Understanding and Applications of Tensors in Ecosystem Services: A Case Study of the Manas River Basin

Abstract

Ecosystem services (ESs) are a multiple whole composed of multiple services and their multiple relations, which can be expressed as tensors (multiple functions of multiple vectors). This study attempts to introduce the concept and method of tensors into ES research to solve problems caused by the multiplicity of ESs, such as multiple descriptions and perceptions of ESs, repeated calculation of ES values, and cascading relationships with the social economy. Taking the Manas River Basin composite ecosystem as an example, we constructed five different types of ES tensors based on different understandings and applications: (1) As multiple vectors, three eigenvectors were extracted from the ES state tensor (ESST), including farmland service (FS), vegetation service (VS) and water service (WS). According to the ES response tensor (ESRT), an increase in FS may lead to a decline in overall services. (2) As multiple functions, the ES value (ESV) of the basin was measured by the ESV metric tensor (ESVMT), with a gross value of 14.8 billion USD and a net value of 10.17 billion USD. From different stakeholders perceptions constructed by the ecosystem services to human well-beings (ES-HW) tensor, the human well-being values (HWV) were ranked as citizens > farmers ≈ herdsmen > public. (3) The HWV output efficiency of different LULC per unit of water use was calculated by a fourth-order mixed tensor constructed by water–LULC–ES–HW multiple cascading relations. Among them, the HWV efficiency of water areas and wetlands was the highest, but the area was the smallest. Cultivated land and unused land had the lowest HWV efficiency and largest area. In general, the ES tensor is the extension and integration of the ES scalars/indicators to the ES vectors/bundles, which can provide tools for the integral expression, objective measurement and multiple perceptions of ESs.

1. Introduction

1.1. Research Progress of Ecosystem Services

Ecosystem services are the benefits that humans obtain directly or indirectly from ecosystems, including provisioning services (e.g., the provision of food and water), regulating services (regulating the atmosphere, soil, hydrology, etc.), cultural services (e.g., spiritual, recreational and cultural) and supporting services (maintaining nutrient cycling and biodiversity, etc.) [1]. Since the concept of ecosystem services was put forward, the understanding and practice of ESs have been enriched and deepened, from concept to definition, classification, quantification, accounting, investigation, evaluation, planning, management, zoning, spatial mapping, spatial and temporal characteristics, service flow, spatial clustering, trade-offs/synergies, supply–demand relationship, as well as the coupling relationships, mechanisms and model simulations with biodiversity, climate change, human activities and human well-being, etc. [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17]. Ecosystem services is a complex multidisciplinary field, involving a series of theories and methods of ecology, geography, social economics, mathematics, computer science and other natural sciences, social sciences, and even philosophy. People in different fields have different understanding perspectives and levels of ESs. The purpose of studying ecosystem services is to better understand and manage ecosystem services and achieve the sustainable and high-quality development of ecological and social economic systems. It can be said that how to understand ecosystem services largely determines how to manage ecosystem services. Human understanding and the management of ecosystem services has gone through the development process from simple to multiple, from mechanical to dialectical, from scattered to systematic. There are complex and dynamic matching and trade-offs/synergies between different services and different needs. In the process of trade-offs/synergies, spatiotemporal differentiation or aggregation occurs, forming different populations, communities, service clusters and demand clusters [18,19,20]. For decision makers and managers, how to coordinate diverse ecosystem services and their diverse perceptions and needs is related to the prosperity, stability and sustainable development of a region. Around the understanding and management of ecosystem services, many studies have proposed various models and modeling methods, such as INVEST model, ARIES model, SolVES model, multi-objective planning, dynamic planning, system dynamics model, etc. [21,22,23]. However, there are great differences between different models and methods, and the results are also very different. It is also difficult to compare and unify different standards. Therefore, for decision makers, they often weigh back and forth between the results of various models, which is difficult to make decisions. It is necessary to find a model that can integrate the results of many parties and objectively and comprehensively understand the overall state of multiple ecosystem services and their internal and external relations.

Biodiversity is the basis of multiple ESs. Many international research programs or organizations also pay close attention to biodiversity and ESs, such as TEEB, IPBES, CICES, MA, etc. [24,25,26,27]. Biodiversity is closely related to ecosystem productivity and stability, but the mechanism is complex and unclear [28]. Some studies have shown that moderately increasing the biodiversity of an ecosystem can improve its productivity and resistance stability [29,30]. For example, using the genetic diversity of field crops can reduce the severity of farm diseases and pests [31]. Moderately increasing the species diversity of sown grassland can improve the productivity and stability of grassland ecosystem and their resilience to disturbance [32]. Some studies also show that a moderate reduction in biodiversity can improve the productivity and resilience stability of ecosystems [33]. However, there is no doubt that as a basic support service, biodiversity maintains and regulates the stability and diversified service functions of the ecosystem. The serious decline of biodiversity will lead to the reduction or even loss of multiple ecosystem service functions and reduce the stability of the ecosystem [33,34]. For example, agriculture-based landscapes can reduce bee phylogenetic diversity and pollination services [35].

There are also many hypotheses to explain the relationship and interaction mechanisms between biodiversity and ESs, such as the specific mechanism of farmland biodiversity for pest control. According to the natural enemy hypothesis, resource concentration hypothesis, seed bank hypothesis, push–pull hypothesis, etc., intercropping/inter-planting, strip harvesting, trap belt planting, and regulating non-crop habitats can increase or regulate the diversity of natural enemies, crop diversity, non-crop diversity and landscape diversity of farmland, provide food and habitat for natural enemies, attract natural enemies to prey on pests, reduce the number of pests, block the spread of pests, interfere with pest foraging, and induce pest migration, improve crop resistance and other ways to control farmland pests and diseases [36,37,38,39]. According to the mid-domain effect, edge effect and spillover effect, through intercropping/inter-planting, rice–fish co-culture, etc., inter-species interaction is used to improve the utilization rate of ecological space and crop yield [37,38,40]. Of course, unreasonable collocation may have little effect or even the opposite effect. In addition, any measure or interference has both advantages and disadvantages, and its ecological effects are two-sided, even multiple. Increasing biodiversity not only suppresses the spread of pests but also affects other ESs. The natural enemy hypothesis is aimed at specific pest control. We should not only consider how to control farmland pests and diseases but also consider how to achieve high and stable yield as well as improve the soil, prevent biological invasion and assist other ecological service functions.

