# Research on the Synergy Measurement for Wetland Ecological-Economic-Social Composite System Based on Fractional Order Dynamic System

Authors: Zhanyong Jin; Huanhuan Cao; Shuang Xia; Qingyue Liu; Menglin Jia; Rui Wang
Journal: Discrete Dynamics in Nature and Society
 (2022)

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## Abstract

For researching the internal structure characteristics and its three subsystems’ evolution of the wetland ecological-economic-social composite system and getting the point that makes wetland ecology, production, and living synergistically develop, in this paper, the coevolution model of wetland complex system based on fractional order dynamic system was established, the genetic algorithm was used to solve the model parameters, and the equilibrium stability of the model was tested by using asymmetric fractional order equilibrium condition. The data of 29 provinces, cities, and districts in China were selected for empirical research, and suggestions for improvement were put forward. The results show that there is a significant nonlinear relationship among the three subsystems, and the stability points exist and change dynamically. This shows that, in practice, only when the local government issues relevant policies, can we achieve the benign coupling and coordinated development of the wetland ecological-economic-social system and achieve the harmonious development of human and nature.


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## Body

## 1. Introduction

Forests, grasslands, rivers, and wetlands are green treasures endowed by nature. As an important part of the three terrestrial ecosystems, wetland is one of the most productive ecosystems on Earth, with irreplaceable ecological functions such as flood control, runoff regulation, pollution control, and climate regulation [1]. However, the destruction of wetland ecology by production and living around wetland not only results in the loss of its ecological function, but also poses a serious threat to the prevention of natural disasters and the safety of industrial and agricultural production in wetland area. In this case, we believe that, in the era of ecological civilization, it is necessary to correctly understand the relationship between ecosystem, economic system, and social system and consider the interactive stress feedback mechanism of wetland eco-economy-social complex system. Only by formulating models and programs for effectively protecting wetland ecology, rationally allocating wetland resources, and reasonably arranging production and life around wetlands can we realize the coupling and coordination of wetland resource protection and utilization, realize the sustainable and healthy development of wetland composite system, and realize the organic unity of wetland economic construction, social construction, and ecological civilization construction.In recent years, scholars from all over the world have carried out research on the codevelopment of wetland ecosystem, economic system, and social system from different disciplines and based on different research directions. The research content covers a wide range and is highly comprehensive. Based on the research direction of wetland health and ecological restoration, scholars analyzed and studied wetland hydrology and water environment, wetland plant community and vegetation, and wetland ecosystem management and evaluation [2, 3]. Based on the research direction of ecosystem service function evaluation, this paper studies how to use model simulation to evaluate wetland ecosystem service in different situations in the future to determine the best management mode [4, 5]. Collaborative development system based on wetland system research direction, research between ecosystem and economic system of the coordinated development of coupling relationship, uses the evaluation index system and a variety of quantitative model to measure evaluation system coupling between collaborative relationship and make the coordinated development of ecological economic research methods constantly improve, with extension of research, from single factor to comprehensive factor, from structural recovery to functional recovery, from problem diagnosis to ecosystem function response and system assessment, with the research trend from two subsystems to multiple subsystems coupling collaborative research transition [6–8]. However, most of the existing literature on the coupling and codevelopment of wetland system focuses on the qualitative or quantitative study of the relationship between ecosystem and economic system and lack of relevant studies on the internal structure characteristics of the complex system, the coevolution model of its three subsystems, and the development and coevolution mechanism.For the study of the issue of synergistically developing wetland ecology, production, and living, the author has constructed wetland ecological-economic-social composite system. In another paper of research on the synergy measurement for wetland ecology, economy, and society composite system based on order parameter, the measurement model of subsystem order parameters and composite system synergetic degree were constructed, and the empirical research was carried out based on the data of 29 provinces and cities in China. Besides, the general nonlinear fractional equation of fractional dynamic system was introduced in this paper, and a coevolution model of wetland complex system was established to study the dynamic evolution characteristics of wetland complex system [9]. This method can better reflect the historical dependence and codevelopment process of system function evolution. In this paper, the parameters of the model are solved by genetic algorithm, and the equilibrium points of the model are obtained. Based on the data of 29 provinces and cities in China, this paper empirically studied the competition and cooperation among the economic, social, and ecological subsystems of wetlands. The key points of wetland ecological-economic-social complex system coordinated development were obtained. On the premise of maintaining the balance of the ecosystem, the paper puts forward some suggestions for the better development of the economic system and social system.


## 2. Feature Analysis of Wetland Ecology, Economy, and Society Composite System

The wetland ecological-economic-social system is studied as a composite system because of its general nature that a system has. In general, the wetland composite system is featured by causality, multiple feedback property, nonlinearity, and system inertia [10].


### 2.1. Causality

The causality of wetland composite system is the foundation for analyzing the wetland ecological-economic-social composite system, as well as its basic laws. Applying external forces on any object in a system will produce certain changes and effects; conversely, any result in a system is driven by some force, or some reason. In the system analysis, it is necessary to figure out and understand what the cause is and what the effect is and whether the effect is positive or negative. Besides, the causality in this composite system includes two types: “one cause, many effects” and “one effect, many causes,” and the duration of causation can also be divided into long-term and short-term. For example, the damage to the ecological environment of wetlands may come from industries of the subsystem economy, while partly from the discharge of domestic pollutants. Economic development not only improves living standards but also facilitates technological transformation and innovation, thus contributing to the protection of ecological environment. Therefore, the analysis of the wetland ecological-economic-social composite system must combine its current situation and the interactions between three subsystems and the duration of those interactions.


