A Conceptual Socio-hydrological Model of the Co-evolution of Humans and Water: Case Study of the Tarim River Basin, Western China

The complex interactions and feedbacks between humans and water are critically important issues but remain poorly understood in the newly proposed discipline of socio-hydrology (Sivapalan et al., 2012). An exploratory model with the appropriate level of simplification can be valuable for improving our understanding of the co-evolution and self-organization of socio-hydrological systems driven by interactions and feedbacks operating at different scales. In this study, a simplified conceptual socio-hydrological model based on logistic growth curves is developed for the Tarim River basin in western China and is used to illustrate the explanatory power of such a co-evolutionary model. The study area is the main stream of the Tarim River, which is divided into two modeling units. The socio-hydrological system is composed of four subsystems , i.e., the hydrological, ecological , economic, and social subsystems. In each modeling unit, the hydrological equation focusing on water balance is coupled to the other three evolutionary equations to represent the dynamics of the social subsystem (denoted by population), the economic subsystem (denoted by irrigated crop area ratio), and the ecological subsystem (denoted by natural vegetation cover), each of which is expressed in terms of a logistic growth curve. Four feedback loops are identified to represent the complex interactions among different subsystems and different spatial units, of which two are inner loops occurring within each separate unit and the other two are outer loops linking the two modeling units. The feedback mechanisms are incorporated into the constitutive relations for model parameters, i.e., the colonization and mortality rates in the logistic growth curves that are jointly determined by the state variables of all subsystems. The co-evolution of the Tarim socio-hydrological system is then analyzed with this conceptual model to gain insights into the overall system dynamics and its sensitivity to the external drivers and internal system variables. The results show a costly pendulum swing between a balanced distribution of socioeconomic and natural ecologic resources among the upper and lower reaches and a highly skewed distribution towards the upper reach. This evolution is principally driven by the attitudinal changes occurring within water resources management policies that reflect the evolving community awareness of society to concerns regarding the ecology and environment.