In addition, the change of biodiversity is also one of the important reasons for the trade-off and synergy of ESs [28]. The ESs cluster is the result of the trade-offs/synergies of ecosystem services in the space–time differentiation [41]. How to express the relationship between biodiversity and ESs, as well as the mechanism of trade-offs/synergies and ESs clusters, is also important for a more scientific management of ecosystem services.

The above questions can be summarized as follows: (1) How do we comprehensively understand and express the multidimensionality, multiplicity, objectivity, relativity and integrity of ESs, as well as the relationship with biodiversity and the overall state response under external effects? (2) How do we understand and express the multiple supply and demand perceptions and trade-offs/synergies between multiple ecosystem services and their different stakeholders? How do we understand and express the relationship and overall characteristics of multiple ecosystem service bundles? (3) How do we understand and integrate the objectivity and relativity of ESVs under different ESs classification indicator systems and value measurement standards, and the problem of repeated calculation of ESVs caused by indicator correlations?

In short, ESs is a multiple whole formed by multiple services and their internal and external relations. Is there a conceptual model or data structure that can qualitatively or quantitatively express such relations or properties and reveal some intrinsic attributes?

1.2. The Introduction of Tensors into Ecosystem Services

In mathematics, “tensor” is a high-order quantity or multidimensional arrays, which can be seen as a natural generalization of scalars (zero-order tensor), vectors/linear transformations (first-order tensor) and matrix (second-order tensor), i.e., multiple linear transformations of multiple vectors [42].

As a concept and mathematical tool, tensors have different understandings and applications in different fields. As a concept, the word “tensor” comes from “tense” or “tent”, which means (of wire or shelter, etc.) stretched tightly or (of a muscle or other part of the body) tight rather than relaxed. Later extended out to “tension”, “extension”, “intension” etc., for example, in mechanics, the word “tension” means the internal force state or response of a material being stretched tight. As a mathematical tool, the term “tensor” was originally proposed in the study of the invariance theory of linear algebra, and researchers studied the invariance under linear transformation [43]. Then, it was introduced into geometry to study the invariance under coordinate transformation. Later, it was introduced into physics to study the invariance under space–time transformation [44,45].

In a narrow sense, tensor emphasizes the invariance under spatial linear transformation, which has a clear mathematical meaning and has broad applications in mathematics and physics [46]. In a broad sense, tensor described the structure of multiple existence or relations, while it weakened the invariance, normally in form of a multidimensional array, which expanded the application field of a tensor, such as computer science, socio-economics, geography, and ecology [47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64]. In essence, tensor is a high-order quantity formed by tensor products of low-order tensors. The tensor product is the algebraization of a Cartesian product, and the Cartesian product is a pairwise combination of elements in two sets; that is where the multiplicity comes from, which is equivalent to many-to-many corresponding relationships. This is why the concept and method of tensor can be introduced into relevant fields to ESs.

By comparing the properties of tensors and ESs, there are some common points between them: (1) Multidimensionality and multiplicity. The multidimensionality means that there are multiple dimensions or indicators of ESs, and a set of ES indicators can constitute a vector. Multiplicity means that each component of ESs is related to multiple dimensions corresponding to multiple vectors or multiple functions. (2) Objectivity and relativity. The objectivity means that the products or functions of a specific ecosystem are objective corresponding to the invariant of tensors. The relativity means that different stakeholders have different needs and perceptions of different ESs, which is equivalent to each of the component values of a tensor under different coordinate systems. (3) Trade-offs and synergies corresponding to contravariant and covariant components of tensors. (4) Dispersivity and integrity. ESs are public, inclusive and dispersed, and they are not targeted to any particular beneficiaries, which just fit the properties of tensor dispersed distribution and spatial anisotropy. Meanwhile, ESs have multiple types of integrity formed by internal relations of biodiversity corresponding to a high-order quantity of tensors in form.

Based on these common points and similarities, this study tried to introduce the concept of “tensor” into ESs research and constructed five different EST types from different understandings and explored their applications and significance in ES states and responses, ESV evaluations and perceptions, as well as the cascading relationships with land–water use, human well-being, etc.

2. Study Area and Data

2.1. Study Area

The Manas River Basin (43°4′–45°20′ N, 84°46′–86°34′ E) is located in the middle of the northern slope of the Tianshan Mountains and the southern margin of the Junggar Basin in the inland arid region of northwest China (Figure 1), with an area of 2.56 × 10⁴ km². It is a typical “mountain–oasis–desert” complex ecosystem in an arid area [65] with an altitude from 250 to 5250 m, including 1.27 × 10⁴ km² of mountainous area, 0.87 × 10⁴ km² of oasis and 0.42 × 10⁴ km² of desert. The huge elevation difference breeds rich natural ecosystems and zonal landscape levels. From mountain to basin, there are landscape bases of ice and snow zone, meadow zone, forest zone, grassland zone, oasis areas, desert areas, and wetlands and water areas ecosystems as patches and corridors.

In social and economic terms, the Manas River Basin includes Shihezi City, Manas County and Shawan County. This is a multi-ethnic settlement; about 32 ethnic groups live here. In 2015, the total population of the basin was about 1.03 million people engaged in agriculture, animal husbandry, industry and service industry, respectively. The GDP was 71.78 billion yuan RMB (equivalent to 11.06 billion USD). The Manas River Basin is a typical agricultural and pastoral region. Animal husbandry is mainly concentrated in the grasslands of the upper and middle reaches. The planting industry is mainly concentrated in the plain area of the downstream oasis. Since 2000, the urbanization is developing rapidly, social and economic well-being has been greatly enhanced, while the ecological well-being has been seriously damaged [66].

The rich natural resources and diverse ecological landscapes provide rich and diverse ecosystem services for the Manas River Basin. For thousands of years, it has not only maintained the livelihood of residents but also is a paradise for the reproduction of various animals and plants. It is also an important habitat for a large number of northern migratory birds. However, over the past 70 years, due to rapid population growth, large-scale development of water and soil resources, and one-sided pursuit of rapid socio-economic development, strong human activities, such as cultivated land expansion, overgrazing, overexploitation, and excessive emissions, have led to dramatic changes in the local ecological landscape and a serious decline in biodiversity, threatening the sustainable development of the local social-economy and ecology [67]. On the other hand, diverse ethnic groups, diverse occupations, diverse groups, and diverse species have diverse needs for ecosystem services, and there are complex trade-offs and synergies between them. These provide favorable conditions for the expressions and applications of the tensors in ESs, which can provide a conceptual model and analysis tool for the development and utilization of water and soil resources, ecosystem service management and human well-being improvement.