### 2.2. Multiple Feedback Property

If there is a causal relationship between the internal structure of a composite system or among subsystem elements, it is called existence feedback relationship, which can be divided into positive feedback and negative feedback in terms of the interaction coefficients of feedback path [11]. Positive feedback strengthens the behavior among systems while negative feedback weakens such behavior and makes it tend to be stable [12]. Through the cycle of positive and negative feedback relationship, the system can evolve and develop, finally being in a stable state. In addition, there is also multiple feedback relationship [13].For the wetland ecological-economic-social composite system, the economic subsystem, social subsystem, and ecological subsystem interact with each other, reflecting the obvious property of multiple feedback. For example, the development of the economic subsystem will promote the construction of social infrastructure and the development of education, culture, and health, whereas it will also generate industrial pollutant emissions into the wetland ecological subsystem, leading to the deterioration of the ecological subsystem. In turn, the decrease of wetland ecological resources will limit the development of economic subsystem, while the improvement of wetland ecological environment will promote the development of society. Furthermore, such relationship also exists in the same subsystem. Taking the economic subsystem as an example, the increase in capital stock and employment will increase GNP and thus R&D investment, which in turn will promote technological innovation, boosting economic development [14]. Therefore, fully considering the multiple feedback property within the composite system and among subsystem elements is central to studying the wetland composite system.


### 2.3. System Nonlinearity

Nonlinearity, which can be found within a system, is one of the basic features of the composite system. The nonlinearity here includes two aspects: first, the nonlinearity of the causality in interaction between subsystems of the composite system; second, the nonlinearity among the elements of the respective systems and between the inputs and outputs. In reflecting the complexity of this system, nonlinearity has a very distinct advantage over linearity. The wetland ecological-economic-social system itself is a complicated composite system, so fully understanding such relationship between the internal structures of the system will play an important role in evaluating its efficiency. Taking economic subsystem and ecological subsystem as an example, it is inappropriate to simply assume that there is a linear relationship between these two subsystems because the economic development boosts the improvement of industrial production technologies, and management models are constantly adapting to the needs of production. Therefore, as economy grows by leaps and bounds, the energy consumption per unit of GDP is gradually decreasing, rather than being a constant. In conclusion, the study of the wetland composite system should take nonlinearity into account and grasp such relationship of this composite system through data analysis.


### 2.4. System Inertia

Originating from physics, the concept of inertia refers to the ability of an object to maintain its original form of motion. The inertia feature, which is generally expressed as a “force of habit” [15], is widespread in a system. As a result of inertia, changes in a system require stronger external forces. In terms of this composite system, the traditional economic growth model has resulted in serious waste of ecological resources and weak environmental protection [16]. Now, with the “two-oriented society” (the society of frugal-resource and friendly environment) and “ecological civilization construction” proposed, the transformation of the existing extensive production model requires knowing the current situation of the low industrial production technology and backward management mode in China. Therefore, achieving the goal of “ecological civilization construction” has a long way to go, rather than overnight.


## 2.1. Causality

The causality of wetland composite system is the foundation for analyzing the wetland ecological-economic-social composite system, as well as its basic laws. Applying external forces on any object in a system will produce certain changes and effects; conversely, any result in a system is driven by some force, or some reason. In the system analysis, it is necessary to figure out and understand what the cause is and what the effect is and whether the effect is positive or negative. Besides, the causality in this composite system includes two types: “one cause, many effects” and “one effect, many causes,” and the duration of causation can also be divided into long-term and short-term. For example, the damage to the ecological environment of wetlands may come from industries of the subsystem economy, while partly from the discharge of domestic pollutants. Economic development not only improves living standards but also facilitates technological transformation and innovation, thus contributing to the protection of ecological environment. Therefore, the analysis of the wetland ecological-economic-social composite system must combine its current situation and the interactions between three subsystems and the duration of those interactions.


## 2.2. Multiple Feedback Property

If there is a causal relationship between the internal structure of a composite system or among subsystem elements, it is called existence feedback relationship, which can be divided into positive feedback and negative feedback in terms of the interaction coefficients of feedback path [11]. Positive feedback strengthens the behavior among systems while negative feedback weakens such behavior and makes it tend to be stable [12]. Through the cycle of positive and negative feedback relationship, the system can evolve and develop, finally being in a stable state. In addition, there is also multiple feedback relationship [13].For the wetland ecological-economic-social composite system, the economic subsystem, social subsystem, and ecological subsystem interact with each other, reflecting the obvious property of multiple feedback. For example, the development of the economic subsystem will promote the construction of social infrastructure and the development of education, culture, and health, whereas it will also generate industrial pollutant emissions into the wetland ecological subsystem, leading to the deterioration of the ecological subsystem. In turn, the decrease of wetland ecological resources will limit the development of economic subsystem, while the improvement of wetland ecological environment will promote the development of society. Furthermore, such relationship also exists in the same subsystem. Taking the economic subsystem as an example, the increase in capital stock and employment will increase GNP and thus R&D investment, which in turn will promote technological innovation, boosting economic development [14]. Therefore, fully considering the multiple feedback property within the composite system and among subsystem elements is central to studying the wetland composite system.


