+y_1\left(k\right)}{2}, k=\mathrm{2,3},\dots ,n. \\ {Z}_2=\left\{z_2\right(2), z_2(3),\dots ,z_2(k),\dots ,z_2(n\left)\right\}, \mathrm{where} z_2\left(k\right)=\frac{y_2(k-1)+y_2\left(k\right)}{2}, k=\mathrm{2,3},\dots ,n. \end{gathered}$$

(14)

Applying the grey method to the Lotka–Volterra system (10), the following approximations are obtained:

$$\left\{\begin{array}{c}{x}_1\left(k\right)\approx z_1\left(k\right)\left(a_1-b_1{z}_1\left(k\right)-c_1{z}_2\left(k\right)\right),\\ x_2\left(k\right)\approx z_2\left(k\right)\left(a_2-b_2{z}_1\left(k\right)-c_2{z}_2\left(k\right)\right).\end{array}\right.$$

(15)

This is the two-dimensional grey Lotka–Volterra model.

Estimators for the coefficients in system (15), such that the sum of the squared errors between the given and predicted values should be minimum, are found into the form:

$$\left(\begin{array}{c}{a}_1^{*}\\ b_1^{*}\\ c_1^{*}\end{array}\right)=({B_1}^T{B}_1{)}^{-1}{B_1}^T{M}_1,\hspace{1em}\left(\begin{array}{c}{a}_2^{*}\\ b_2^{*}\\ c_2^{*}\end{array}\right)=({B_2}^T{B}_2{)}^{-1}{B_2}^T{M}_2,$$

(16)

where the matrices $M_1,B_1,M_2,B_2$ are given by:

$$\begin{gathered} M_1=\left(\begin{array}{c}{x}_1\left(2\right)\\ ⋮\\ x_1\left(n\right)\end{array}\right),B_1=\left(\begin{array}{ccc}{z}_1\left(2\right)& -z_1^2\left(2\right)& -z_1\left(2\right)z_2\left(2\right)\\ ⋮& ⋮& ⋮\\ z_1\left(n\right)& -z_1^2\left(n\right)& -z_1\left(n\right)z_2\left(n\right)\end{array}\right), \\ {M}_2=\left(\begin{array}{c}{x}_2\left(2\right)\\ ⋮\\ x_2\left(n\right)\end{array}\right),B_2=\left(\begin{array}{ccc}{z}_2\left(2\right)& -z_1\left(2\right)z_2\left(2\right)& -z_2^2\left(2\right)\\ ⋮& ⋮& ⋮\\ z_2\left(n\right)& -z_1\left(n\right)z_2\left(n\right)& -z_2^2\left(n\right)\end{array}\right). \end{gathered}$$

Thus, estimators of the values from the sequences (12) are obtained as:

$$x_i^{*}(k+1)=y_i^{*}(k+1)-y_i^{*}\left(k\right), k=1, 2,\dots ,$$

(17)

where, estimators for $y_i\left(k+1\right)$ can be obtained, using (11) with $h=1$ and $t=k$, as follows:

$$\begin{gathered} y_1^{*}\left(k+1\right)=\frac{{\alpha }_1{y}_1\left(k\right)}{1+{\beta }_1{y}_1\left(k\right)+{\gamma }_1{y}_2\left(k\right)}, \mathrm{with} {\alpha }_1=e^{a_1^{*}},{\beta }_1=\frac{b_1^{*}}{a_1^{*}}\left({\alpha }_1-1\right),{\gamma }_1=\frac{c_1^{*}}{a_1^{*}}\left({\alpha }_1-1\right), \\ {y}_2^{*}\left(k+1\right)=\frac{{\alpha }_2{y}_2\left(t\right)}{1+{\beta }_2{y}_1\left(k\right)+{\gamma }_2{y}_2\left(k\right)}, \mathrm{with} {\alpha }_2=e^{a_2^{*}},{\beta }_2=\frac{b_2^{*}}{a_2^{*}}\left({\alpha }_2-1\right),{\gamma }_2=\frac{c_2^{*}}{a_2^{*}}\left({\alpha }_2-1\right), \end{gathered}$$

(18)

Finally, the mean absolute percentage error ($MAPE$) is computed for both sequence:

$$MAP{E}_i=\frac{100}{n}\sum _{k=2}^n\left|\frac{x_i\left(k\right)-x_i^{*}\left(k\right)}{x_i\left(k\right)}\right|, i=1, 2.$$

(19)

The relationship between the variables depending on the signs of the coefficients $c_1$ and $b_2$ is emphasized in

Table 2. Relationship between the variables of the Lotka–Volterra model [12].

${\mathit{c}}_1$	${\mathit{b}}_2$	Type of Relationship	Explanation
+	+	Pure competition	Both species suffer from each other’s existence
−	+	Predator–Prey	$y_1$—$\mathrm{predator}, \mathrm{destroys} \mathrm{the} \mathrm{prey} y_2$
+	−	Prey–Predator	$y_2$—$\mathrm{predator}, \mathrm{destroys} \mathrm{the} \mathrm{prey} y_1$
−	−	Mutualism	Symbiosis or a win-win situation
+	0	Amensalism	$y_1$—$\mathrm{suffers} \mathrm{from} \mathrm{the} \mathrm{existence} \mathrm{of} y_2$, who is impervious to what is happening
0	+	Amensalism	$y_2$—$\mathrm{suffers} \mathrm{from} \mathrm{the} \mathrm{existence} \mathrm{of} y_1$, who is impervious to what is happening
−	0	Commensalism	$y_1$—$\mathrm{benefits} \mathrm{from} \mathrm{the} \mathrm{existence} \mathrm{of} y_2$, who is impervious to what is happening
0	−	Commensalism	$y_2$—$\mathrm{benefits} \mathrm{from} \mathrm{the} \mathrm{existence} \mathrm{of} y_1$, who is impervious to what is happening
0	0	Neutralism	No interaction

.

3. Generalization of the Grey Method for a $\mathit{s}$-Dimensional Lotka–Volterra Model

Because in most of real-world situations, more than two variables interact, a generalization of the $GLV\left(2\right)$ model is useful. An extension of the grey method to a three-dimensional quadratic Lotka–Volterra system can be found in [13].

In this section the grey method for the Lotka–Volterra model, exposed in Section 2, is generalized for a $s$-dimensional Lotka–Volterra model.

Consider the $s$-dimensional quadratic Lotka–Volterra system of differential equations:

$$\left\{\begin{array}{c}{\stackrel{.}{y}}_1=y_1\left(a_1-b_{11}{y}_1-b_{12}{y}_2-\dots -b_{1s}{y}_s\right),\\ \dots \\ {\stackrel{.}{y}}_i=y_i\left(a_i-b_{i1}{y}_1-b_{i2}{y}_2-\dots -b_{is}{y}_s\right),\\ \dots \\ {\stackrel{.}{y}}_s=y_s\left(a_s-b_{s1}{y}_1-b_{s2}{y}_2-\dots -b_{ss}{y}_s\right),\end{array}\right.$$

(20)

where $a_i$, $b_{ij}$, $i,j=\overline{1,s}$, are real parameters.

In the following we generalize the results in [10] for the $s$-dimensional system (20).

Theorem 3. 

The functions $y_1,y_2,\dots ,y_s$ satisfying for every $i=1,\dots ,s$:

$$y_i(t+h)=\frac{{\alpha }_i{y}_i\left(t\right)}{1+{\beta }_{i1}{y}_1\left(t\right)+{\beta }_{i2}{y}_2\left(t\right)+\dots +{\beta }_{is}{y}_s\left(t\right)},$$

(21)

with

$${\alpha }_i=e^{a_{i}h},{\beta }_{i1}=\frac{b_{i1}}{a_i}\left({\alpha }_i-1\right), \dots ,{\beta }_{is}=\frac{b_{is}}{a_i}\left({\alpha }_i-1\right),$$

(22)

are solutions of system (20).