2.2. Data Source and Processing

The data involved in this paper are mainly land use and land cover (LULC) data from 2000, 2005, 2010, and 2015 from the Chinese Academy of Sciences Resource and Environment Data Center (http://www.resdc.cn) (accessed on 29 May 2022) based on Landsat (30 m) image interpretation, which are mainly used to evaluate the ESV of the LUCC. The DEM data are the ASTER GEDM V3 version from NASA with a spatial resolution of 30 m.

The main data processing tools of data include ArcGIS 10.2 for ESV calculation, map-ping and sample extraction, SPSS 19 for factor analysis, dimension reduction and 3D scatter plot display, MATLAB R2012b and Excel 2013 for tensor analysis and 3D modeling.

3. Methods

3.1. Theoretical Framework

According to the generalized definition of tensor, an n-order tensor (n = r + s) can be expressed as a tensor formed by the tensor product of s elements in vector space V and r elements in its dual space V*. According to the relationship with the base transformation of the coordinate system, the elements in V are generally called contra-variant or upper indices, and the elements in V* are called covariant or lower indices. The so-called (r, s)-type tensor $T_r^s$ is also called the r-order covariant s-order contra-variant tensor, which is equivalent to the r-fold linear transformation of the s-fold vector in the V space, mapping the elements in the vector space V and the elements in its dual space V^* to a real number R (Formula (1)).

$$T_r^s:\underset{r}{\underbrace{V^{*}\otimes \cdots \otimes V^{*}}}\otimes \underset{s}{\underbrace{V\otimes \cdots \otimes V}}\to R$$

(1)

where $T_r^s$ is a tensor of type (r, s); V is the element in the vector space; V* is the element in the dual space; R is the element in the real number field; and V and V* are dual variables with respect to the real number R.

According to the agreement, the upper index ‘s’ generally represents the contra-variant, namely, the multiple vectors; the lower index ‘r’ is generally covariant, namely, multiple linear transformations. When r = 0, it is generally called the s-order contra-variant tensor. For example, the stress tensor and diffusion tensor are generally (0,2)-type contra-variant tensors, i.e., double vectors; when s = 0, they are generally called r-order covariant tensors. For example, the metric tensor is generally a (2,0)-type covariant tensor, i.e., double linear transformation. If r and s are not zero, it is called a mixed tensor of order r + s, such as the curvature tensor and elastic tensor.

Tensors are just a higher-order quantity that can represent either multiple vectors or multiple functions as well as multiple functions of multiple vectors. ESs are also a relatively abstract and comprehensive concept that can be either products, functions, values or human subjective perceptions. Therefore, this study proposed the concept of the “ecosystem service tensor (EST)” referring to the mathematical expression of multiple services and multiple internal and external relations of an ecosystem.

Based on the definition and understanding above, this study constructed five types of ESTs (Figure 2), including the (0,2), (2,0), (1,1), and (3,1) type tensors. To facilitate classification and description, they are discussed here separately to find and establish the correspondence with ESs.

3.2. Understanding and Construction of the (0,2)-Type Tensor

The (0,2)-type tensor $T^{\mathrm{ij}}$ is also called the second-order contra-variant tensor or double vector, such as the stress tensor and diffusion tensor. The stress tensor can describe the spatial anisotropy of the stress distribution. The essence of the stress tensor is an internal response of the intermolecular force in the material to the external force. The diffusion tensor describes the spatial anisotropy of the fluid diffusion velocity in all directions.

For an ecosystem, an ES is a multiple entity composed of multiple service functions and their internal relations. By analogy to the stress tensor, we can study the state or response of different internal components to external interference or demand, thus describing the integral state of multiple service functions of an ecosystem, which is called the ES tensor. The ES tensor reflects the results of internal trade-offs and synergies of different ES indicators. In the ES space constructed with various ES indicators as the coordinate system, each ES sample corresponds to a sample point in the coordinate system. Ideally, (samples are fully and randomly mixed), the morphology of the scatter cloud is shown as a high-dimensional ellipsoid (Figure 3), representing all possible ES state distributions of an ecosystem. Of course, it is impossible to enumerate all possible combinations in reality. Generally, it is only a part or section of an ellipsoid, while each component obtained by measurements or calculations of a certain ES contains all ES indicators or dimensional effects.

3.3. Understanding and Construction of the (2,0)-Type Tensor

The (2,0)-type tensor $T_{\mathrm{ij}}$, also known as the second-order covariant tensor or double linear transformation, has two main applications in ES: one is the response of ES supply to demand, and the other is used as a metric tensor for ESV de-duplicated calculation.

(1) Response of ES supply to demand

The multiservice state of a composite ecosystem mentioned above can be described as a state tensor, which is analogous to the stress tensor reflecting the internal stress state of an object, and there is strain when there is stress. The strain tensor reflects the deformation of the object under the external force, and the stress is generated by the object to resist the deformation. Similarly, the structure and state of an ecosystem can be altered by an external disturbances of climate change and the demands of human activity, with a certain degree of resistance, resilience, and elasticity determined by the steady state of the ecosystem. By analogy with the strain tensor, the response of ecosystem services to external demands or disturbances can be calculated and simulated, which is manifested in the change in ES scatter cloud morphology, corresponding to the change in the covariance matrix. Assuming an undisturbed ES ideal state or selecting the current ES state as the reference state, the scatters are approximately spherically distributed, and they are equivalent to a unit matrix. However, when subjected to external action or interference, different ES components grow and decline with each other, the distribution of scatter points gradually changes from spherical to ellipsoid (Figure 4), and the matrix also changes from unit matrix to diagonal matrix or symmetric matrix, reflecting the change of ES state and the response to demand, which is called an ES response tensor (ESRT). The component of an ESRT represents the load of each demand on each supply or the response of each supply to each demand. The demand is a vector, while the supply is a tensor: that is, the stress distribution or strain shape generated by the interaction within the system under the action of external directional demand. The ideal state after standardization is equivalent to an ellipsoid.