## 2.3. System Nonlinearity

Nonlinearity, which can be found within a system, is one of the basic features of the composite system. The nonlinearity here includes two aspects: first, the nonlinearity of the causality in interaction between subsystems of the composite system; second, the nonlinearity among the elements of the respective systems and between the inputs and outputs. In reflecting the complexity of this system, nonlinearity has a very distinct advantage over linearity. The wetland ecological-economic-social system itself is a complicated composite system, so fully understanding such relationship between the internal structures of the system will play an important role in evaluating its efficiency. Taking economic subsystem and ecological subsystem as an example, it is inappropriate to simply assume that there is a linear relationship between these two subsystems because the economic development boosts the improvement of industrial production technologies, and management models are constantly adapting to the needs of production. Therefore, as economy grows by leaps and bounds, the energy consumption per unit of GDP is gradually decreasing, rather than being a constant. In conclusion, the study of the wetland composite system should take nonlinearity into account and grasp such relationship of this composite system through data analysis.


## 2.4. System Inertia

Originating from physics, the concept of inertia refers to the ability of an object to maintain its original form of motion. The inertia feature, which is generally expressed as a “force of habit” [15], is widespread in a system. As a result of inertia, changes in a system require stronger external forces. In terms of this composite system, the traditional economic growth model has resulted in serious waste of ecological resources and weak environmental protection [16]. Now, with the “two-oriented society” (the society of frugal-resource and friendly environment) and “ecological civilization construction” proposed, the transformation of the existing extensive production model requires knowing the current situation of the low industrial production technology and backward management mode in China. Therefore, achieving the goal of “ecological civilization construction” has a long way to go, rather than overnight.


## 3. Building a Coevolution Model for Wetland Composite System Based on Fractional Order Dynamic System

### 3.1. Correlation Theoretical Analysis

Based on the general nonlinear fractional order system model of fractional order differential dynamic system, this paper studies and constructs the coevolution model of wetland complex system. In recent years, the theory of fractional calculus has been widely concerned by more and more scholars, which has achieved great success in many natural disciplines such as fractal dynamics, fluid mechanics, physics, bionics, and so on. Fractional differential equation is a generalization of the classical integral differential equation by applying fractional calculus [17]. As an important part of the qualitative theory of differential equations, fractional order differential equation is more accurate than integer order differential equation to describe the microscopic world and reflect some properties of nature, and the fractional order differential to describe the process of memory and genetic properties of different materials provides a very effective means, in terms that material performance of the model has a good advantage. In many scientific fields, such as fractal, biomechanics, engineering, automatic control, physics, and so on, all involve the application of fractional differential equations. Especially for physics and engineering problems related to fractal dimension, biological system related research, etc., fractional differential equation can more accurately describe the law of change and essential attributes of things.The study of wetland composite systems should consider nonlinearity and emphasize historical inheritance, multiple feedback, nonlinearity, and dynamic evolution, while the fractional order system composed of fractional order equations has memory function and is more stable, which can better reflect the historical dependence process of system function development [18]. Therefore, it has significant advantages to introduce a general nonlinear fractional order system model of fractional order dynamic system to study the dynamic evolution characteristics of wetland composite systems. Based on relevant experience, the stability of fractional order differential equations with unknown parameters is studied, and the equilibrium point of the model is analyzed to evolve to a higher-level development state [19]. Facts have proved that, with the change of fractional order, the wetland ecological economic and social composite system has a certain evolution method and achieves dynamic balance.


### 3.2. Model Construction of Wetland Composite System

The coevolutionary process of the wetland ecological-economic-social composite system will undergo a dynamic evolutionary process of birth, growth, maturity, and death and eventually form a balanced state at a certain level of development. Its trajectory is in line with the S-Curve [20] and can be described by the logistic growth model [21, 22]. Based on this model, the following coevolution model for wetland composite system is constructed to analyze the synergy measurement of its three subsystems: society, economy, and ecology.The settings represent wetland ecological subsystem, economic subsystem, and social development subsystem, respectively, and in order to portray the competition and cooperation relationship of this composite system composed of three subsystems in the process of coevolution, parameters are introduced, which are called subsystem-to-subsystem influence coefficients and then a coevolution model for the wetland composite system is obtained as follows:(1)dX1dt=α1X11−X1−β12X2−β13X3,dX2dt=α2X21−X2−β21X1−β23X3,dX3dt=α3X31−X3−β31X1−β32X2.The above formulas indicate the order degree and multiplication factors of three subsystems, respectively, reflecting the development degree of each subsystem in the wetland composite system.The model parameters are analyzed as follows: (1) Ifαii=1,2,3>0, the subsystem is in a state of evolution as a whole; if αii=1,2,3<0, the subsystem i is in a state of degradation. (2) When βiji,j=1,2,3>0, it means that the subsystem j and the subsystem i are in competition with each other, and the evolution of the subsystem j is not conducive to the development of the subsystem; when βiji,j=1,2,3<0, it means that the subsystem j and the subsystem i are in cooperation with each other, and the evolution of the subsystem j is conducive to the development of the subsystem i.A general nonlinear fractional order system model based on fractional order differential dynamic system [23]:(2)Dtα10xt=c1+a11xt+a12yt+a13zt+a14xtyt+a15xtzt+a16ytzt+a17xt2+a18yt2+a19zt2,Dtα10yt=c2+a21xt+a22yt+a23zt+a24xtyt+a25xtzt+a26ytzt+a27xt2+a28yt2+a29zt2,Dtα10zt=c3+a31xt+a32yt+a33zt+a34xtyt+a35xtzt+a36ytzt+a37xt2+a38yt2+a39zt2.In the model above,xt,yt,zt represent the order parameter of economic subsystem, social subsystem, and ecological subsystem, respectively. For simplicity of expression, definitions are made as follows:(3)g1xt,yt,zt,A1=c1+a11xt+a12yt+a13zt+a14xtyt+a15xtzt+a16ytzt+a17xt2+a18yt2+a19zt2,g2xt,yt,zt,A2=c2+a21xt+a22yt+a23zt+a24xtyt+a25xtzt+a26ytzt+a27xt2+a28yt2+a29zt2,g3xt,yt,zt,A3=c3+a31xt+a32yt+a33zt+a34xtyt+a35xtzt+a36ytzt+a37xt2+a38yt2+a39zt2,Ai=ci,ai1,ai2,⋯,ai9,i=1,2,3.Then, model (2) can be expressed as(4)Dtα10xt=g1xt,yt,zt,A1,Dtα10yt=g2xt,yt,zt,A2,Dtα10zt=g3xt,yt,zt,A3.