Proof. 

As $y_i(t+h)=\frac{{\alpha }_i{y}_i\left(t\right)}{1+\frac{{\alpha }_i-1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]}$ we get

$$y_i(t+h)-y_i\left(t\right)=y_i\left(t\right)\left[\frac{{\alpha }_i}{1+\frac{{\alpha }_i-1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]}-1\right]=y_i\left(t\right)\left({\alpha }_i-1\right)\left[\frac{1-\frac{1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]}{1+\frac{{\alpha }_i-1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]}\right]$$

As $\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{\alpha }_i=1$, it follows that $\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\alpha }_i-1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]=0$. In addition, $\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{{\alpha }_i-1}{h}=\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{e^{a_{i}h}-1}{h}=a_i,$ and so we have:

$$\begin{array}{ll}\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}\frac{y_i\left(t+h\right)-y_i\left(t\right)}{h}& =\underset{h\to 0}{\mathrm{l}\mathrm{i}\mathrm{m}}{y}_i\left(t\right)\frac{{\alpha }_i-1}{h}\left[\frac{1-\frac{1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]}{1+\frac{{\alpha }_i-1}{a_i}\left[b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\right]}\right]\\ & =y_i\left(t\right)a_i[1-\frac{1}{a_i}(b_{i1}{y}_1\left(t\right)+b_{i2}{y}_2\left(t\right)+\dots +b_{is}{y}_s\left(t\right)\left)\right].\end{array}$$

Thus, ${\stackrel{.}{y}}_i=y_i(a_i-b_{i1}{y}_1-b_{i2}{y}_2-\dots -b_{is}{y}_s)$ that is system (20) is verified. □

Taking $h=1$ into (21), (22) it follows:

Corollary 1. 

The discrete system corresponding to (20) is:

$$\left\{\begin{array}{c}{y}_1(t+1)=\frac{{\alpha }_1{y}_1\left(t\right)}{1+{\beta }_{11}{y}_1\left(t\right)+{\beta }_{12}{y}_2\left(t\right)+\dots +{\beta }_{1s}{y}_s\left(t\right)}, \\ \dots \\ y_i(t+1)=\frac{{\alpha }_i{y}_i\left(t\right)}{1+{\beta }_{i1}{y}_1\left(t\right)+{\beta }_{i2}{y}_2\left(t\right)+\dots +{\beta }_{is}{y}_s\left(t\right)}, \\ \dots \\ y_s(t+1)=\frac{{\alpha }_s{y}_s\left(t\right)}{1+{\beta }_{s1}{y}_1\left(t\right)+{\beta }_{s2}{y}_2\left(t\right)+\dots +{\beta }_{ss}{y}_s\left(t\right)}, \end{array}\right.$$

(23)

with

$${\alpha }_i=e^{a_i},{\beta }_{i1}=\frac{b_{i1}}{a_i}\left({\alpha }_i-1\right), \dots ,{\beta }_{is}=\frac{b_{is}}{a_i}\left({\alpha }_i-1\right).$$

(24)

Next, we generalize the grey method for the $s$-dimensional system Lotka–Volterra (20).

Consider $s$ data sequences:

$$X_i=\left\{x_i\right(1), x_i(2),\dots ,x_i(k),\dots ,x_i(n\left)\right\}, i=\mathrm{1,2},\dots ,s,$$

(25)

the accumulating sequences

$$Y_i=\left\{y_i\right(1), y_i(2),\dots ,y_i(k),\dots ,y_i(n\left)\right\}, \mathrm{where} y_i\left(k\right)=x_i\left(1\right)+x_i\left(2\right)+\dots +x_i\left(k\right), k=1,\dots ,n,$$

(26)

and the mean generation of consecutive neighbors sequences:

$$Z_i=\left\{z_i\right(2), z_i(3),\dots ,z_i(k),\dots ,z_i(n\left)\right\}, \mathrm{where} z_i\left(k\right)=\frac{y_i(k-1)+y_i\left(k\right)}{2}, k=\mathrm{2,3},\dots ,n.$$

(27)

For $t\in (k-1,k), \mathrm{with} k=2,\dots ,n$, we make the following approximations:

$$y_i\left(t\right)\approx \frac{y_i(k-1)+y_i\left(k\right)}{2}=z_i\left(k\right), i=1,\dots ,s.$$

(28)

Applying the Lagrange theorem on the interval $[k-1,k]$, there exists ${\xi }_{ik}\in (k-1,k)$ such that ${\stackrel{.}{y}}_i\left({\xi }_{ik}\right)=y_i\left(k\right)-y_i\left(k-1\right)=x_i\left(k\right).$ Approximating $\stackrel{.}{y_i}\left(t\right)$ for $t\in (k-1,k)$ by ${\stackrel{.}{y}}_i\left({\xi }_{ik}\right)$, we get:

$$\stackrel{.}{y_i}\left(t\right)\approx x_i\left(k\right), k=2,\dots ,n, i=1,\dots ,s.$$

(29)

Replacing (27) and (28) into system (20) we get for every $k=2,\dots ,n$:

$$\left\{\begin{array}{c}{x}_1\left(k\right)\approx z_1\left(k\right)\left[a_1-b_{11}{z}_1\left(k\right)-b_{12}{z}_2\left(k\right)-\dots -b_{1s}{z}_s\left(k\right)\right],\\ \dots \\ x_i\left(k\right)\approx z_i\left(k\right)\left[a_i-b_{i1}{z}_1\left(k\right)-b_{i2}{z}_2\left(k\right)-\dots -b_{is}{z}_s\left(k\right)\right],\\ \dots \\ x_s\left(k\right)\approx z_s\left(k\right)\left[a_s-b_{s1}{z}_1\left(k\right)-b_{s2}{z}_2\left(k\right)-\dots -b_{ss}{z}_s\left(k\right)\right].\end{array}\right.$$

(30)

Model (30) will be called $s$-dimensional grey Lotka–Volterra model, and it will be denoted $GLV\left(s\right)$.

Theorem 4. 

The least square estimates $a_i^{*},b_{ij}^{*},i,j=\overline{1,s}$, of the grey Lotka–Volterra model (30) satisfy

$$\left(\begin{array}{c}{a}_i^{*}\\ b_{i1}^{*}\\ ⋮\\ b_{is}^{*}\end{array}\right)=({B_i}^T{B}_i{)}^{-1}{B_i}^T{M}_i,$$

(31)

where $B_i$ and $M_i$ are given by

$$M_i=\left(\begin{array}{c}{x}_i\left(2\right)\\ ⋮\\ x_i\left(n\right)\end{array}\right),\hspace{1em}{B}_i=\left(\begin{array}{cccccc}{z}_i\left(2\right)& -z_i\left(2\right)z_1\left(2\right)& \dots & -z_i(2{)}^2& \dots & -z_i\left(2\right)z_s\left(2\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ z_i\left(n\right)& -z_i\left(n\right)z_1\left(n\right)& \dots & -z_i(n{)}^2& \dots & -z_i\left(n\right)z_s\left(n\right)\end{array}\right).$$

(32)

Proof. 

