At present, there is a growing understanding of ecosystem services and a growing need to incorporate ecosystem services into policy decisions. It has become a growing trend to integrate urban ecosystem services into policy making, in particular, for a rapidly developing city that is highly dependent on scientific planning and decision-making. Models play an important role in realizing efficient urban ecosystem service assessment and explicit spatial representation, since the research on modelling urban ecosystem services has emerged. In order to apply models to better make urban management strategies and promote urban sustainable development, various modelling techniques and model frameworks have been continuously applied in the urban ecosystem service models. Based on the logic of models, the existing classification of modelling techniques mainly includes correlative models, expert-based models, and process-based models ^[1][30].

Correlative models usually use land use types or other ecological parameters to represent the value of ecosystem services based on relationships between ecosystem services and ecological parameters or their statistical correlations ^[2][31]. The majority of them often integrate various data, including social and economic data, field survey data, and the attribute data of hydrology, soil, and vegetation, to simulate the value of ecosystem services, and the spatial scopes vary from global to cities. For instance, the Matrix model is a kind of correlative models for quickly mapping ecosystem services by using rows and columns to represent land cover types and ecosystem service classes respectively ^[3][32]. de Groot et al. ^[4][33] summarized the results of ecosystem services assessment from over 300 cases around the world and derived the global average value. Although correlative models are simple and intuitive, and easy to apply in the preliminary assessment, there are still some limitations. It is difficult to show the characteristics, and temporal and spatial heterogeneity for a specific area, to reflect the scale effects on ecosystem services supply ability of different land cover types, and to illustrate the influence of external changes on the relationship between social system and ecosystem ^[5][6][34,35]. In particular, the correlative model applied in urban areas often has great uncertainty with low credibility and weak applicability, due to such reasons as insufficient data accuracy, high heterogeneity, and effects of external factors.

Assessing urban ecosystem services often involves multiple systems with a high degree of interactions, but the understanding of the complex system often lacks theory and enough data to support; expert-based models are semi-quantitative ways which can simulate and predict it with interdisciplinary knowledge ^[7][36]. Social ecological scenarios analysis and Bayesian network are common methods. The former focuses on dynamic change of relationship between ecosystem services and human well-being. Involving driving force of ecological changes into modeling can improve its effectiveness ^[8][37] and provide referenced solutions for stakeholders to face the changes in ecosystem services and human well-being. As for future prediction, it is also useful to analyze tradeoffs of multiple ecosystem services and illustrate their relevance to spatial pattern ^[9][38]. The latter combines the qualitative probability of experts’ prior knowledge with the quantitative relationship of model variables to simulate the flows of ecosystem services. Balbi et al. ^[10][39] used the ARIES model to integrate several kinds of data to effectively predict people’s dependence on natural resources in rapidly urbanized areas, further demonstrating that expert knowledge can offset data limitation. However, the expression of relationship by semi-quantitative method still exists difficulties in describing the feedback process of ecosystem services, and the lack of clear practical guidance limited their widely usage ^[11][40].

Process-based models can establish the flow process of natural capitals and ecosystem services with the understanding of ecosystem functions and biophysical processes ^[12][41]. According to the different theories and modeling purposes, it can be divided into specific models and comprehensive models. Specific models are designed for a particular service, and accentuate its responses to externally driven changes, while comprehensive models more concentrate on multiple ecosystem services and aim to analyze their trade-offs and synergies. Compared with the limitations of the former two modeling techniques, process-based models are more applicable to reflect the actual supply of ecosystem services, the feedback interactions of social system and ecosystem, and the objective logical correlation between ecosystem services ^[13][29]. In addition, using analysis of sensitivity and uncertainty to calibrate and verify the model results can also increase its credibility. In spite of this, process-based models are relatively difficult to develop and limited in their openness; they still benefit from exploring the mechanism of interactions between human and ecosystem during long-term ^[14][42], and can better settle the issues that lack dynamic interactions and feedback mechanisms ^[15][43].

Process-based models were initially derived from the needs of application for forestry, hydrology, agronomy, and edaphology for assessing ecosystem services generated by natural ecosystem. These models mainly focused on the description of vegetation productivity, the carbon cycle, hydrological processes, vegetation dynamics, soil erosions, and other processes, respectively. However, there was little attention paid to urban ecosystems and few studies targeted urban biophysical processes. With the increase of research on urban ecosystem services, these process-based models have also been applied to the practice of urban area, and mainly focus on the regulation services due to their similarity with natural ecosystems.