## 3.1. Correlation Theoretical Analysis

Based on the general nonlinear fractional order system model of fractional order differential dynamic system, this paper studies and constructs the coevolution model of wetland complex system. In recent years, the theory of fractional calculus has been widely concerned by more and more scholars, which has achieved great success in many natural disciplines such as fractal dynamics, fluid mechanics, physics, bionics, and so on. Fractional differential equation is a generalization of the classical integral differential equation by applying fractional calculus [17]. As an important part of the qualitative theory of differential equations, fractional order differential equation is more accurate than integer order differential equation to describe the microscopic world and reflect some properties of nature, and the fractional order differential to describe the process of memory and genetic properties of different materials provides a very effective means, in terms that material performance of the model has a good advantage. In many scientific fields, such as fractal, biomechanics, engineering, automatic control, physics, and so on, all involve the application of fractional differential equations. Especially for physics and engineering problems related to fractal dimension, biological system related research, etc., fractional differential equation can more accurately describe the law of change and essential attributes of things.The study of wetland composite systems should consider nonlinearity and emphasize historical inheritance, multiple feedback, nonlinearity, and dynamic evolution, while the fractional order system composed of fractional order equations has memory function and is more stable, which can better reflect the historical dependence process of system function development [18]. Therefore, it has significant advantages to introduce a general nonlinear fractional order system model of fractional order dynamic system to study the dynamic evolution characteristics of wetland composite systems. Based on relevant experience, the stability of fractional order differential equations with unknown parameters is studied, and the equilibrium point of the model is analyzed to evolve to a higher-level development state [19]. Facts have proved that, with the change of fractional order, the wetland ecological economic and social composite system has a certain evolution method and achieves dynamic balance.


## 3.2. Model Construction of Wetland Composite System

The coevolutionary process of the wetland ecological-economic-social composite system will undergo a dynamic evolutionary process of birth, growth, maturity, and death and eventually form a balanced state at a certain level of development. Its trajectory is in line with the S-Curve [20] and can be described by the logistic growth model [21, 22]. Based on this model, the following coevolution model for wetland composite system is constructed to analyze the synergy measurement of its three subsystems: society, economy, and ecology.The settings represent wetland ecological subsystem, economic subsystem, and social development subsystem, respectively, and in order to portray the competition and cooperation relationship of this composite system composed of three subsystems in the process of coevolution, parameters are introduced, which are called subsystem-to-subsystem influence coefficients and then a coevolution model for the wetland composite system is obtained as follows:(1)dX1dt=α1X11−X1−β12X2−β13X3,dX2dt=α2X21−X2−β21X1−β23X3,dX3dt=α3X31−X3−β31X1−β32X2.The above formulas indicate the order degree and multiplication factors of three subsystems, respectively, reflecting the development degree of each subsystem in the wetland composite system.The model parameters are analyzed as follows: (1) Ifαii=1,2,3>0, the subsystem is in a state of evolution as a whole; if αii=1,2,3<0, the subsystem i is in a state of degradation. (2) When βiji,j=1,2,3>0, it means that the subsystem j and the subsystem i are in competition with each other, and the evolution of the subsystem j is not conducive to the development of the subsystem; when βiji,j=1,2,3<0, it means that the subsystem j and the subsystem i are in cooperation with each other, and the evolution of the subsystem j is conducive to the development of the subsystem i.A general nonlinear fractional order system model based on fractional order differential dynamic system [23]:(2)Dtα10xt=c1+a11xt+a12yt+a13zt+a14xtyt+a15xtzt+a16ytzt+a17xt2+a18yt2+a19zt2,Dtα10yt=c2+a21xt+a22yt+a23zt+a24xtyt+a25xtzt+a26ytzt+a27xt2+a28yt2+a29zt2,Dtα10zt=c3+a31xt+a32yt+a33zt+a34xtyt+a35xtzt+a36ytzt+a37xt2+a38yt2+a39zt2.In the model above,xt,yt,zt represent the order parameter of economic subsystem, social subsystem, and ecological subsystem, respectively. For simplicity of expression, definitions are made as follows:(3)g1xt,yt,zt,A1=c1+a11xt+a12yt+a13zt+a14xtyt+a15xtzt+a16ytzt+a17xt2+a18yt2+a19zt2,g2xt,yt,zt,A2=c2+a21xt+a22yt+a23zt+a24xtyt+a25xtzt+a26ytzt+a27xt2+a28yt2+a29zt2,g3xt,yt,zt,A3=c3+a31xt+a32yt+a33zt+a34xtyt+a35xtzt+a36ytzt+a37xt2+a38yt2+a39zt2,Ai=ci,ai1,ai2,⋯,ai9,i=1,2,3.Then, model (2) can be expressed as(4)Dtα10xt=g1xt,yt,zt,A1,Dtα10yt=g2xt,yt,zt,A2,Dtα10zt=g3xt,yt,zt,A3.