For every $i=\overline{1,s}$, consider ${\stackrel{-}{x}}_i\left(k\right)=z_i\left(k\right)\left[a_i-b_{i1}{z}_1\left(k\right)-b_{i2}{z}_2\left(k\right)-\dots -b_{is}{z}_s\left(k\right)\right]$ the approximate value of $x_i\left(k\right)$ and the errors ${\epsilon }_{ik}=x_i\left(k\right)-{\stackrel{\_}{x}}_i\left(k\right)$, for $k=\overline{2,n}$. Thus:

$${\epsilon }_{ik}=x_i\left(k\right)-a_i{z}_i\left(k\right)+b_{i1}{z}_i\left(k\right)z_1\left(k\right)+b_{i2}{z}_i\left(k\right)z_2\left(k\right)+\dots +b_{ii}{z}_i{\left(k\right)}^2+\dots +b_{is}{z}_i\left(k\right)z_s\left(k\right).$$

The error vectors read:

$${\epsilon }_i=\left(\begin{array}{c}{\epsilon }_{i2}\\ ⋮\\ {\epsilon }_{in}\end{array}\right)=\left(\begin{array}{c}{x}_i\left(2\right)\\ ⋮\\ x_i\left(n\right)\end{array}\right)-\left(\begin{array}{cccccc}{z}_i\left(2\right)& -z_i\left(2\right)z_1\left(2\right)& \dots & -z_i(2{)}^2& \dots & -z_i\left(2\right)z_s\left(2\right)\\ ⋮& ⋮& & ⋮& & ⋮\\ z_i\left(n\right)& -z_i\left(n\right)z_1\left(n\right)& \dots & -z_i(n{)}^2& \dots & -z_i\left(n\right)z_s\left(n\right)\end{array}\right)\left(\begin{array}{c}{a}_i\\ b_{i1}\\ ⋮\\ b_{is}\end{array}\right),$$

equivalent to:

$${\epsilon }_i=M_i-B_i\left(\begin{array}{c}{a}_i\\ b_{i1}\\ ⋮\\ b_{is}\end{array}\right).$$

In order to have the minimum of $f_i(a_i,b_{i1},\dots ,b_{is})={{\epsilon }_i}^T{\epsilon }_i={\epsilon }_{i2}^2+{\epsilon }_{i3}^2+\dots +{\epsilon }_{in}^2$, we put $\nabla f_i=0$. As

$$\begin{gathered} \frac{\partial f_i}{\partial a_i}=-2{\epsilon }_{i2}{z}_i\left(2\right)-\dots -2{\epsilon }_{in}{z}_i\left(n\right), \\ \frac{\partial f_i}{\partial b_{i1}}=2{\epsilon }_{i2}{z}_i\left(2\right)z_1\left(2\right)+\dots +2{\epsilon }_{in}{z}_i\left(n\right)z_1\left(n\right), \\ \dots \\ \frac{\partial f_i}{\partial b_{ii}}=2{\epsilon }_{i2}{z}_i(2{)}^2+\dots +2{\epsilon }_{in}{z}_i(n{)}^2, \\ \dots \\ \frac{\partial f_i}{\partial b_{is}}=2{\epsilon }_{i2}{z}_i\left(2\right)z_s\left(2\right)+\dots +2{\epsilon }_{in}{z}_i\left(n\right)z_s\left(n\right), \end{gathered}$$

it follows that

$$\nabla f_i=-2B_i^T{\epsilon }_i=-2B_i^T\left[M_i-B_i\left(\begin{array}{c}{a}_i\\ b_{i1}\\ ⋮\\ b_{is}\end{array}\right)\right]=-2B_i^T{M}_i+2B_i^T{B}_i\left(\begin{array}{c}{a}_i\\ b_{i1}\\ ⋮\\ b_{is}\end{array}\right).$$

Solving the equation $\nabla f_i=0$, estimators of the parameters in system (30) are obtained into the form (31). □

Corollary 2. 

Using (23), estimators of $y_i\left(k+1\right)$ can be obtained as follows:

$$y_i^{*}\left(k+1\right)=\frac{{\alpha }_i{y}_i\left(k\right)}{1+{\beta }_{i1}{y}_1\left(k\right)+{\beta }_{i2}{y}_2\left(k\right)+\dots +{\beta }_{is}{y}_s\left(k\right)}, k=2,\dots ,n,$$

(33)

while, according to (24), the coefficients are given by

$${\alpha }_i=e^{a_i^{*}},{\beta }_{i1}=\frac{b_{i1}^{*}}{a_i^{*}}\left({\alpha }_i-1\right), \dots ,{\beta }_{is}=\frac{b_{is}^{*}}{a_i^{*}}\left({\alpha }_i-1\right), \mathrm{for} \mathrm{all} i=\overline{1,s}.$$

(34)

As a consequence, estimators of the values from the sequence (25) are given by:

$$x_i^{*}(k+1)=y_i^{*}(k+1)-y_i^{*}\left(k\right), k=1, 2,\dots$$

(35)

Finally, the mean absolute percentage error ($MAPE$) is computed for every sequence as:

$$MAP{E}_i=\frac{100}{n}\sum _{k=2}^n\left|\frac{x_i\left(k\right)-x_i^{*}\left(k\right)}{x_i\left(k\right)}\right|, i=1, 2,\dots ,s.$$

(36)

The accuracy of the model is given by the values of $MAPE$ as specified in Table 1.

4. Case Study: Application to GDP, Export, Import and Investment for the European Union

In this section a $GLV\left(4\right)$ model is used to analyze the competition and cooperation relationship between four macroeconomic indicators, namely GDP, Investment, Import and Export, and to obtain short-time forecasting for them. The data used in the study correspond to values of these four economic variables for the European Union (EU), covering two periods of time, namely 2005–2022 and 2011–2022.

The case study pursues the following objectives:

-

To assess the estimation accuracy of the $GLV\left(4\right)$ model and to compare the estimated values and forecasts with the ones obtained using the grey model $GM\left(\mathrm{1,1}\right)$;

-

To determine, using the $GLV\left(4\right)$ model, the type of cooperation or competition relationships between pairs of economic variables;

-

To compare the empirical relationships obtained for the two different time periods, on one hand, and with the ones resulted using the $GLV\left(2\right)$ model for pairs of two variables, on the other hand;

-

To derive economic interpretations for the empirically obtained results and to compare them with the ones identified in the literature.

4.1. Economic Background Analysis

Gross domestic product (GDP) measures economic activity, indicating the strength of an economy, by determining the value of all final goods and services produced within an economy, in a given period of time. GDP is defined as the monetary value of all final goods and services produced within the borders of a specific country in a given time period [14].

GDP is calculated using three methods: the value added method (or the production method), the expenditure method (or the use of final production) and the income method. We chose to work with the GDP determined by the expenditure method, expressed in value at market prices (millions of euros), being the only possibility to sum up the heterogeneous goods realized at the level of the national economy. The expression of GDP, in current market prices, eliminates the differences between the price level in different countries and allows it to be used as a tool for comparing living standards, or for monitoring the convergence process in the European Union.

The expenditure approach calculates GDP using total expenditure on domestic goods, respectively, expenditure on private consumption, investment, government spending and net exports. In this sense, starting from this method, the following variables were considered in the study: GDP, investments, import and export. This approach was motivated by the lack of a cumulative analysis of the 4 indicators. In general, other research considered the analysis of only certain types of expenditures on GDP.

A survey of the specialized literature, showed that international trade (import and export) as well as investment have a significant impact on the GDP of a nation.

Investments are acquisitions of goods that will be utilized to create wealth in the future. Foreign investment are transfers of investment made by natural or legal persons from one nation to another nation or nations [15]. Foreign direct investment can bring innovations in the activity of firms through technology transfer having a strong impact on GDP [26]. Thus, for Central and Eastern Europe, it was found that the growth of foreign investments is the determining factor in the development and coherence of the economy [17]. Public and private investment play a central role in production functions, providing the capital needed for development [16]. Investment can stimulate a country’s economic growth in two ways: it provides host countries with capital that stimulates the expansion of investment and production; through the advanced technology and the experience brought, it causes the enterprises of the host country to optimize their production efficiency [27]. Various studies have demonstrated the positive and significant effect of investment on economic growth [17,18,19,20,21,22,23,24,25,28,29,30,31,32]. At the same time, although investments have a causal impact on economic growth, causality from GDP to investments is not confirmed [19].