The demand vector can be written in the form of a diagonal matrix, and the internal trade-offs and synergies of multiple ESs can be represented by the correlation coefficient matrix R. The mathematical expression is as follows:

$$ESRT=D\cdot R\cdot D$$

(2)

$$D=\left(\begin{array}{ccc}es{d}_{11}& & \\ & \ddots & \\ & & es{d}_{nn}\end{array}\right)\begin{array}{ccc}& & \end{array}R=\left(\begin{array}{ccc}1& \dots & r_{1n}\\ ⋮& \ddots & ⋮\\ r_{n1}& \cdots & 1\end{array}\right)$$

(3)

where ESRT is the ES response tensor, which is equivalent to the covariance matrix; D is the demand matrix, which is equivalent to the standard deviation matrix; esd is the demand component; and R is the correlation coefficient matrix between ES indicators.

(2) Duplicated calculation of the ESVs

The assessment of the ESV often has the problem of duplicated calculation, which is caused by the internal correlation of the ESVs. Measuring the ESVs more effectively has always been an awkward problem. In addition, different studies have used different measurement standards, making it difficult to put the comparison between ESVs into a unified measurement system; however, these problems can be avoided to some extent by using the metric tensor. First, for the problem of repeated calculation of the ESVs, the metric tensor is introduced to convert the module length of the ESVs in the oblique coordinate system to the rectangular coordinate system to obtain relatively objective ESVs. Second, for the problem of different measurement standards adopted by different studies, the ES value tensor (ESVT) can be decomposed into contra-variant components and covariant bases. Choosing different measurement standards is equivalent to choosing different covariant bases. Assuming that the ESV is an objective quantity, the contravariant/covariant components under different covariant/contra-variant bases are also different, and the ESV tensor is constant, which is independent of the choice of coordinate bases. The specific calculation formula is as follows:

$${\mathrm{esv}}_0=es{v}^{\mathrm{i}}\cdot G_{\mathrm{ij}}^{}\cdot es{v}^{\mathrm{j}}=es{v}^{\mathrm{i}}\cdot G_{\mathrm{i}}^{\mathrm{j}}\cdot es{v}_{\mathrm{j}}=es{v}_{\mathrm{i}}\cdot G^{\mathrm{ij}}\cdot es{v}_{\mathrm{j}}$$

(4)

where esv₀ is the value volume or modulus of ESs; esv is the ES value vector, which is composed of ES value scalars; and G is the metric tensor, which can be expressed by the correlation coefficient matrix R of the ESV indicators. Of course, the metric tensor can be covariant, contra-variant, or mixed. It is covariant between different value reference systems, contra-variant between different measurement units in the same value reference system, and mixed between different measurement units in different value reference systems. Here, because the ESV in the oblique coordinate system is converted to its dual rectangular coordinate system, the reference system has changed; therefore, the second-order covariant metric tensor G_ij is used here to calculate the value modulus or volume of the ESV.

In this paper, the ecosystem service value (ESV) equivalent table of the Manas River Basin was formulated based on the ESV equivalent table of China’s ecosystem service value revised by Xie et al. (2015), combined with the average grain yield of the basin in the same period, and referring to the evaluation results of local related ESV studies (

Table 2. ESV equivalent table of the Manas River Basin (¥·hm⁻²·a⁻¹).

Ecosystem Service	Cultivated Land	Woodland	Grassland	Wetland	Water Area	Ice–Snow Land	Unused Land
Food production	3892	1145	1969	2335	2060	0	46
Material production	6227	2656	1557	2289	1053	0	137
Water supply	0	1374	870	11,859	37,957	9890	92
Gas regulation	3068	19,780	5540	8699	3526	275	824
Climate regulation	1648	26,144	14,606	16,483	39,102	50,732	458
Hydrology regulation	1236	18,727	10,714	110,941	137,177	32,646	824
Environmental purification	458	7646	4808	16,483	83,240	733	962
Soil conservation	4716	18,406	6731	11,401	4258	5494	504
Biodiversity	595	20,650	6135	36,034	20,925	46	458
Aesthetic landscape	275	9524	3983	21,657	8654	9066	1099

) [68,69,70].

3.4. Understanding and Construction of the (1,1)-Type Tensor

The (1,1)-type tensor $T_{\mathrm{i}}^{\mathrm{j}}$, called the first-order covariant and first-order contra-variant mixed tensor, is equivalent to the tensor product of a row vector and a column vector. Its geometric meaning can be, for example, the derivation of the first-order vector, which is the so-called Jacobian matrix, which is often used in surface differential geometry. It can also be understood as a linear transformation of a vector space, which is equivalent to the coefficient matrix ‘A’ of affine transformation y = Ax + b.

The ES-HW tensor is equivalent to a covariant tensor that projects the ES vector from the ES space to the HW space. In short, the ES-HW tensor reflects different stakeholder perceptions of different ESs. The ESs here are equivalent to an objective quantity; that is, the ecosystem services generated by the ecosystem are objectively and independently calculated, which is irrelevant to the subjective perceptions of different beneficiary groups, while the latter is based on different weights given by the former. Since the subjective preferences of different stakeholders are different, the feelings are also different. The ES-HW tensor is essentially a multi-linear transformation, of which the HW subjective perception of different beneficiary groups can be calculated according to the ES objective quantity. In this paper, the preference weights of four different stakeholders for four ecosystem services were constructed by the expert scoring method and normalized (

Table 3. Preference weights of different stakeholders for different ES.

ES Type	Farmers	Herdsmen	Citizens	Public	Total
Provisioning	0.66	0.21	0.04	0.09	1.00
Regulating	0.38	0.07	0.39	0.15	1.00
Cultural	0.07	0.07	0.69	0.17	1.00
Supporting	0.07	0.17	0.19	0.57	1.00

) as the ES-HW relationship tensor.

