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

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.