Some studies identify a mutually beneficial relationship between GDP and investment in the long run, but in the short run investment limits economic development [1].

Trade plays a significant role among the factors of economic growth [33]. Imports and exports are essential components for estimating the GDP of a country, being referred to as “Net Exports”. Net exports are calculated as the difference between the estimated total exports and the estimated total imports. A positive value of net exports indicates a trade surplus, and a negative value a trade deficit. International trade is basically represented by net exports in the expenditure model for calculating GDP. In this approach, since GDP measures domestic production, imports should have no impact on GDP. However, there is an important influence of imports on economic sustainability [34].

Economic growth is significantly boosted by greater openness to trade. Accumulation of imports can be a sign of a growing economy, as higher demand for foreign goods may be an indication of an increase in income and consumer spending [35].

Exports are goods and services that are produced domestically but sold to buyers in other countries which brings an inflow of funds into the country. Thus, exports have a positive impact on economic growth, and international trade can strongly increase incomes by improving living standards [36,37]. Exporting stimulates economic growth from the demand side and produces an efficiency gain on the supply side [38]. Therefore, nations that implement export-oriented strategies continuously absorb the flow of foreign currency, producing more output [39].

Investment plays a critical role in economic development and export growth [17,23,24]. Due to the visible benefits of investment to the national economy, countries adopt various policies that facilitate the flow of investment, such as investment incentives and facilitation of foreign trade procedures [40].

The results of the investigation into the relationship between economic growth, investment and exports were mixed. Thus, for developing countries exports and foreign direct investment are of great importance for economic development [18]. Some research has found that there is no long-run equilibrium relationship between the growth rate of real gross domestic product (GDP), investment and exports but only between exports and GDP [41]. Others have identified that investment and export lead to economic growth [38] but the reciprocal is not true [42]. In general, countries are open to international trade to achieve high economic growth, as there is a long- and short-run causal relationship between investment, GDP and exports [43]. According to [44], there is bidirectional causality between exports and GDP and investment has unidirectional effects on GDP directly and indirectly through exports. The trade war has two consequences: on the one hand, it causes a decline in exports and a rise in GDP, but on the other hand, it creates an opportunity for investment to emerge that could benefit the economy [45]. At the same time, the decrease in exports can lead to a decrease in GDP, because it can be affected by several economic variables [46].

Imports are the goods and services purchased from other countries which implies an outflow of funds from the country. Although the value of imports seems that they should not be taken into consideration in the analysis of the influence on GDP (it measures domestic production), a bidirectional causality is found between imports and GDP, which highlights the significant influence of imports on economic growth, but also the importance of GDP, whose growth causes an increase in imports.

Imports due to the transfer of technology and innovation enable high productivity, formation and accumulation of capital and thus lead to economic growth. Thus, for developing countries, imports allow domestic industries to obtain advanced technologies, tools and machinery from developed countries which contributes to their economic growth [34].

Analysis of the relationship between exports, imports and GDP indicates that imports significantly promote economic growth, while GDP growth stimulates export growth [47]. At the same time, some studies have found that imports have a greater effect on production growth than exports [48,49]. The increase in exports contributes to the increase in imports because it builds up foreign exchange reserves used to pay for imports. GDP has an important role in the growth of exports and is significantly affected by imports [34].

In the long run, GDP and import growth are driven by exports. Thus, the increase in exports accelerates not only economic growth, but also the capacity to import [34]. Significant economic growth is seen in countries where investment and trade have increased [50].

The different results of the studies that analyzed the relationship between the economic indicators under analysis are due to the models used in the analysis. Most of the research focuses on statistical methods and artificial intelligence and to a lesser extent on grey models.

The role of the investment factor in economic growth is highlighted by econometric models on groups of countries [51,52,53,54], identifying different types of causality between investment and GDP (bidirectional, unidirectional and no causality) according to the countries and periods analyzed [29,34,55]. Using the autoregressive distributed lag (ARDL) test some studies find that there is a unidirectional long-run relationship from investment to economic growth [20], and others find that there is no significant causal relationship between them [56].

Although statistical and AI-based approaches are the main time series prediction techniques, they are not accurate for nonlinear problems. Moreover, these models need a large number of samples and are too complex to be used in predicting future values [57]. In such conditions, grey models are more effective methods for time series prediction [58].

Some authors have used the generalized method of moments (GMM) to estimate the positive interaction between investment and economic growth [25]. The use of neural networks [59] and grey Lotka–Volterra models [1] shows improved forecast accuracy. Analysis of factors influencing GDP through GMM demonstrated the positive influence of investment and trade, showing that exports increase economic fluctuations, while imports decrease them [60].

Grey Lotka–Volterra models were used to test trade relations (import–export) [61].

4.2. Data

The empirical study was performed for the European Union. Four macroeconomic indicators were taken into account: gross domestic product (GDP), Investment, Export and Import expressed in billions of euros. The values of the indicators are taken from the Euro-stat database for two periods 2005–2022 and 2011–2022. The indicators are expressed in billions of euros [62].

The construction of the database was achieved by using SDMX web services, at the level of the Eurostat database, based on the ETL (Extract–Transform–Load) process. The algorithms described by the methodology were implemented in source codes using the scientific software products Wolfram Research Mathematica 9.0 (Wolfram, Champaign, IL, USA) and Maple 18 (Waterloo Maple (Maplesoft), Waterloo, ON, Canada), which were also used for the data processing.

As illustrated in Figure 1, for the analyzed periods, the macroeconomic indicators subjected to the analysis have an increasing tendency. For the period 2011–2022, there is an increase in them with a syncope in 2020 due to the COVID-19 pandemic. The period 2005–2022 is characterized by the presence of two crises, the financial crisis and the COVID crisis, so that we have two moments in which there are decreases in the values of the analyzed macroeconomic indicators (2009 and 2020).

4.3. Methods and Methodology

The estimation of the evolution of the considered indicators is performed using three models.

Firstly, the grey forecasting model $GM\left(\mathrm{1,1}\right)$ is used in order to estimate and forecast short and long-time the individual evolution of each indicator, independently. For each of the four considered economic indicators, the following steps are performed: step (1) start with the initial data $X=\left\{x\right(1),x(2),\dots ,x(k),\dots ,x(n\left)\right\}$; step (2) generate the accumulation sequence, as in Formula (2); step (3) generate the adjacent mean value sequence by Formula (3); step (4) compute the estimators of the $a,b$ parameters by means of Formula (5); step (5) compute the estimated values $x^{*}\left(k\right),k\ge 2$, using Formula (8).

In the above algorithm, $n=12$ for the data corresponding to 2011–2022, and $n=18$ for the data corresponding to 2005–2022.

The estimation accuracy is evaluated using the mean absolute percentage error ($MAPE$), given by (9), and the mean squared error ($MSE$), given by:

$$MSE=\frac{1}{n}\sum _{k=1}^n{\left(x\left(k\right)-x^{*}\left(k\right)\right)}^2.$$

(37)

The $MSE$ is used to ascertain the performance of the models and to appreciate how close estimates or forecasts are to the actual values. The lower the $MSE$, the closer the forecast is to the actual values. The $MSE$ can be used as a baseline to see if the model’s accuracy is improving over time or not.