3.5. Understanding and Construction of the (3,1)-Type Tensor

The tensor of type (3,1) is a mixed tensor of third-order covariant, first-order contra-variant. For example, the multiple correspondences between LULC, W, ES, and HW can form a fourth-order tensor, in which the W-LULC-ES-HW tensor is equivalent to a fourth-order type (3,1) tensor (Formula (5)); that is, a triple linear transformation of a vector, which is a four-dimensional matrix in form, and tensor product of lulc, w, es, and hw in the algorithm, and essentially a multiple mapping relationship of four kinds of indicators (Figure 5).

$$T_{\mathrm{ijl}}^{\mathrm{k}}=h{w}_{\mathrm{i}}\otimes e{s}_{\mathrm{j}}\otimes lul{c}^{\mathrm{k}}\otimes w_{\mathrm{l}}$$

(5)

where T is the fourth-order tensor, hw is the human well-being vector, es is the ecosystem service vector, lulc is the land use and land cover vector, and w is the water use or water yield vector.

Here, LULC is taken as a vector, and W, ES, and HW are taken as the multi-linear functions of LULC. Given the water consumption equivalent and ES equivalent of different LULCs and different HW stakeholders’ perceptions of different ES indices, such a fourth-order tensor can be constructed. Its significance lies in the following aspects: (1) the changes in water, ES and HW can be calculated according to LUCC; (2) the virtual water, water footprint and ecological footprint of different ES or HW can be calculated flexibly; and (3) given the HW demand, the response of ESs and their impacts on water use and land use can be calculated. This study mainly examined the HWV output and corresponding water use efficiency of different LULC types.

4. Results

4.1. ES State Tensor (ESST) and Response Tensor (ESRT)

Taking the Manas River Basin as an example, the spatial distribution of 10 kinds of ESs was obtained by using the ESV of different LULC types, and 930 samples of 5 km × 5 km were extracted. Three rotating principal components, namely, common factors, were extracted by factor analysis: farmland service (FS), water service (WS) and vegetation service (VS). According to the factor score, the samples of the factors were obtained, and correlation analysis was conducted. The correlation coefficient matrix (

Table 4. Matrix representation of the ESST in the Manas River Basin.

Factor	FS	WS	VS
FS	1	−0.351	−0.292
WS	−0.351	1	0.677
VS	−0.292	0.677	1

Note: FS: farmland service. WS: water service. VS: vegetation service.

) here is equivalent to a state tensor or structure tensor (Figure 6). There is a significant negative correlation between farmland services and the other two types of services, indicating that the expansion of farmland has a trade-off relationship with other services.

The correlation coefficient matrix of the ES components (FS, WS, and VS) is taken as the state tensor or structure tensor, and the demands for the three types of ESs are taken as the demand vector. If the demand for each of the three types of services increases by one unit, it can be seen from

Table 5. Response rate of three types of ESVs to unit demand.

Component	FS	WS	VS	Total
FS	1	−0.633	−0.800	−0.433
WS	−0.633	1	0.756	1.123
VS	−0.800	0.756	1	0.956

Note: FS: farmland services. WS: water services. VS: vegetation services.

that under the same demands, for each additional unit of farmland services, the total service decreased by 0.433 units; if one unit of water service was added, the total service increased by 1.123 units. An increase of one unit in vegetation services resulted in an increase of 0.956 units in total services. Under comprehensive effects, the total service response is 2.323 units (sum of the upper/lower triangle), which is 0.677 units less than the total demand of 3 units. That is to say, the expansion of farmland services would lead to a decline in overall services.

4.2. ESV Assessment and Perception

4.2.1. ESV Assessment: ESV Metric Tensor (ESVMT)

Taking the Manas River Basin as an example, 10 ESV indicators were selected to calculate the ESV from 2000 to 2015 and then converted into the de-duplicated ESV (DESV) by the metric tensor (ESVMT) according to Formula (4). The results are as follows, and it can be seen that the total ESV of the basin was 14.8 billion USD on average, while the net value (or value modulus) was 10.17 billion USD on average, approximately 68.68% of the total value (

Table 6. GDP, ESV and DESV of the Manas River Basin (10⁹ ¥).

Year	GDP	ESV	DESV	Proportion (%)
2000	10.17	108.36	74.69	68.93
2005	16.78	107.32	73.78	68.75
2010	39.90	108.22	74.27	68.63
2015	71.78	107.01	73.20	68.41
Average	34.66 (4.76)	107.73 (14.80)	73.99 (10.17)	68.68

Note: (1) Proportion is the percentage of deduplicated ESV (DESV) in the ESV. (2) The data in brackets are denominated in USD. The average exchange rate of RMB against USD from 2000 to 2015 is about 7.28.

). From 2000 to 2015, the ESV of the basin did not change much, with a slight fluctuating decline. The reason for this is that the whole LUCC of the basin did not change much since 2000, while the GDP of the basin increased by approximately seven-fold in 15 years. By 2015, it was equivalent to the de-duplicated ESV of the basin.

4.2.2. ESV Perception: ES-HW Tensor

ES products and functions are objective quantities. They can be material or functional quantities and can be translated into common values (i.e., universal equivalent) as well. In this paper, for the sake of a unified measurement, we took the ESV calculated by using the ESV equivalent table as the relatively objective value, which did not consider different subjective perceptions of HWV stakeholders.

First, from the perspective of the structure and dynamics of ESVs, regulating services accounted for the largest proportion, which was followed by supporting services, because the mountain area was in a dominant position (

Table 7. ESV of the Manas River Basin from 2000 to 2015 (10⁸ ¥).

Year	Provisioning	Regulating	Cultural	Supporting	Total
2000	119.00	673.03	78.87	212.70	1083.59
2005	120.06	663.56	78.10	211.44	1073.16
2010	123.96	668.89	77.55	211.80	1082.20
2015	126.12	658.54	76.24	209.21	1070.10
Average	122.28	666.00	77.69	211.29	1077.26

). In the oasis area, the provisioning services took the largest part because farmland took the largest area. From 2000 to 2015, the provisioning services gradually increased, and other services showed a downward trend, with the greatest decline in 2005 (Figure 7). The reason may be that 2005 was a relatively dry year, coupled with the rapid expansion of land use, resulting in a significant increase in provisioning services and a relative decline in other services. Indeed, the total ESV only showed a slightly fluctuating downward trend because the LUCC did not change much.

Second, different HW stakeholders have different needs and perceptions for different ESs, which can be translated into human well-being value (HWV) by using the preference weight table (Table 3). According to ESVs of the Manas River Basin, the changes in the HWV from 2000 to 2015 were analyzed (

Table 8. HWV of different stakeholders in the basin from 2000 to 2015 (10⁸ ¥).