Introduction
In the emergent Anthropocene, the competition for water between humans and ecosystems leads to complex interactions between hydrologic and social systems.They play fundamental roles in the co-evolutionary history of coupled human-nature systems as well as their possible future trajectories.Furthermore, the nature of such interactions is always changing as social and natural conditions change.Over time, they generate new connections and, in particular, additional significant feedbacks that need to be understood, assessed, modeled and predicted (Montanari et al., 2013).These feedback mechanisms between the hydrological and social systems are often ignored in traditional hydrology.For example, water consumption activities and landscape changes driven by humans are usually prescribed as external forcings in hydrologic models.The underlying assumption here is stationarity (Milly et al., 2008;Peel and Blöschl, 2011) in spite of the fact that water-related human actions turn out to be internal (endogenous) processes of the coupled socio-hydrologic system and evolve constantly.Also, human actions tend to be treated as static and externally prescribed in traditional water resources planning and management.A prominent example is the "scenario-based" approach used to represent future state(s) of the coupled socio-hydrological system in the science of integrated water resources management.Consequently, possible evolutionary trajectories of human-water systems cannot be fully explored or predicted.To address this deficiency, a new trans-disciplinary science of sociohydrology has been proposed which aims at understanding and predicting the dynamics and co-evolution of coupled human-water systems (Sivapalan et al., 2012).
There are three avenues through which socio-hydrology can advance (Sivapalan et al., 2012), i.e., historical sociohydrology, comparative socio-hydrology, and process sociohydrology.Besides, motivated by the success of hydrological modeling in traditional hydrology towards recognizing the limits of our understanding of hydrological processes, the research method of numerical modeling should be introduced into socio-hydrology as well to light up the three avenues of socio-hydrologic inquiry mentioned above.Models could be used to explore new knowledge and to test the limits of existing knowledge (as included in models) about the complex interactions between hydrologic and social systems.We acknowledge that the interactions and associated feedback mechanisms between hydrological and social processes remain largely unexplored and poorly understood (Di Baldassarre et al., 2013a) at the current state of development of socio-hydrology.In a river basin context, we are not yet in a position where the description of social processes in a coupled socio-hydrologic model can match the level of detail in traditional hydrologic models such as SWAT (Arnold et al., 1998) and THREW (Tian et al., 2006(Tian et al., , 2008)).However, there is considerable value in the development and use of simpler, coupled models to improve our understanding of such complex systems.The simplification is aimed at capturing the most important inter-dependent relationships, and leaves out much of the (perhaps unnecessary) detail.This is a practice widely adopted in many other inter-disciplinary fields.For example in the case of ecology, Levins and Culver (1971) introduced the logistic growth function to describe vegetation dynamics, which is an idea borrowed from population growth models (Tsoularis and Wallace, 2002).Baudena et al. (2007) introduced the role of soil moisture into the colonization and extinction rates of vegetation in the form of a logistic function and, in this way, the soil moisture dynamics was coupled to vegetation dynamics.Lin et al. (2013) developed a simplified ecohydrological model with pulsed atmospheric forcing to analyze non-trivial dynamic behaviors, both qualitatively and numerically, and confirmed the existence of multiple stationary states.In the case of social sciences, many researchers have intensively studied the interactions and feedbacks between human society and natural resources by using simple constitutive relations.For example, Brander and Taylor (1998) presented a model of renewable resource and population dynamics in the form of a predator-prey model, with humans as the predator and resources as the prey.They applied this model to the historical situation in Easter Island to show that plausible parameter values generate a "feast and famine" pattern of cyclical adjustment in population and resource stocks that may have occurred there.D' Alessandro (2007) studied the long-term dynamic interactions between the exploitation of natural resources and population growth by the harvesting production function and found a multiplicity of steady states, which made it possible to consider the effects of technological advances, and cultural and climate changes on the resilience of existing development pathways.Good and Reuveny ( 2009) coupled an ecological-economic model of human-resource interaction with endogenous coupling of population growth to economic growth.Good and Reuveny (2009) used this model to study the abrupt collapse of the Sumerian, Maya, Rapanui and Anasazi peoples and attributed their breakdown to anthropogenic environmental degradation; however, in this case resource use was not explicitly incorporated in their model.In the area of socio-hydrology itself, there have been a couple of pioneering studies that have shown considerable potential in this direction.For example, Di Baldassarre et al. (2013a, b) developed a simple dynamic model to represent the interactions and feedbacks between hydrological and social processes in the case of flooding, and found that a simple conceptual model is able to reproduce reciprocal effects between floods and people and the generation of emergent patterns from the coupled system dynamics.Along with the simple conceptual model, several researchers have focused on the human response to environmental change.For example, Elshafei et al. (2014) proposed a prototype framework for models of socio-hydrology with the concept of community sensitivity as a core for feedback between environmental and socio-economic systems, and van Emmerik et al. (2014) simulated the co-evolution of humans and water and adopted the concept of environmental awareness to explain dominant features of the pendulum swing observed in the Murrumbidgee River basin in Australia.
In this study we attempt to develop a simple conceptual model of the co-evolution of the socio-hydrological system in an arid inland oasis area.The Tarim River basin in the western part of China is chosen for this study.The main stream of the Tarim River is located in an inland hyper-arid area, with runoff principally from the headwaters (Zhou et al., 2012).In this area, humans are heavily engaged in agricultural production (other industries will be ignored here because their scales are small compared to agriculture).In the long history of Tarim, human populations and their agricultural activities have depended exclusively on water from the Tarim River, and constantly moved with the river as it migrated in response to climatic variations (Liu et al., 2014).In the last 60 years, due to the dramatic increase of irrigated agriculture, the lower reach of the Tarim River has nearly dried up (Deng, 2009), causing the degradation of the riverine ecosystem.In order to restore the downstream ecological system, the water reallocation policy was introduced and more water has been increasingly released into the lower reach since 2000 (Chen et al., 2010;Liu et al., 2012b).This adjustment of the water allocation policy can be seen as a response of the social system to the change of ecohydrological system and thus represents a negative feedback.On the longer timescale, streamflow, vegetation cover, human population and irrigated area could be exchanged between the upper and lower reaches, which are the key co-evolutionary processes associated with the socio-hydrologic system.Coevolution of the hydrological and associated systems (including society, economy and ecology) needs to be recognized and incorporated within a suitable modeling approach, in order to predict their reaction to future human or environmental changes (Montanari et al., 2013), which is the aim of this study.
The remainder of the paper is organized as follows.In Sect.2, the study area and the data used in the modeling are introduced, which is followed by the details and justification of the conceptual model adopted for the Tarim River basin, which are presented in Sect.3. Results of the model calibra-tion and validation are presented in Sect.4, along with the results of sensitivity analysis with the model.The paper concludes with the main results and recommendations for future research.