Secondly, the $GLV\left(2\right)$ model is used to estimate the competition and cooperation relationship between each of the six different pairs of the considered economic indicator. For each pair of the four considered economic indicators, the following steps are performed: step (1) start with two sequence of data $X_1, X_2$, as in (12); step (2) generate the accumulation sequences $Y_1, Y_2$, using Formula (13); step (3) generate the adjacent mean value sequences $Z_1, Z_2$ by Formula (14); step (4) compute the estimators of the coefficients $a_i^{*},b_i^{*}, c_i^{*}, i=\mathrm{1,2}$, using (16); step (5) compute the estimated values ${x_1}^{*}\left(k\right),{x_2}^{*}\left(k\right),k\ge 2$, using Formula (17), with ${y_1}^{*}\left(k\right),{y_2}^{*}\left(k\right),k\ge 2$, given by (18).

The algorithm is applied for each pair of indicators, for the two series of data corresponding to 2011–2022, and to 2005–2022, respectively. The relationship between each pair of indicators is established by taking into account the interpretations in Table 2.

Finally, the $GLV\left(4\right)$ model is used to estimate the qualitative interactions between the four indicators.

For each of the two periods of time, 2005–2022 and 2012–2022, the following steps are performed: step (1) start with four sequence of data $X_i, i=\overline{\mathrm{1,4}}$, as in (25); step (2) generate the accumulation sequences $Y_i, i=\overline{\mathrm{1,4}}$, using Formula (26); step (3) generate the adjacent mean value sequences $Z_i, i=\overline{\mathrm{1,4}}$ by Formula (27); step (4) compute the estimators of the $a_i^{*},b_{ij}^{*}, i,j=\overline{\mathrm{1,4}}$, parameters in (30) using Formulae (31); step (5) compute the estimated values of the accumulation sequences ${y_i}^{*}\left(k\right),k\ge 2$,$i=\overline{\mathrm{1,4}}$ using Formula (33), and the estimators ${x_i}^{*}\left(k\right),k\ge 2$,$i=\overline{\mathrm{1,4}}$, given by (35).

The estimation accuracy is evaluated using the mean absolute percentage error for each indicator (${MAPE}_i, i=\overline{\mathrm{1,4}}$), given by (36), and the mean squared error (${MSE}_i, i=\overline{\mathrm{1,4}}$), while the relationship between each pair of indicators $(i,j)$ is established by taking into account the interpretations in Table 2, considering $c_1=b_{ij}^{*}$ and $b_2=b_{ji}^{*}, i,j=\overline{\mathrm{1,4}}$.

The results of the empirical estimations obtained using the three mentioned models, with data corresponding to two time periods are compared. The performance of the model and the results are discussed.

4.4. Empirical Estimations and Results

4.4.1. Estimations Using the $GM\left(\mathrm{1,1}\right)$ Model

The estimated values for EU GDP, Export, Import and Investment based on data collected for the two periods 2005–2022 and 2011–2022, using $GM\left(\mathrm{1,1}\right)$ model are given in

Table 3. Estimated values for GDP, Export, Import and Investment EU for 2011–2022, with $GM\left(\mathrm{1,1}\right)$ model; data for 2011–2022, forecasting 2023.

Year	GDP		Export		Import		Investment
	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values
2011	11,328	11,328.00	4893	4893.00	4702	4702.00	2357	2357.00
2012	11,396	11,512.30	5103	5136.21	4776	4665.51	2317	2280.53
2013	11,516	11,871.20	5178	5392.67	4774	4919.83	2276	2386.68
2014	11,782	12,241.30	5380	5661.94	4937	5188.02	2333	2497.77
2015	12,215	12,622.80	5747	5944.65	5215	5470.82	2470	2614.02
2016	12,548	13,016.30	5853	6241.49	5307	5769.04	2566	2735.69
2017	13,074	13,422.10	6314	6553.14	5759	6083.52	2715	2863.02
2018	13,533	13,840.50	6648	6880.35	6131	6415.14	2863	2996.28
2019	14,018	14,271.90	6908	7223.91	6430	6764.84	3115	3135.73
2020	13,461	14,716.80	6244	7584.62	5758	7133.60	2971	3281.68
2021	14,532	15,175.60	7323	7963.34	6784	7522.46	3205	3434.43
2022	15,806	15,648.60	8842	8360.97	8592	7932.52	3592	3594.28
2023		16,136.40		8778.45		8364.93		3761.57

and

Table 4. Estimated values for GDP, Export, Import and Investment EU for 2005–2022, with $GM\left(\mathrm{1,1}\right)$ model; data for 2005–2022, forecasting 2023.

Year	GDP		Export		Import		Investment
	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values
2005	9560	9560.00	3576	3576.00	3423	3423.00	2100	2100.00
2006	10,112	10,235.50	4003	4001.04	3878	3789.41	2296	2171.15
2007	10,738	10,493.30	4370	4185.08	4233	3958.19	2510	2228.21
2008	11,085	10,757.50	4556	4377.58	4459	4134.49	2570	2286.78
2009	10,587	11,028.30	3841	4578.94	3663	4318.64	2247	2346.88
2010	10,980	11,306.00	4428	4789.56	4253	4510.99	2271	2408.57
2011	11,328	11,590.70	4893	5009.87	4702	4711.91	2357	2471.87
2012	11,396	11,882.50	5103	5240.31	4776	4921.79	2317	2536.84
2013	11,516	12,181.70	5178	5481.35	4774	5141.00	2276	2603.52
2014	11,782	12,488.40	5380	5733.48	4937	5369.99	2333	2671.95
2015	12,215	12,802.90	5747	5997.20	5215	5609.17	2470	2742.18
2016	12,548	13,125.20	5853	6273.06	5307	5859.00	2566	2814.26
2017	13,074	13,455.70	6314	6561.61	5759	6119.96	2715	2888.22
2018	13,533	13,794.50	6648	6863.42	6131	6392.55	2863	2964.14
2019	14,018	14,141.80	6908	7179.13	6430	6677.28	3115	3042.05
2020	13,461	14,497.90	6244	7509.35	5758	6974.68	2971	3122.00
2021	14,532	14,862.90	7323	7854.76	6784	7285.34	3205	3204.06
2022	15,806	15,237.20	8842	8216.06	8592	7609.83	3592	3288.27
2023		15,620.80		8593.98		7948.78		3374.70

. A forecast for the values of these economic indicators, based on the $GM\left(\mathrm{1,1}\right)$ model, for 2023 is also provided in the last line in Table 3 and Table 4. Note that the $GM\left(\mathrm{1,1}\right)$ model can be used to obtain long-term forecasts of the variables, using the estimation Formula (8).

The estimated values from Table 3 and Table 4 show that these values are very close to the actual values for both periods.

The forecasting values for 2023 show an increase for all variables. The increase is more moderate for the forecast made for the longer data series (18 years).

4.4.2. Estimations Using $GLV\left(2\right)$, for Pairs of the Indicators

The application of the $GLV\left(2\right)$ model for pairs of two indicators, following the algorithm described in Section 4.3, provides the values of the estimating coefficients for the periods 2011–2022 and 2005–2022, given in

Table 5. Estimated coefficients for pairs of indicators, using the $GLV\left(2\right)$ model; data for 2011–2022.

Pairs	${\mathit{a}}_1^{\mathit{*}}$	${\mathit{b}}_1^{\mathit{*}}$	${\mathit{c}}_1^{\mathit{*}}$	${\mathit{a}}_2^{\mathit{*}}$	${\mathit{b}}_2^{\mathit{*}}$	${\mathit{c}}_2^{\mathit{*}}$
GDP—Export	0.481071	0.038386	−0.075548	0.508415	0.044208	−0.087654
GDP—Import	0.434929	0.028035	−0.058736	0.453128	0.033742	−0.071771
GDP—Investment	0.479432	0.042232	−0.188849	0.464598	0.039213	−0.175220
Export—Import	0.310138	0.223155	−0.238060	0.301404	0.231536	−0.247324
Export—Investment	0.286787	0.023806	−0.047702	0.286792	0.028734	−0.058689
Import—Investment	0.297552	−0.048475	0.107353	0.300810	−0.025403	0.059587

and

Table 6. Estimated coefficients for pairs of indicators, using the $GLV\left(2\right)$ model; data for 2005–2022.