Various process-based models focus on urban carbon storage and capacity of carbon sequestration. The biome-biogeochemistry (Biome-BGC) model, mainly used for simulating carbon, nitrogen, and water flux, flows of natural ecosystems at large scale ^[16][44]. Milesi et al. ^[17][45] applied it in urban areas to simulate the potential carbon and water flux of turf grass in the United States under different management situations, and pointed out that the demand for irrigation of urban turf grass is much higher than crops, which lead to the increasing pressure on fresh water for many cities. In addition to large-scale urban studies, Brown et al. ^[18][46] applied it in the urban garden of the university of Maryland, simulated the net biome production per unit area per year from 1978 to 2008, and proved its effectiveness in local scale with the comparison with actual calculation. Another large-scale model, Carnegie-Ames-Stanford approach (CASA), is also used in urban areas. For instance, Zhou et al. ^[19][47] assessed carbon sequestration in China’s Guanzhong–Tianshui economic zone by integrating net primary production (NPP), grain yield, and DMSP/OLS night light data, and quantitatively analyzed the relationship between urbanization and urban ecosystem services. Tripathi et al. ^[20][48] used CASA model to simulate the NPP of urban arboretum in India at finer, local scale and studied the capacity of carbon sequestration for artificial forests. In addition, applying biomass models is an easier way to assess the urban carbon storage, but it is worth noting that these biomass models need be corrected to avoid deviation rather than using them directly due to the differences between urban and natural vegetation ^[21][49].

Cities are the main sources of air pollution, while urban vegetation provides an air purification service due to its rough surface, which is more beneficial in depositing air pollutants compared to the smooth artificial surface ^[22][50]. Most models use pollutant deposition model combined with leaf area index (LAI) of vegetation to calculate the removal capacity of air pollutants ^[23][51]. Janhäll ^[24][52] summarized the studies on removal of several major air pollutants by urban vegetation, including PM_2.5, PM₁₀, and ozone, and found that better design and selection of urban vegetation with understanding of pollutant deposition and dispersion had significant impacts on the improvement of urban vegetation air purification services. Specific to urban forests, Bottalico et al. ^[22][50] calculated the pollutant removal efficiency of different urban forest types in Florence, Italy, based on high-resolution remote sensing data and field-measured LAI. It was found that the removal efficiency of evergreen broad-leaved forests and coniferous forests was higher than that of other types of forest, and urban forest could contribute 6% to 13% in total to urban air pollution mitigation. Similarly, in the Mediterranean region, Fusaro et al. ^[25][53] identified the relative role of urban and peri-urban forests in ameliorating air quality through stomatal uptake of O₃, by using the process-based model Growth of Trees is Limited by Water (GOTILWA+). It was found that the influence of different management practices on urban forest structure will change its ability to remove air pollutants, for example, increasing irrigation in summer can improve the absorption of ozone by trees.

The risk of urban flooding is significantly booming due to the increase of urban impervious surfaces, so that many models used to evaluate water and soil conservation services of natural ecosystem are increasingly applied in urban areas ^[26][54]. Marques et al. ^[27][55] used a rainfall–runoff model combined with urban planning scheme and decisions of land use conversion to evaluate the value of urban hydrological regulation, and maximize the land use efficiency of urban floods control. Land use changes affect the vulnerability of urban flood risk. Chang et al. ^[28][56] assessed the flood risk of different regions in Taiwan with a grid-based, spatial land use change model. It was found that urbanization aggravated the risk of surrounding regions, which was much higher than that of the central city. However, in view of the inner cities, Zölch et al. ^[29][57] analyzed the ability of different green infrastructures (trees, green roofs, etc.) of flood prevention by scenario simulation of residential area in Munich, Germany by the MIKE-SHE model. It was found that urban vegetation can effectively mitigate the disturbance in urban hydrological cycle, regulate the surface runoff, and slow down the local rainstorm risk.

The current process-based models for specific ecosystem service can be well applied in urban areas, but the evaluation of such single type of urban ecosystem service cannot meet the needs of urban sustainable development. Increasingly more policy decisions rely on the assessment of synergies and tradeoffs of multiple urban ecosystem services. Moreover, using these models in cities still lacks the description of the impact of urban human activities, social environment, and other factors. The comprehensive ecosystem service model frameworks for urban ecosystem have been proposed intensively.