## 4. Model Solving Based on DNLPSO Algorithm and Equilibrium Point Stability Analysis

### 4.1. Solution of Model Parameters Based on DNLPSO Algorithm

In order to estimate the parameters of model (2), approximate estimation of the fractional order is performed by the method of numerical computational. Combining Podlubny [21]and the short-term memory principle, the explicit approximate solution of the GL fractional order derivative is(5)Dtkαk−Lm/hft≈h−α∑i=0kciαftk−i,in which Lm is the memory length; ciα is the binomial coefficient satisfying c0α=1,ciα=1−1+α/ici−1α. At this point, the numerical approximate solution of the fractional order system model (4) is(6)xtk=g1xtk,ytk,ztk,A1hα1−∑j=v1kcjα1xtk−j,ytk=g2xtk,ytk,ztk,A2hα2−∑j=v1kcjα2ytk−j,ztk=g3xtk,ytk,ztk,A3hα3−∑j=v3kcjα3ztk−j,0<αi<1,2,3.Whenk<Lm/h,vi=1; when k≥Lm/h,vi=k−Lm/h.Assume thatxt0,yt0,zt0t=1,2,…,T denotes the sequence of true values, the following optimization problem is constructed to describe the trajectory of the system in a more refined way.(7)minHα,A=∑t=1T−1xt−xt+102+yt−yt+102+zt−zt+102,xtk=g1xtk,ytk,ztk,A1hα1−∑j=v1kcjα1xtk−j,ytk=g2xtk,ytk,ztk,A2hα2−∑j=v1kcjα2ytk−j,ztk=g3xtk,ytk,ztk,A3hα3−∑j=v3kcjα3ztk−j,0<αi<1,2,3,in which α=α1,α2,α3;A=A1,A2,A3;xt0=x10,yt0=y10,zt0=z10.Then, considering that the model above is a nonlinear equation, it is not easy for the commonly applied methods for solving the model parameters, such as least square method and maximum likelihood estimate, to achieve satisfying accuracy. Moreover, the maximum likelihood estimate method requires the exact distribution that the parameters obey. Based on this, this project plans to solve the model parameters by applying the modified particle swarm algorithm method [24] proposed by Huang et al. The dynamic neighbor and local search-based particle swarm algorithm (DNLPSO) [25] can overcome the shortcomings of the traditional PSO algorithm which is liable to fall into local optimal solution and prematurity and has advantages such as less computation, high accuracy, and simple parameter setting. Assume that the general optimization problem is(8)minfθ1,θ2,…,θqaj≤θj≤bjj=1,2,…,q,in which θj is the set of q variables; aj,bj is the initial variation interval; and f is the nonnegative optimization criterion function. In the process of iterative algorithm, n historical optimal solutions of all particles are randomly selected as the starting point of the Quasi-Newton Method every L generation, and then local search is performed so as to generate m optimal solutions. Finally, the equivalent number of historical optimal solutions is submitted by these m optimal solutions. The experiment is repeated until the optimal criterion function of the optimal individual is lower than one set value or a set number of cycles.


### 4.2. Equilibrium Point Stability Analysis

After solving the model parameters with the method of genetic algorithm, the stability of the model is analyzed as follows: if the equilibrium points obtained after solving the system of (4) are noted as Q∗=x∗,y∗,z∗, then the Jacobi matrix at each equilibrium point obtained is as follows:(9)J=j11j12j13j21j22j23j31j32j33=∂g1∂x∂g1∂y∂g1∂z∂g2∂x∂g2∂y∂g2∂z∂g3∂x∂g3∂y∂g3∂z.Assume that the order of model (4) is α^=α^1,α^2,α^3, and α^i=vi/uii=1,2,3. Let m be the least common multiple of ui, and mi=mα^ii=1,2,3. Then the characteristic equation of each equilibrium point corresponding to the Jacobi matrix is(10)λm1+m2+m3−j22λm3+m1−j33λm2+m1−j11λm3+m2+b1λm1+b2λm2+b3λm3+b4=0,in which(11)b1=j22j33−j23j32;b2=j11j33−j13j31;b3=j11j22−j12j21,b4=−j11j22j33−j23j32+j12j33j21−j23j31−j13j21j32−j31j22.The results are analyzed as follows: (1) Denoteω as the root of equation (9); if π/2m−minargω<0, then the equilibrium point is stable. (2) Based on the analysis of the stability of the equilibrium points, the coevolution process and direction of the synergistic development of the wetland ecological-economic-social composite system can be obtained, which means the path and model of achieving the harmonious unification of wetland ecology, production, and life are also obtained.