Pairs	${\mathit{a}}_1^{\mathit{*}}$	${\mathit{b}}_1^{\mathit{*}}$	${\mathit{c}}_1^{\mathit{*}}$	${\mathit{a}}_2^{\mathit{*}}$	${\mathit{b}}_2^{\mathit{*}}$	${\mathit{c}}_2^{\mathit{*}}$
GDP—Export	0.406540	0.016394	−0.032990	0.418528	0.016657	−0.033528
GDP—Import	0.346135	0.014218	−0.030756	0.365365	0.015980	−0.034980
GDP—Investment	0.189891	0.011389	−0.050649	0.176769	0.010638	−0.047512
Export—Import	0.213402	−0.017028	0.019794	0.209160	−0.019252	0.022113
Export—Investment	0.534930	−0.048989	0.113308	0.569645	−0.056978	0.131063
Import—Investment	0.555943	−0.068918	0.146265	0.549513	−0.070067	0.148583

, respectively.

The analysis of the empirical estimation results of the $GLV\left(2\right)$ model in Table 5 and Table 6 shows that the variation in the accumulated GDP sequence is positively influenced by the variation in the other three indicators.

The variation in the accumulated Export is influenced positively by Investment and Import and negatively by GDP.

Exports are negatively influenced by GDP for both periods. The influence of Investment and Import is positive on Export for the 12-year period, and negative for the 18-year period. Import is negatively influenced by GDP and Investment for both analyzed periods, while the influence of Export is negative for the 12 year-period and positive for the 18-year period. The Investments are positively influenced by Import and negatively by Export and GDP for the 12-year period, while for the 18-year period, there are positively influenced by Import and Export and negatively by GDP.

According to the interpretation given in Table 2 to the signs of the coefficients $c_1$ and $b_2$, the relationships between the accumulated values for pairs of the 6 indicators for the analyzed periods are presented in

Table 7. The type of relationship between indicators, as results of the estimation using the $GLV\left(2\right)$ model, data for 2011–2022 and 2005–2022.

Pairs	Type of Relationship 2011–2022	Type of Relationship 2005–2022
GDP—Export	Predator–Prey	Predator–Prey
GDP—Import	Predator–Prey	Predator–Prey
GDP—Investment	Predator–Prey	Predator–Prey
Export—Import	Predator–Prey	Prey–Predator
Export—Investment	Predator–Prey	Prey–Predator
Import—Investment	Prey–Predator	Prey–Predator

.

Between the pairs GDP—Export, GDP—Import and GDP—Investment it is observed that for both periods there is a Predator–Prey type relationship. Practically these pairs condition each other. GDP growth is based on Investment and trade.

Between the pairs Export—Investment and Export—Import for the 12-year period the relationship is Predator–Prey, which means that Export benefits from the interaction with Investment and Import. For the 18-year period, the relationship is of the Prey–Predator type, which means that for longer periods Export can be affected by Investment and Import.

The relationship between Import—Investment for both periods is Prey–Predator, which means that Investment can benefit from Import.

The interpretation in Table 7 shows a change in the bilateral relations corresponding to the two periods in the case of some of the pairs, as is also highlighted in the paper [11].

4.4.3. Estimations Using the $GLV\left(4\right)$ Model

The actual values and the empirical estimations for the four indicators, obtained as results of the application of the $GLV\left(4\right)$ model presented in Section 3, for the data from 2011 to 2022, are presented in

Table 8. Data for GDP, Export, Import and Investment EU, and estimated values with $GLV\left(4\right)$ model data for 2011–2022, forecasting 2023.

Year	GDP		Export		Import		Investment
	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values
2011	11,328	11,328.00	4893	4893.00	4702	4702.00	2357	2357.00
2012	11,396	8969.14	5103	3969.51	4776	3744.94	2317	1819.08
2013	11,516	13,682.60	5178	6220.92	4774	5752.11	2276	2755.08
2014	11,782	10,948.70	5380	4977.47	4937	4534.31	2333	2144.68
2015	12,215	9968.27	5747	4618.10	5215	4207.51	2470	1979.15
2016	12,548	12,463.00	5853	5907.48	5307	5356.77	2566	2554.73
2017	13,074	12,186.40	6314	5668.42	5759	5127.49	2715	2530.94
2018	13,533	13,937.30	6648	6769.43	6131	6218.64	2863	2951.04
2019	14,018	13,692.10	6908	6865.90	6430	6411.34	3115	2964.72
2020	13,461	14,899.90	6244	7440.33	5758	6955.46	2971	3328.09
2021	14,532	10,863.60	7323	4833.37	6784	4351.87	3205	2400.23
2022	15,806	18,892.00	8842	9571.09	8592	8963.65	3592	4171.79
2023		8969.14		8813.82		9122.48		3066.37

, while the estimations corresponding to the data in the 2005–2022 period are given in

Table 9. Data for GDP, Export, Import and Investment EU, and estimated values with $GLV\left(4\right)$ model; data for 2005–2022, forecasting 2023.

Year	GDP		Export		Import		Investment
	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values	Data	Estimated Values
2005	9560	9560.00	3576	3576.00	3423	3423.00	2100	2100.00
2006	10,112	5405.82	4003	2019.22	3878	1985.05	2296	1239.80
2007	10,738	13,898.40	4370	5535.19	4233	5378.07	2510	3161.29
2008	11,085	13,074.30	4556	5381.69	4459	5193.10	2570	3021.37
2009	10,587	11,978.50	3841	5042.74	3663	4905.77	2247	2706.80
2010	10,980	9658.27	4428	3380.85	4253	3119.66	2271	1917.66
2011	11,328	10,122.80	4893	4208.51	4702	3985.96	2357	1998.56
2012	11,396	10,514.80	5103	4796.99	4776	4577.18	2317	2125.43
2013	11,516	11,140.90	5178	5179.18	4774	4812.26	2276	2237.26
2014	11,782	11,427.80	5380	5267.27	4937	4823.72	2333	2259.44
2015	12,215	11,832.10	5747	5510.15	5215	5035.20	2470	2367.13
2016	12,548	12,919.40	5853	6078.92	5307	5506.89	2566	2659.65
2017	13,074	12,828.30	6314	5940.81	5759	5358.66	2715	2663.48
2018	13,533	14,073.80	6648	6778.97	6131	6206.87	2863	2992.04
2019	14,018	14,578.40	6908	7257.53	6430	6755.50	3115	3161.55
2020	13,461	14,013.60	6244	7088.54	5758	6644.92	2971	3133.78
2021	14,532	11,082	7323	5090.10	6784	4617.54	3205	2417.88
2022	15,806	15,640	8842	7993.94	8592	7512.59	3592	3501.00
2023		20,242.10		12,563.80		12,646.90		4756.79

. The last line in Table 8 and Table 9 contains the forecasting values for the year 2023.

The estimated values for the coefficients $a_i$, $b_{ij}$ and $b_{ji}, i,j=\overline{\mathrm{1,4}},$ obtained empirically applying the $GLV\left(4\right)$ model (30), are given in

Table 10. Estimated coefficients using the $GLV\left(4\right)$ model; data for period 2011–2022.