DPSIR (drivers, pressures, the state, impact, and response model of intervention) has been widely used for constructing integrated models, which is a causal framework for describing the human impact on the environment and vice versa ^[30][58]. Integrated models concentrated on several key issues, such as interactions of multiple urban ecosystem services, biophysical processes in urban ecosystem, and feedbacks of human activities. Nassl and Löffler ^[31][59] coupled DPSIR on ecosystem service cascades, enhanced its presentation of complex causality, and constructed a closed cycle of ecosystem services including social feedback, which can capture more potential interactions. The DPSIR framework is often used to develop natural-based solutions, offers professional perspectives for local municipalities and other policy makers to improve urban resilience to climate change, and has better support for management applications Lafortezza and Sanesi ^[32][60]. Currently, DPSIR has been mainly applied in urban wetlands due to its obvious pressure response process. On a small scale, vulnerability assessment of urban wetland integrated the impact of human activities on wetland ecosystem services promotes the development of appropriate management strategies ^[33][61]. It was also used to evaluate the ecological benefit improvement in the treatment and restoration process of polluted rivers ^[34][62]. While on a larger scale, the modeling of coastal ecosystems in coastal cities also integrates people’s demand for urban ecosystem services, such as resource supply, coastal protection, leisure, and entertainment in coastal cities, to quantifies the pressure responses and reflects obvious social and ecological dynamics ^[35][63]. Due to the complexity of the socio-ecosystem, DPSIR is also combined with the system dynamic (SD) model to jointly describe the relationship between the ecosystem and human stress. Ingram et al. ^[36][64] used this model to simulate the process of social and ecological interactions in Hawaii and found that the local resource management strategy had a great impact on the pressure of the ecosystem, especially the cultural services, and there was an urgent need to develop the strategic deployment of the sustainable development of the ecosystem.

In order to better understand the internal interactions of complex urban ecosystem and the effectiveness of decisions, SD models have been popularly used to promote future sustainable development. SD is predictive tool to simulate the biophysical processes within an environmental system. Xi and Poh ^[37][65] established a comprehensive tool with SD and Analytic Hierarchy Process framework to assess the risk of urban flooding in Singapore and support decision making for water resources management. It is argued that the initial proposed strategy cannot mitigate the risk, but desalination and recycled water should be priority measures. Analyzing future landscape change process under scenarios by SD, and simulating the ecosystem service change, it is useful to provide effective decision-making opinions ^[38][66]. SD is a holistic framework to examine feedback interactions in socio-ecosystem. Tan et al. ^[39][67] established an SD model, which composed of four subsystems, economy, society, environment, and resources, to simulate the performance of urban sustainable development in Beijing with the complex interactions. Lopes and Videira ^[15][43] also emphasized that the management of urban ecosystem services is particularly rely on the feedbacks, which supported the identification of the interrelationships among different ecosystem services and provided key indicators for management decisions. SD is also a platform for participatory modeling to involve stakeholders and make them have better understandings. Cavender-Bares et al. ^[40][68] built a sustainable SD framework and provide stakeholders with the tool to make decisions by integrating the ecological mechanism, the biophysical tradeoffs and inherent limitations, the preferences and values of stakeholders, and the response to future needs with time changes. Liu et al. ^[41][69] used SD and data envelopment analysis (DEA) to analyze the synergy between greening and urbanization in Tianjin. It indicates that greening is the essential pursuit of economic development and provides decision support for sustainable urban development. In settling environmental issues with SD, effective design can be further promoted to achieve more reasonable models, including clarifying the modeling purpose and scope in conceptualization stage, emphasizing the calibration of quantitative relationship and feedback loops and validating in various aspects ^[42][70].

Due to the participation of people, social organizations, and government in urban ecosystems, ecological problems often change with agent behaviors, and the feedback effect of human activities on ecosystem services is often affected by policies and behavioral preferences. Involving basic human elements into decision-making process, such as integrated stakeholder perceptions into quantitative simulation through a series of numerical methods, is helpful in solving complex social-ecosystems issues ^[43][71]. The change of society, system, individual behaviors, and ecosystem is the key to simulating the evolution of the social-ecosystem ^[44][72]. It is necessary to illuminate the complex relationships between humans and the environment in order to better understand and manage urban ecosystem services. Miyasaka et al. ^[45][73] established an agent-based model composed of heterogeneous social and ecological components and feedback mechanisms at multiple scales. The model evaluated UES tradeoffs with typical characteristics of the system, such as cross-scales feedback loops, time-delay effects, and threshold changes. The results showed that the policy of returning farmland to forest promoted vegetation and land restoration in the semi-arid areas of northeast China, but caused further land degradation beyond the implementation areas. Agent-based models can help us to understand how cross-scale processes contribute to social-ecosystem, which are often combined with spatial explicit model, land use, and biophysical model and economic drivers to explore the influence of human disturbance and policy adjustment on system results ^[46][74]. Although the agent-based model is a powerful tool that can represent the interaction of human actions and ecosystem, it still needs interdisciplinary cooperation to remedy its limits, such as complexity and difficult practicability, and improve the availability of experiential data ^[47][75].