Year	Farmers	Herdsmen	Citizens	Public	Total
2000	356.43	116.23	360.75	250.18	1083.59
2005	353.36	115.49	356.34	247.96	1073.16
2010	357.96	116.72	358.26	249.26	1082.20
2015	355.15	115.88	352.92	246.16	1070.10
Average	355.72	116.08	357.07	248.39	1077.26

). It can be seen that the HWV of citizens was the largest, taking 38%, which was equivalent to the sum of farmers and herdsmen (Figure 8). The HWV of farmers and herdsmen was comparable. From 2000 to 2015, the HWV of different stakeholders showed a fluctuating downward trend. The HWV of farmers and herdsmen increased significantly in 2010 because the local ESV increased in the wet year in approximately 2010.

4.3. HWV Water Use Efficiency: W-LULC-ES-HW Tensor

There are multiple cascading relationships between ecosystem services, land–water use and human well-being. A mixed tensor can be constructed through multiple correspondences between any two of them. Conversely, if a multidimensional data volume is composed of relevant indicators for all samples, the relationships between ecosystem services, land–water use and human well-being can also be resolved through tensor decomposition.

Taking the Manas River Basin as an example, the W-LULC-ES-HW mixed tensor was constructed according to the water quota and service equivalent of different LULC and the perception of different HWs on different ESs. On this basis, the HWV outputs and water use efficiencies of different LULCs were calculated. The results are as follows (Figure 9 and Figure 10): In general, the HWV output value per unit area of water area was the highest, which was followed by wetland, woodland, ice–snow land, grassland, cultivated land, unused land, etc. From the perspective of single HWV output value, cultivated land played a greater role in the well-being of farmers and herdsmen because the provisioning services of cultivated land were relatively high, while other land types contributed more to citizens and public well-being because of their higher regulating, supporting and cultural services.

Taking 2000 as the base year, the relative changes in ES, HW and water use were calculated based on LUCC in the Manas River Basin from 2000 to 2015. The results are shown in Figure 11. The provisioning service (ESV1) increased year by year, and the water consumption also expanded rapidly. Regulating, supporting and cultural services all showed a downward trend, and the decline in regulating services was the largest. Among them, there was a slight rebound in 2010 compared with 2005. The reason for this is that 2010 was a wet year, and the increase in water volume caused the regulating services to ease and improve.

5. Discussion

5.1. Significances and Innovations of Tensor in ESs Research

5.1.1. Significances of Tensor in ESs Research

The laws of mathematical physics are often simple, while the real world is complex. As the stress tensor, strain tensor and elastic tensor are introduced into the continuum mechanics, Hooke’s law, which describes the relationship between elastic force and elastic deformation in one-dimensional space is extended to three-dimensional space to study the internal force and deformation of materials or fluids [71,72]. The real world is objective, and human perception is subjective. Just as in general relativity, the problem of space–time metric conversion between different parts of four-dimensional curved space–time can be solved by introducing a metric tensor, including describing the relative relationship between an electric field and magnetic field in electromagnetic tensor, in which the speed of light c is an important invariant [73]. In addition, the relationship or internal structure between many things cannot be directly observed, such as the distribution structure of cardiovascular and cerebrovascular and nerve fibers. Through nuclear magnetic resonance technology and a diffusion tensor imaging algorithm, intuitive medical images can be obtained [56].

Ecosystem services is a comprehensive and complex field, with both objective, invariant and observable components and subjective, changing and unobservable components. Many traditional researches are based on indicators or scalars: some are objective, invariant and observable, and some are subjective, variable and unobservable. Simple indicators cannot reflect the full and original appearance of ESs, indicating the need to study the internal relationship between different indicators, such as ESs trade-off/synergy based on correlation analysis, and integrate these relationships into ESBs through principal component analysis or cluster analysis, which can reduce the dimensions of indicators and extract main features [74]. However, these relationships are indicator dependent, and these indicators are dependent on observation and statistics, with a certain degree of subjectivity and randomness. Even for the same research object, different indicators and different measurement standards are often difficult to compare and unify with each other. On the other hand, due to the intersection of different disciplines, even the views on the same issue are multidimensional and multiple. Each indicator and each pair of relations can only reflect a part or a certain aspect of the ESs. We need to know the overall appearance of the ESs and the relative position of these indicators or relations in the ESs. This is the reason and purpose of introducing tensors to better understand and manage ESs. Definitely, these problems have been noticed by researchers and different theoretical models and solutions have been proposed, such as:

(1)

Using the analytic hierarchy process (AHP): For a complex system composed of many interrelated and mutually restricted factors, a complex multi-objective decision-making problem is regarded as a system. The target is decomposed into multiple objectives or criteria and then decomposed into several levels of multiple indicators. Then, the method of solving the eigenvector of the judgment matrix is used to calculate the weight of each layer to the previous layer. Finally, the weighted sum is obtained, and the maximum weight is the optimal solution. In ecological economics, for example, Wan et al. (2009) selected comprehensive profit (CP) and comprehensive cost (CC) as targets to emphasize the sustainable development of agriculture in pest management [75]. From the perspective of various people (researchers, farmers, economists, decision makers, environmentalists, etc.), the RCCCP model was established. We used AHP to evaluate the advantages of different strategies in pest management in horticultural reserves. AHP is a goal-oriented method, which is convenient for goal decision making. Its advantages are systematic, concise and practical. The disadvantage is that it has large directivity and subjectivity in hierarchy design and indicator selection. The social economic ecological composite system is often not hierarchical but distributed and networked. The tensor model is more objective and comprehensive, and there is more freedom for analysis. The judgment matrix in AHP is actually relative to a simple second-order perception tensor. Of course, constructing tensor models requires enough indicators and observation samples, and the more the better.

(2)

The SolVES model can calculate the social value of ESs and use the social value transfer function and the maximum entropy model to evaluate the social value of ESs in different regions according to the corresponding relationship between different environmental variables and social values [76,77]. The social value transfer function is equivalent to a second-order linear transformation. The maximum entropy model has the characteristics of uniform distribution and equal probability. It is fair without any subjective presupposition, while the probability distribution of the objective world is often not uniform. In addition, entropy is based on statistics, which measures the complexity and information of the system. It is scalar; tensor describes the overall state of the system, and the connotation is more abundant.