$\mathit{i}$	${\mathit{a}}_{\mathit{i}}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}1}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}2}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}3}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}4}^{\mathit{*}}$
1	0.940031	0.148687	−0.745659	0.627457	−0.308759
2	0.954818	0.150889	−0.733742	0.600043	−0.288888
3	0.947082	0.150987	−0.736913	0.594714	−0.271475
4	0.926801	0.145842	−0.749555	0.624984	−0.281964

Notes: Index 1 corresponds to GDP, index 2 to the Export, index 3 to Import, while index 4 corresponds for the Investment.

and

Table 11. Estimated coefficients using the $GLV\left(4\right)$ model; data for period 2005–2022.

$\mathit{i}$	${\mathit{a}}_{\mathit{i}}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}1}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}2}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}3}^{\mathit{*}}$	${\mathit{b}}_{\mathit{i}4}^{\mathit{*}}$
1	0.51906	0.0134973	−0.0820738	0.0426608	0.0354589
2	0.519238	0.0129581	−0.0554651	0.00965442	0.0466704
3	0.533936	0.0136645	−0.0556277	0.005304	0.0525463
4	0.543433	0.0122718	−0.087783	0.0424383	0.0540872

Notes: Index 1 corresponds to GDP, index 2 to the Export, index 3 to Import, while index 4 corresponds for the Investment.

.

Following the analysis of the sign of the empirically estimated coefficients for the period 2011–2022 in Table 10, it is found that the variation in GDP accumulation is positively influenced by the accumulated value of Export and Investment and negatively by that of Import. For the 18-year period, the variation in GDP accumulation is positively influenced only by the accumulated value of Export and negatively by the accumulated value of Import and Investment.

The variation in Export accumulation for the period 2011–2022 is influenced negatively by the value of GDP and Import accumulation and positively by that of Investment. For the period 2005–2022, it is negatively influenced by the value of the accumulation of the other three indicators: GDP, Import and Investment.

The analysis of the variation in Import accumulation for the 12-year period is positively influenced by the accumulation of Export and Investment and negatively by that of GDP. For the 18-year period, the variation in Import accumulation is positively influenced only by the Export accumulation and negatively by that of GDP and Investment. Regarding the variation in accumulated Investment for both analyzed periods, it is found that it is positively influenced by the accumulation of Export and negatively by that of GDP and Import.

According to the interpretation given in Table 2 to the signs of the coefficients $c_1$ and $b_2$, applied for pairs of coefficients $b_{ij}^{*}$ and $b_{ji}^{*}$, the relationships between the accumulated sequences for pairs of the six indicators for the two analyzed periods are presented in

Table 12. The type of relationship between indicators according to the $GLV\left(4\right)$ model.

Pairs	2011–2022	2005–2022
GDP—Export	Predator–Prey	Predator–Prey
GDP—Import	Competition	Competition
GDP—Investment	Predator–Prey	Competition
Export—Import	Prey–Predator	Prey–Predator
Export—Investment	Mutualism	Prey–Predator
Import—Investment	Predator–Prey	Competition

.

In establishing the type of relationship between the pairs of indicators by using the $GLV\left(4\right)$ model, the impact of the evolution of the other indicators is also taken into account. Thus, analyzing of the types of relationships between indicators with the $GLV\left(4\right)$ model for the two periods, the following are found:

-

Between the GDP—Export pair, for both periods there is a Predator–Prey relationship, that is, GDP benefits from Export;

-

Between GDP—Import there is a competition relationship for both periods. This type of relationship shows that the two indicators coexist, both can lead to economic growth in the conditions where Import are used for economic development (add value to domestic production);

-

The pair GDP—Investment exhibits a Predator–Prey relationship for the shorter period, respectively, GDP can benefit from the Investment. For the period of 18 years, it can be observed that there is a Competition-type relationship, thus making them coexist;

-

Between the Export—Import pair there are Prey–Predator relationships in both periods. Thus, the Import benefits from the realization of Export, because a greater source of income from Export can be used for the realization of Import;

-

The relationship between Export—Investment for the period 2011–2022 is of Mutualism type. This demonstrates the mandatory relationship between the two indicators, as a result of which both indicators are positively affected, not being able to exist separately. For the period 2005–2022, the relationship is of the Prey–Predator type, the respective Investment benefit from Export;

-

Between the pair Import—Investment for the period of 12 years there is a Predator–prey relationship, Investment may suffer as a result of the allocation of funds for Import. For the longer period (18 years) the 2 indicators are in a competition relationship, which shows that both indicators need funds for development and the best solution is to coexist.

4.4.4. The Accuracy of the Estimations

To assess the accuracy of the models, $MAPE$ and $MSE$ were used. According to

Table 13. Values for MAPE and MSE.

Model	Year	Estimated Values—GDP		Estimated Values—Export		Estimated Values—Import		Estimated Values—Investment
		2011–2022	2005–2022	2011–2022	2005–2022	2011–2022	2005–2022	2011–2022	2005–2022
$GM\left(\mathrm{1,1}\right)$	$MAPE$	3.04783	3.37139	5.63536	6.12806	6.8327	7.33507	4.53289	7.11028
	$MSE$	0.25270	0.23043	0.24719	0.20482	0.29551	0.24693	0.02329	0.04356
$GLV\left(4\right)$	$MAPE$	11.328	10.4777	12.0394	13.6619	12.0123	14.2087	11.5727	11.2306
	$MSE$	3.536	3.081	1.033	0.883	0.924	0.878	0.161	0.163

, all $MAPE$ values are in the interval [3,8] when using the $GM\left(\mathrm{1,1}\right)$ model (so inferior to 10), thus the $GM(\mathrm{1,1}$) model shows “High accuracy”.

The $MAPE$ coefficients for the estimations obtained with the $GLV\left(4\right)$ model show a “Good accuracy”, from theoretical point of view (as $MAPE<20$). Nevertheless, the expected values for the year 2023 predicted by the two simulations are quite different and far from the values predicted by the $GM\left(\mathrm{1,1}\right)$ model. This shows that, for this set of economic indicators, the $GLV\left(4\right)$ model is not quite appropriate for forecast. Yet, it offers good indications between the interdependence between indicators.

Regarding the $MSE$ analysis, it can be seen that the values for the 18-year period are lower, for all indicators, except for Investment, which means that the accuracy of the model is improving when larger time-series data are considered.

The $MAPE$ and $MSE$ values prove the level of accuracy of the proposed model for establishing the types of relationships between indicators.

4.5. Comparison between the Types of Relationships Identified with GLV(2) vs. GLV(4)

Comparing the type of relationships identified by the $GLV\left(2\right)$ model and the $GLV\left(4\right)$ model estimations, the following is found.

-

A Predator—Prey relationship was identified between GDP and Export, with both models and for both periods. Thus, it is found that Export have a positive impact on economic growth, results also found by other authors [38,39]. Through the $GLV\left(4\right)$ model, it is found that the increase in Export accelerates not only economic growth, but also the ability to Import and make Investment, results also found by other studies [34,41,49].

-

In what concerns the GDP—Import pair, the $GLV\left(2\right)$ model shows a Predator–Prey relationship for both periods, which confirms the influence of Import on economic growth [34]. The estimations obtained using the $GLV\left(4\right)$ model lead to a Competition-type relationship, which demonstrates that the evolution of the other indicators influences the relationship between economic growth and Import [34,47].

-

For the GDP and Investment pair, a Predator–Prey relationship is identified for the 12-year period by both models. This confirms the important role of Investment in production functions, providing the necessary capital for development [16,20,21,22]. For the period of 18 years, the type of relationship identified with the $GLV\left(4\right)$ model is Competition, which proves that in the long term the coexistence between economic growth and Investment is also influenced by the evolution of Import and Export, results confirmed by the results of other works [38,40,43,45]. There is thus a mutually beneficial relationship between GDP and long-term Investment [1].