## 4.1. Solution of Model Parameters Based on DNLPSO Algorithm

In order to estimate the parameters of model (2), approximate estimation of the fractional order is performed by the method of numerical computational. Combining Podlubny [21]and the short-term memory principle, the explicit approximate solution of the GL fractional order derivative is(5)Dtkαk−Lm/hft≈h−α∑i=0kciαftk−i,in which Lm is the memory length; ciα is the binomial coefficient satisfying c0α=1,ciα=1−1+α/ici−1α. At this point, the numerical approximate solution of the fractional order system model (4) is(6)xtk=g1xtk,ytk,ztk,A1hα1−∑j=v1kcjα1xtk−j,ytk=g2xtk,ytk,ztk,A2hα2−∑j=v1kcjα2ytk−j,ztk=g3xtk,ytk,ztk,A3hα3−∑j=v3kcjα3ztk−j,0<αi<1,2,3.Whenk<Lm/h,vi=1; when k≥Lm/h,vi=k−Lm/h.Assume thatxt0,yt0,zt0t=1,2,…,T denotes the sequence of true values, the following optimization problem is constructed to describe the trajectory of the system in a more refined way.(7)minHα,A=∑t=1T−1xt−xt+102+yt−yt+102+zt−zt+102,xtk=g1xtk,ytk,ztk,A1hα1−∑j=v1kcjα1xtk−j,ytk=g2xtk,ytk,ztk,A2hα2−∑j=v1kcjα2ytk−j,ztk=g3xtk,ytk,ztk,A3hα3−∑j=v3kcjα3ztk−j,0<αi<1,2,3,in which α=α1,α2,α3;A=A1,A2,A3;xt0=x10,yt0=y10,zt0=z10.Then, considering that the model above is a nonlinear equation, it is not easy for the commonly applied methods for solving the model parameters, such as least square method and maximum likelihood estimate, to achieve satisfying accuracy. Moreover, the maximum likelihood estimate method requires the exact distribution that the parameters obey. Based on this, this project plans to solve the model parameters by applying the modified particle swarm algorithm method [24] proposed by Huang et al. The dynamic neighbor and local search-based particle swarm algorithm (DNLPSO) [25] can overcome the shortcomings of the traditional PSO algorithm which is liable to fall into local optimal solution and prematurity and has advantages such as less computation, high accuracy, and simple parameter setting. Assume that the general optimization problem is(8)minfθ1,θ2,…,θqaj≤θj≤bjj=1,2,…,q,in which θj is the set of q variables; aj,bj is the initial variation interval; and f is the nonnegative optimization criterion function. In the process of iterative algorithm, n historical optimal solutions of all particles are randomly selected as the starting point of the Quasi-Newton Method every L generation, and then local search is performed so as to generate m optimal solutions. Finally, the equivalent number of historical optimal solutions is submitted by these m optimal solutions. The experiment is repeated until the optimal criterion function of the optimal individual is lower than one set value or a set number of cycles.


## 4.2. Equilibrium Point Stability Analysis

After solving the model parameters with the method of genetic algorithm, the stability of the model is analyzed as follows: if the equilibrium points obtained after solving the system of (4) are noted as Q∗=x∗,y∗,z∗, then the Jacobi matrix at each equilibrium point obtained is as follows:(9)J=j11j12j13j21j22j23j31j32j33=∂g1∂x∂g1∂y∂g1∂z∂g2∂x∂g2∂y∂g2∂z∂g3∂x∂g3∂y∂g3∂z.Assume that the order of model (4) is α^=α^1,α^2,α^3, and α^i=vi/uii=1,2,3. Let m be the least common multiple of ui, and mi=mα^ii=1,2,3. Then the characteristic equation of each equilibrium point corresponding to the Jacobi matrix is(10)λm1+m2+m3−j22λm3+m1−j33λm2+m1−j11λm3+m2+b1λm1+b2λm2+b3λm3+b4=0,in which(11)b1=j22j33−j23j32;b2=j11j33−j13j31;b3=j11j22−j12j21,b4=−j11j22j33−j23j32+j12j33j21−j23j31−j13j21j32−j31j22.The results are analyzed as follows: (1) Denoteω as the root of equation (9); if π/2m−minargω<0, then the equilibrium point is stable. (2) Based on the analysis of the stability of the equilibrium points, the coevolution process and direction of the synergistic development of the wetland ecological-economic-social composite system can be obtained, which means the path and model of achieving the harmonious unification of wetland ecology, production, and life are also obtained.


## 5. Empirical Studies

### 5.1. Theoretical Analysis

In research on the synergy measurement for wetland ecology, economy and society composite system based on order parameter, degree of order subsystem evaluation model and degree of synergy composite system calculation model of the wetland ecological-economic-social composite system have been established, and the empirical studies were conducted based on the data from 29 provinces, cities, and districts in China, based on which, in this section the author will have an analysis of coevolution of the wetland ecological-economic-social composite system according to the fractional order dynamic system model established in the foregoing.In studying the evolution and having stability analysis of wetland ecology-economy, the author holds that it is more appropriate to use the first-order differential of the original data because in the regional development of wetlands, the concept of system stability does not simply stay at a certain level of development but continues growing at a fixed rate.