(3)

A matrix model is a kind of special correlation model based on LULC. The supply capacity and service demand of ecosystem services are simulated according to different land cover types by using the matrix model with geospatial units and ecosystem services as rows and columns, respectively, and the model modifies it based on socio-economic statistical data to build a matrix template applicable to various biological communities and social–ecosystems [22]. The matrix model is very popular in the environmental assessment of regional policies because it can quickly map the demand and supply distribution of ecosystem services, and the evaluation results are intuitive and easy to understand. However, the lack of further integration of different data does not reflect the trade-off/synergy between ESs supply and demand and the overall characteristics. The introduction of a tensor can construct an ES tensor on the supply side, HW tensor on the demand side, and ES-HW tensor of supply and demand relationship, which can more comprehensively reflect the supply and demand situation of ESs.

(4)

In terms of ESV de-duplication calculation, there are also many models and methods, some from the process point of view, which can distinguish between intermediate services and final services [78,79] or distinguish between static services and dynamic services [80]. Meanwhile, the metric tensor ESVMT distinguishes internal value and external value from the perspective of relationship on the basis of existing value evaluation. In general, a tensor is not only a quantity, but also an analytical tool and method. Compared with other concepts, models and methods, a tensor has its own unique expression, and it also can be integrated into other models or synthesize the results of other models.

5.1.2. The Relationship between Biodiversity and ESs from the Perspective of Tensor

As mentioned in the beginning, the relationship between biodiversity and ESs is complex and the mechanisms are unclear. Biodiversity is manifested in multiple scales such as genes, species, ecosystems, and landscapes, which can be measured by indexes of α, β, and γ diversity [33]. There are many theories and hypotheses to study how biodiversity is formed and maintained at different scales, such as niche theory, neutral theory, mid-domain effect, edge effect, modern climate hypothesis, geological history hypothesis, and negative density-dependent hypothesis (Janzen–Connell hypothesis) [40,81,82,83,84]. These hypotheses are opposite, complementary, and deepening integration. For example, since Grinnell (1917) proposed niche to describe the spatial position of birds in the ecosystem, Lack (1947) proposed niche relationships and explained the formation of biodiversity. Clarke (1954) defined niche as the functional ecological relationship of species in the community. Hutchinson (1957) proposed the hyper-volume niche of multidimensional resource space, which is the basic niche under ideal conditions, while the real niche is more complex and can be regarded as a subset of the basic niche under ideal conditions. MacArthur (1970) proposed the resource utilization function niche. In the same year, Wuenscher (1970) thought that the space of species niche can be regarded as a multidimensional vector space. Vectors can be used to represent the habitat and living conditions in the species niche. Wang (1984) defined the generalized species niche as a set of correlated vectors corresponding to the description of environmental factors. GraVel (2006) integrated the niche theory and the neutral theory to propose the neutral-niche continuum hypothesis, which holds that the maintenance of species diversity has both the role of random ecological drift and the deterministic process of niche [81]. These lay the foundation for the description of tensor in biodiversity.

Compared with studies of biodiversity, which takes the ecosystem as the center, the ESs focus is people-oriented, examining the functional diversity of ecosystems and their relationship with services for human well-being [85,86,87,88,89]. Because biodiversity and ecosystem function have many dimensions, representing different aspects of property and meaning, the relationship between biodiversity and ecosystem function is very complex. There is no simple one-to-one correspondence between functions and services. Each service may be produced by multiple functions, but the relative importance of each function may be different. That is to say, there are multiple relationships and trade-offs/synergies between biodiversity and ecosystem services. Changes in biodiversity will lead to the changes in ecosystem functions and services.

Diversity is universal, whether it is habitat diversity, biodiversity, functional diversity, or service diversity, the essence of diversity is different aspects of the same kinds of things. There are both connections and differences between things. There are both intrinsic natures and external manifestations of things. The intrinsic nature of ecosystem services is an ecological function. A species or a function of the ecosystem can participate in the formation of one or more ecosystem services, and a certain ecosystem service may come from a certain function or a combination of multiple functions of an ecosystem. Therefore, there is a many-to-many relationship between ecosystem functions and services [90]. Meanwhile, the relationship is relative, including the relationship between different objects and different perspectives. From the perspective of a tensor, a tensor describes a multiple integrity from multiple perspectives, such as describing the state of food web in an ecosystem, and the multiple functions, roles and status of each species in the ecosystem as well as the trade-offs and synergies between different species.

Changes in biodiversity can lead to changes in the composition, structure and status of food webs, which in turn affect ecological functions and characteristics such as ecosystem stability, primary productivity, and landscape characteristics, and then affect ESs through a combination of functions. For example, rice–fish co-culture can provide rich food and nutrition, and it can also extend the food chain, improve resource utilization, and control pests [37]. In addition, biodiversity itself is also an important support service and cultural service, such as improving soil to improve primary productivity, providing existence value, heritage value, tourism value and scientific research value [91]. These service functions are not independent but rather interdependent and interactive [29]. We should not just pay attention to a certain ecological service function of a certain species or only pay attention to biodiversity itself or simply focus on human beings, protect for the sake of protection, or develop for the sake of development. We should regard natural ecosystems as a whole and systematically consider the dialectical relationship between human and nature.

5.1.3. The Relationship of ESs Trade-off/Synergy, ESB and EST

There are many types of trade-off/synergy of ESs, including spatial and temporal, direct and indirect, reversible and irreversible. There are many reasons for trade-off/synergy, including supply side, demand side, and between supply side and demand side [5]. Due to the diversity and selectivity of ESs, in the process of trade-off/synergy, the parts of the trade-off are mutually exclusive, and the parts of the synergy are attracted to each other, gathering or differentiating in time and space, resulting in service clusters and demand clusters. If these service clusters or demand clusters are drawn in the service space constructed by different ESs indicators, the size and direction of the arrows will be different. Each arrow represents a service eigenvector or a demand eigenvector. All the eigenvectors and their correlations formed an EST, and the ESBs are equivalent to the eigenvectors of the EST.