-

The relationship between the Export and Import indicators for the period 2011–2022 using the $GLV\left(2\right)$ model is of Predator–Prey type. For the 18-year period, this relationship is of the Prey–Predator type, being identical to the one resulting when applying the $GLV(4$) model for both periods. The existence of this type of relationship shows that for longer periods, Export can be affected by Import, results also found by other studies [62]. At the same time, according to the $GLV\left(4\right)$ model, the relationship is also influenced by the evolution of Investment and GDP. Import are important factors for Export, Export can accelerate the realization of Import in the long-term facilitating the realization of Investment and the growth of GDP [34].

-

The type of relationship between Export—Investment pair for the analyzed period of 12 years with the $GLV\left(2\right)$ model is Predator–Prey, and with $GLV\left(4\right)$ the type of relationship is Mutualism, which proves that the other indicators compete for the existence of a positive relationship between Export and Investment. For the period 2005–2022 with both models, a Prey–Predator relationship between the indicators is identified, demonstrating that both the studied period and the other indicators influence the type of relationship, Export being affected by the evolution of Investment. Thus, Investment have an important impact on Export potential, results confirmed by other authors [17,23].

-

Regarding the type of relationship between Import—Investment, there are major differences between the results of the two models. If, following the $GLV\left(2\right)$ model, the relationship is of the Prey–Predator type for both periods, applying the $GLV\left(4\right)$ model, the results are different. For the period 2011–2022 there is a relationship of type Predator–Prey, while for 2005–2022 the relationship is of type Competition. It is found, as in other studies, that in the long term the coexistence between Import and Investment is also influenced by the evolution of Export and economic growth [49].

The results of empirical estimations obtained using the $GLV\left(4\right)$ model confirm the results of other research studies. According to the study presented in this paper, there is a non-linear relationship between financial development and economic growth. The findings are in accordance with the research provided by [21,25,28,29,30,32,54] who indicated that Investment has a positive and direct impact on economic growth, and greater openness to international trade can significantly influence economic growth. At the same time, Investment and Export lead to economic growth, which is in agreement with other researches [18,38,42]. Import have a significant effect on economic growth [48] and GDP growth stimulates Export growth [47].

The results show that, compared to other models, the generalized $GLV\left(s\right)$ model can describe more accurately the type of nonlinear relationships at the level of macroeconomic indicators, thus improving the prediction accuracy of real data series. At the same time, due to the good interpretability of the model, the relationship between the indicators can be analyzed from a qualitative point of view.

5. Conclusions

The relevance of the study consists in the fact that it adds value to existing research by achieving two main objectives.

Firstly, a generalization of the $GLV\left(2\right)$ model was realized, by applying the grey method to a $s$-dimensional quadratic Lotka–Volterra system. The generalized grey Lotka–Volterra model can be applied for an arbitrary number of variables in any field.

Secondly, the accuracy and applicability of the model was tested in a particular case, using four economic indicators. In this study, the reliability of the model was tested by identifying the significance of the relationships between four macroeconomic indicators, namely, GDP, import, export and investment.

The $GLV\left(4\right)$ model and, more generally, the $GLV\left(s\right)$ model provide a new methodological research paradigm for identifying the relationship between economic indicators.

This research presents a number of practical and theoretical implications that may be relevant.

5.1. Theoretical Implications

Various research studies have shown that grey Lotka–Volterra models are effective and suitable for handling difficult problems, possessing the ability to adapt to uncertainty and insufficient information needed for decision making. Starting from the fact that in the real world several variables interact, in this study a generalization of the two-dimensional grey Lotka Volterra model was performed.

The new derived model was tested on a system of four macroeconomic indicators. This research used the proposed multidimensional grey Lotka–Volterra model, designed to analyze the effect of long-term competition and cooperation between the four considered indicators. The choice was motivated by the fact that in recent years the economic development process has been marked by uncertainty (the COVID-19 pandemic, the Russian–Ukrainian war) more than ever, which makes the accuracy of establishing the relationships between the indicators critical.

The study has a series of theoretical implications arising from the identification of the relationship between the four macroeconomic indicators at the EU level, based on the generalized grey Lotka–Volterra model applied in this particular case.

Much of the specialized literature was oriented towards identifying the relationship between GDP and investment or GDP and international trade, or GDP and export and investment, but they did not study the relationship between the four macroeconomic indicators. At the same time, research has focused on determining the role of investments or international trade on GDP using statistical methods or neural networks in most cases.

The current research contributes to the specialized literature by identifying cooperative or competitive relationships between the studied indicators. Previous studies have only highlighted the existence of some relationships between some of the indicators and the role they play in economic development.

This research adds value to existing research by approaching the $GLV$(4), which identifies the meaning of the relationship between indicators, this being a novelty in the specialized literature. This study has the potential to change the perspectives of EU countries regarding GDP, Import, Export and Investment at the national and international level, and can be a valuable guide for researchers.

5.2. Practical Implications

We believe that the study has a series of practical implications that can be relevant considering that the analyzed indicators are of significant importance for the European Union to achieve a significant development that ensures a better future for community members.

The practical implications of the empirical research at the EU level confirmed that over different periods of time GDP, Investment, Import and Export can have different, positive or negative influences on each other. Thus, the results of the study can be used for other research, having an important role in establishing economic policy strategies at the national and implicitly European level to stimulate economic growth.

The 2019 coronavirus pandemic has caused declines in the values of the analyzed macroeconomic indicators, so the accuracy of forecasts and relationships between indicators based on existing data may not be the most reliable. In this sense, the accuracy of the identification of the types of relationships made on the basis of $GLV\left(4\right)$ is noteworthy. Due to the good interpretability of the $GLV\left(4\right)$ model, the relationship between different types of economic indicators can be analyzed qualitatively.

The experimental results of analyzing GDP, Import, Export and Investment show that the accuracy of the $GLV\left(4\right)$ model is high and the results are reasonable. The type of relationships established between indicators with this model shows their interdependence.

The results of this study could play an essential role in establishing economic policy strategies at the European level to stimulate economic growth and to implement stimulus programs at the level of the economic development of all member states.

In order for the EU and its member countries to establish optimal development plans, a clear understanding of the relationship between economic indicators and their development trend is essential. In this sense, the results of the analysis made with the $GLV\left(4\right)$ model could guide government decision making.

The findings from this study have the potential to indirectly affect the economic performance of the European Union and implicitly the member states and to increase the perspective of decision makers related to investments and trade. Identifying relationships between the studied variables can guide decision makers in planning strategic actions. Thus, based on the type of relationships (Predator–Prey, Prey–Predator, Competition or Mutualism) identified between the economic indicators, a series of political implications for international trade and investments can be determined. In this sense, at the level of the European Union, the importance of exports in its sustainable economic growth must still be taken into account. The political factors at the level of the member states of the European Union should continuously exploit the implications of the export expansion strategy in the other states on the other continents for the perspectives of economic development. Trade remains an engine of economic growth, and domestic demand and consumption effectively drive economic prosperity. In the face of potentially negative shocks or economic threats (the COVID-19 pandemic, the Russian–Ukrainian war), state governments must focus on both domestic and international circulation. This research also highlights the importance of developing investor-friendly policies to attract investors to the country. Therefore, this research indirectly encourages decisions that can make countries more competitive and productive.

5.3. Limitations and Further Developments

The limitation of this study is that the research sample is limited to the European Union, and the situation of the member countries can show significant differences with each other and with the states of other continents. At the same time, only four were considered macroeconomic indicators in the generalized grey Lotka–Volterra model.

Consequently, in the future we aim to apply the multidimensional model to a larger sample of countries. Moreover, this research model can be enriched by adding other variables. We encourage other researchers to conduct research by considering multiple quantitative indicators related to the state of the economy.