### 5.2. Data Application and Parameter Analysis

Based on the above analysis, the study variables are taken from the first-order differential terms of the wetland ecological, wetland economic, and wetland social subsystems. Now there are 116 samples. Through the software Matlab and genetic algorithm, the author estimates the parameters of model (9). The results are shown in Table 1.Table 1
Coevolution model parameters estimation.
Parameters to be estimated/subsystemsWetland ecological subsystemsWetland economic subsystemsWetland social subsystemsαi0.5800.4780.495ci0.9111.2071.537ai1−12.955−10.122−67.004ai218.102−8.07951.919ai395.718−86.708−70.295ai4−29.52235.94493.560ai516.289−27.066−57.756ai62.3390.89235.617ai72.533−29.511−44.516ai830.86162.61027.360ai93.525−17.241−33.056The results of the study are shown in Table1. (1) There are significant nonlinear relationships between the three subsystems: wetland ecology, wetland economy, and wetland society. (2) There is a two-way competition between the wetland ecology subsystem and the wetland economy subsystem. Protecting the wetland environment and reducing the industrial pollutant emission to the wetland environment are required for developing wetland ecology in an orderly way [26], which, to some extent, inhibit the development of wetland economy. On the contrary, when economic development technology falls behind, wetland regional economy will inevitably develop at the cost of regional environment, damaging the wetland ecological environment. (3) There is a two-way competition between the wetland economic subsystem and the wetland social subsystem. The reason for this is that the development of regional social life requires a large amount of capital investment by the local government, which suppresses the current capital investment-led regional economic development model. (4) As for the wetland social subsystem and wetland ecological subsystem, the development of the latter hinders the development of the former, but the development of the former promotes the orderly development of the latter. It is because the orderly development of the wetland ecological subsystem blocks the rapid development of the wetland economic subsystem, which in turn makes the local government lack sufficient funds to promote the development of social undertakings; meanwhile, the development of the wetland social subsystem means the improvement of national quality, which will enhance the awareness of wetland protection among residents and the society, promoting the orderly development of the wetland ecological subsystem.


### 5.3. Model Equilibrium Point Solving and Result Analysis

Solve for the equilibrium points of the estimated system of equation (9), and the Jacobi matrices at each equilibrium point are obtained. Next, observe the state of the equilibrium point according to the discriminant criterion of the stable points of the asymmetric fractional order system. The detailed results are shown in Table 2.Table 2
Results of stability of the equilibrium points.
Equilibrium pointsDiscriminant criterionStability(2.637, −7.026, −3.309)0.016Unstable(0.842, 2.103, 1.332)0.016Unstable(−1.609, −2.385, 1.264)−0.012Unstable(0.113, 0.068, 0.070)−0.029Stable(0.592, 0.346, 0.362)−0.025Stable(−5.742, −4.511, −3.150)0.016UnstableThe results of the study are shown in Table2. It is clearly shown in Table 2 that there are six equilibrium points in the asymmetric fractional order system corresponding to the wetland ecological-economic-social composite system, but only the equilibrium points (0.113, 0.068, 0.070) and (0.592, 0.346, 0.362) are stable. (1) It can be concluded that the stable points of the wetland ecological-economic-social composite system are dynamically changing. (2) The continuous changes in external conditions, such as local wetland policies, major environmental events, and economic events, will serve new impacts on the wetland composite system, which in turn form new stimulus for development, pushing the wetland composite system to evolve synergistically towards a new stable point. (3) Through the analysis of the equilibrium points, it is clear that the three subsystems of wetland ecology, economy, and society are in synergistic state, i.e., moving towards a higher orderly development. This indicates that, in practice, wetland ecology, production, and life can synergistically develop with the intervention and adjustment of local policies.


## 5.1. Theoretical Analysis

In research on the synergy measurement for wetland ecology, economy and society composite system based on order parameter, degree of order subsystem evaluation model and degree of synergy composite system calculation model of the wetland ecological-economic-social composite system have been established, and the empirical studies were conducted based on the data from 29 provinces, cities, and districts in China, based on which, in this section the author will have an analysis of coevolution of the wetland ecological-economic-social composite system according to the fractional order dynamic system model established in the foregoing.In studying the evolution and having stability analysis of wetland ecology-economy, the author holds that it is more appropriate to use the first-order differential of the original data because in the regional development of wetlands, the concept of system stability does not simply stay at a certain level of development but continues growing at a fixed rate.


## 5.2. Data Application and Parameter Analysis

Based on the above analysis, the study variables are taken from the first-order differential terms of the wetland ecological, wetland economic, and wetland social subsystems. Now there are 116 samples. Through the software Matlab and genetic algorithm, the author estimates the parameters of model (9). The results are shown in Table 1.Table 1
Coevolution model parameters estimation.
Parameters to be estimated/subsystemsWetland ecological subsystemsWetland economic subsystemsWetland social subsystemsαi0.5800.4780.495ci0.9111.2071.537ai1−12.955−10.122−67.004ai218.102−8.07951.919ai395.718−86.708−70.295ai4−29.52235.94493.560ai516.289−27.066−57.756ai62.3390.89235.617ai72.533−29.511−44.516ai830.86162.61027.360ai93.525−17.241−33.056The results of the study are shown in Table1. (1) There are significant nonlinear relationships between the three subsystems: wetland ecology, wetland economy, and wetland society. (2) There is a two-way competition between the wetland ecology subsystem and the wetland economy subsystem. Protecting the wetland environment and reducing the industrial pollutant emission to the wetland environment are required for developing wetland ecology in an orderly way [26], which, to some extent, inhibit the development of wetland economy. On the contrary, when economic development technology falls behind, wetland regional economy will inevitably develop at the cost of regional environment, damaging the wetland ecological environment. (3) There is a two-way competition between the wetland economic subsystem and the wetland social subsystem. The reason for this is that the development of regional social life requires a large amount of capital investment by the local government, which suppresses the current capital investment-led regional economic development model. (4) As for the wetland social subsystem and wetland ecological subsystem, the development of the latter hinders the development of the former, but the development of the former promotes the orderly development of the latter. It is because the orderly development of the wetland ecological subsystem blocks the rapid development of the wetland economic subsystem, which in turn makes the local government lack sufficient funds to promote the development of social undertakings; meanwhile, the development of the wetland social subsystem means the improvement of national quality, which will enhance the awareness of wetland protection among residents and the society, promoting the orderly development of the wetland ecological subsystem.