Compared with ES trade-offs/synergies and ESB, the EST is more in-depth and integrated. ES trade-offs/synergies mainly focus on the positive or negative relationship between the apparent indicators of ESs without going deep into the essential factors [6]. ESBs are clusters based on the distribution of ES samples or the principal components extracted according to the covariance or correlations of ESs indicators, which can reduce the dimensions of the ESs [92]. However, some overall information has been lost without considering the relationships and transition intervals between clusters or bundles; after all, there are many ESs without obvious trade-off and synergy. The EST not only contains the internal structure and essential characteristics but also retains the overall information of the sample population, which can not only reflect the overall state of ESs and the response to external demand but can also conduct internal measurement and external perceptions of ESVs.

5.2. Shortcomings and Prospects of This Study

This study only preliminarily examined several understandings and applications of tensors in regard to ESs, because the concept of ESs is very broad, and the mathematical connotation of tensors is very deep, it is really very difficult to introduce tensors into ESs and establish effective correspondence. In addition, the amount of data is limited, so this study is more focused on the discussion of concepts and methods. There are still many problems to be examined in depth, and there is still some room to expand the understanding and application of tensors.

(1) In regard to taking EST as multiple vectors

To understand the ES tensor from the perspective of multiple vectors, the state tensor in ES space is essentially similar to principal component analysis and factor analysis, but it is different from principal component or factor decomposition; tensor decomposition can decompose data into tensor products of several rank-one tensors. As mentioned in Section 3.4 and Section 4.3, the type (3,1) mixed tensors can be expressed as tensor products of LULC, W, ES, and HW vectors. Conversely, tensor decomposition can extract multiple ES structures of an ecosystem. Certainly, the analysis here is based on the ESV as an example, which is also applicable to other ES calculations, such as a variety of ESs calculated in terms of material quantity and functional quantity, which in essence can be understood as a linear or nonlinear combination of some base parameters (LULC, NDVI, NPP, P/ET, DEM, etc.), and these parameters are often not independent; therefore, common factors can be extracted. Based on the sample statistical data of multiple indicators, the ES multiple structures of an ecosystem can be theoretically extracted through tensor decomposition and integration.

(2) In regard to taking EST as multiple linear functions

To understand the ES tensor from the perspective of multiple functions, the ES correlation matrix is used here. As a metric tensor, although it can play a role in the de-duplicated calculation, the measurement of value is generally linear and additive rather than in the form of the vector modulus. Necessarily, as a measurement method, it still has a certain reference value. For example, for an ecosystem, if the components of its value tensor are highly correlated with each other, the modulus is relatively small, indicating that the value connotation of the system is relatively simple; if the components are relatively independent, the calculated modulus will be larger, which shows that the value connotation of the system is relatively rich.

On the other hand, when used as a bilinear transformation, it can be used to calculate the response of ES supply to demand. Since the demand is oriented to specific objects, it can be expressed as a vector, while supply is not oriented to specific objects; thus, it can be expressed as a tensor. Combining the ES-HW type tensor in Section 3.3 and Section 4.2.2, a type-(2,1) third-order mixed tensor ES-ES-HW can be constructed, which means that HW demand is can first be converted into ES demand according to the ES-HW tensor. Then, according to the ES-ES tensor, the ES demand vector is converted into the response tensor ESRT on the ES supply side.

(3) In regard to taking EST as multiple linear functions of multiple vectors

The generalized tensor is equivalent to a multidimensional matrix formed by multiple vectors or functions through tensor products. It is an extension of linear algebra from vector to tensor, thus expanding its application field. Of course, this requires that the system is a linear system. In practice, a system often has both linear components and non-linear components, while nonlinear components can sometimes be transformed into linear components to some extent by finding invariants in the transformation process.

(4) Prospects of tensors in ES studies

Definitely, the EST is just an idealized model. According to Figure 6 in Section 4.1, the actual ES-ES scatter may not be filled with all possible directions of the ES state space but often only as a part of it, because a basin in reality cannot contain all possibilities, and many rasterized socio-economic data are incomplete or difficult to obtain. Thus, the complete tensor data structure has not been completely established. If the data are rich and the indicators are diverse, then the high-order tensor based on big data can be constructed, and the tensor decomposition method can be used for feature extraction, dimension perspective, data compression, etc. Of course, there is also a neural network model called “TensorFlow” that is directly based on tensor data [63], which can be seen as the combination of generalized tensor and neural network models. In addition, for large spatiotemporal field data, the tensor field may be considered, which may also have some application potential in multiple spatiotemporal ES evolutionary research.

6. Conclusions

Ecosystem services are a multiple integrity composed of multiple services and their internal and external relations, which can be described by tensors (multiple functions of multiple vectors). As multiple vectors, the ES tensor can represent the structure or state of ESs. As multiple functions, it can represent the internal relationships and intrinsic measurement of an ESs. As multiple functions of multiple vectors, it can represent the internal state of ESs and external relationships with land-water use and human well-being.

This study constructed five kinds of ES tensors based on different understandings, namely ES state tensor (ESST), ES response tensor (ESRT), ES value metric tensor (ESVMT), ES-HW tensor, and W-LULC-ES-HW tensor. Taking the Manas River Basin as an example for comprehensive application, we found the following: (1) According to the ESST and the ESRT, three eigenvectors of ESST—namely, farmland services (FS), water services (WS) and vegetation services (VS) —were extracted, among which FS were trade-offs with WS and VS. The increase in FS will reduce WS and VS, thus leading to the decline in overall services. (2) According to the ESVMT, the ESVs of the basin before and after de-duplication were 14.8 billion USD and 10.17 billion USD. (3) According to the ES-HW tensor, the HWVs of different stakeholders were ranked as citizens > farmers ≈ herdsmen > public. From 2000 to 2015, the well-being of farmers and herdsmen showed an upward trend due to farmland expanding, while the well-being of citizens and public showed a fluctuating downward trend. (4) According to the W-LULC-ES-HW tensor, the HWV efficiency of water areas and wetlands was the highest, while the areas were the smallest. Cultivated land and unused land had the lowest HWV efficiency and the largest area.

The introduction of tensor analysis provided a mathematical tool for studies of multiple ESs state and response, ESV assessment and perceptions, land-water use-ecosystem service-human well-being cascading relationships. The concept of EST takes into account the multiplicity, integrity, objectivity and subjectivity of ESs, which can help people from different perspectives better understand ESs.