## 5.3. Model Equilibrium Point Solving and Result Analysis

Solve for the equilibrium points of the estimated system of equation (9), and the Jacobi matrices at each equilibrium point are obtained. Next, observe the state of the equilibrium point according to the discriminant criterion of the stable points of the asymmetric fractional order system. The detailed results are shown in Table 2.Table 2
Results of stability of the equilibrium points.
Equilibrium pointsDiscriminant criterionStability(2.637, −7.026, −3.309)0.016Unstable(0.842, 2.103, 1.332)0.016Unstable(−1.609, −2.385, 1.264)−0.012Unstable(0.113, 0.068, 0.070)−0.029Stable(0.592, 0.346, 0.362)−0.025Stable(−5.742, −4.511, −3.150)0.016UnstableThe results of the study are shown in Table2. It is clearly shown in Table 2 that there are six equilibrium points in the asymmetric fractional order system corresponding to the wetland ecological-economic-social composite system, but only the equilibrium points (0.113, 0.068, 0.070) and (0.592, 0.346, 0.362) are stable. (1) It can be concluded that the stable points of the wetland ecological-economic-social composite system are dynamically changing. (2) The continuous changes in external conditions, such as local wetland policies, major environmental events, and economic events, will serve new impacts on the wetland composite system, which in turn form new stimulus for development, pushing the wetland composite system to evolve synergistically towards a new stable point. (3) Through the analysis of the equilibrium points, it is clear that the three subsystems of wetland ecology, economy, and society are in synergistic state, i.e., moving towards a higher orderly development. This indicates that, in practice, wetland ecology, production, and life can synergistically develop with the intervention and adjustment of local policies.


## 6. Conclusion

Firstly, by studying the internal structural characteristics of the wetland eco-economic and social complex system and analyzing the coevolution of the three subsystems, this paper creatively constructed the coevolution model of the wetland eco-economic and social complex system based on the fractional dynamic system by applying the fractional equation theory, the coevolution theory, and other relevant theories. The model not only overcomes the shortcomings of local correlation and low fitting accuracy of traditional integer order difference models, but also reflects the historical dependence process of system function development better. Secondly, genetic algorithm is applied to solve the parameters of the coevolution model of wetland complex system, and the equilibrium point of the model and the corresponding Jacobian matrix are obtained. By analyzing the stability of the equilibrium point according to the stability criterion of the asymmetric fractional order equilibrium point, the process and direction of the coevolution of the wetland eco-economic and social complex system can be obtained, and the path and mode to realize the harmonious unity of wetland ecology, production, and life can be obtained. Finally, through the application of the model and correlation analysis, the coevolution of wetland eco-economic and social complex system is empirically studied based on the interprovincial data. The results show that (1) there is a significant nonlinear relationship among the three subsystems of wetland ecology, wetland economy, and wetland society. (2) There is a two-way competition between the wetland ecological subsystem and the wetland economic subsystem. The former hinders the development of the latter, and the latter develops at the expense of the former. (3) There is a two-way competition between wetland economic subsystem and wetland social subsystem, and government investment in the development of regional social life will suppress the development of regional economy dominated by capital investment. (4) There is one-way competition between the wetland social subsystem and the wetland ecological subsystem. The development of the wetland ecological subsystem hinders the economic development, which leads to less capital investment in the social subsystem. The development of the wetland social subsystem means the improvement of national quality, which will enhance the awareness of wetland protection of residents and society. That is to say, the development of the latter hinders the development of the former, but the development of the former promotes the orderly development of the latter.In a word, the stable point of wetland eco-economic and social complex system is dynamic. At the stable point, wetland ecology, production, and life are all evolving to a higher level of development. This shows that, in practice, wetland ecology, production, and life can develop in tandem with local policy interventions and adjustments. One will develop ecological benefit as the center of gravity, in order to realize ecological benefit, economic benefit, and social benefit coordinated development, invest a large amount of capital and technology, and with strict rules and flexible management control as a guarantee, vigorously promote wetland compound system orderly development, promote the ecosystem and economy, and realize the coordinated development of benign coupling and social system. In order to ease the competition between wetland ecological balance and economic development, promote wetland restoration and protection projects, speed up the construction of circular economy, and implement environment-friendly economic system. In order to promote the common development of regional economy and social life, it is necessary to allocate social capital scientifically, develop advanced science and technology, train new industrial talents, and adjust industrial structure. In order to ease the one-way competition between wetland ecosystem and social system development, we should constantly innovate science and technology, improve the level of ecological management, and fully mobilize the enthusiasm of ecological protection, so as to improve the ecological environment and make the regional social life more comfortable. To sum up, in order to promote the construction of ecological civilization, improve the level of regional people’s material and cultural life, and achieve more efficient economic development and social progress, government needs to issue relevant policies, which can promote the coordinated development of the wetland ecological-economic-social system, keep the dynamic balance of wetland ecosystem, and realize the harmonious development of man and nature.



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*Source: 1934271-2022-04-13.xml*