Study area
Tarim River basin (TRB) is located in western China and experiences a hyper-arid climate with an annual precipitation of 50-100 mm only.It is the largest inland basin in China with an area of 1 100 000 km 2 , most of which is covered by the Taklimakan Desert.There are four tributaries of the Tarim River, namely the Aksu River, Yarkand River, Hotan River, and Kongqi River.The four tributaries serve as the main source of water for the main stream of the Tarim River, which originates from the point of union of the Aksu, Yarkand and Hotan rivers (near the city of Aral in western Xinjiang) and empties into a terminal lake (Taitema Lake).The overview of the Tarim River basin and its river system are shown in Fig. 1.For more details about the TRB, including especially the historical development of the coupled socio-hydrological system within the TRB, please refer to Chen et al. (2010) and Liu et al. (2014).
In this study, we focus our modeling efforts on the main stream of the Tarim River, i.e., from Aral to Taitema Lake which is divided into two modeling units, i.e., the upper reach, from Aral (40 • 31 41 N, 81 • 16 12 E) to Yingbazha (41 • 10 28 N, 84 • 13 45 E), and the "middle and lower" reach (although shortened as the lower reach hereafter in the paper), from Yingbazha to Taitema Lake.See Fig. 2 for  more details about the discretization of the main stream of the Tarim River into these two units.

Data
In this study the modeling period is from 1951 to 2010.Daily precipitation and pan evaporation data from Aral and Tikanlik (40  Tarim Petroleum Annual, andXinjiang Fifty Years (1955-2005).
The NDVI (normalized difference vegetation index) time series data of reference points near the main channel of the Tarim River are employed as the reference values to the simulated regional vegetation cover.The NDVI data are extracted from MODIS products, MODIS/Terra Vegetation Indices 16-Day L3 Global 250 m SIN Grid V005 (MOD13Q1), with quality control.The vegetation reference point in the upper reach (VPU) is located at 40 • 57 40 N, 82 • 25 0 E, near Aral as shown in Fig. 2. The vegetation reference point (VPL) in the middle and lower reach is at 41 • 1 10 N, 86 • 14 0 E, near Qiala.

General description of the socio-hydrological system
The socio-hydrological system associated with the Tarim River basin is a complex network of multiple inter-connected processes and, in order to undertake the modeling with thestate-of-the-art understanding of the system, we assume that 1. the land use types are composed of irrigated crop field, naturally vegetated land and bare desert; 2. the water requirement of natural vegetation in the lower reach is mainly fed by the streamflow released from the upper reach; 3. the released discharge from the upper reach is determined by inflow into the upper reach, degradation of natural vegetation in the lower reach and the regional water resources management policy.
The modeling framework for the co-evolution of the TRB socio-hydrological system is shown in Table 1.Each modeling unit includes a hydrological system, an ecological system, a social system, and an economic system.Principally, TRB is an agricultural society and other economic sectors (e.g., industrial) are neglected in the economic system.The state variables of each unit are as follows: 1. Water storage (W ), in cubic meters.W represents allocatable water resources of the modeling unit.
2. Vegetation cover (V C ), dimensionless, in [0, 1].V C represents the natural vegetation cover, which is determined by the available water.It is defined as the ratio of the area covered by natural vegetation to the area of the modeling unit.
3. Irrigated crop area ratio (R I ), dimensionless, in [0, 1].R I is defined as the ratio of irrigated crop area to the area of the modeling unit.In each modeling unit, four ordinary differential equations are used to describe the dynamics of the hydrological sub-system represented by water storage, ecological subsystem represented by natural vegetation cover, economic sub-system represented by irrigated crop area, and social sub-system represented by human population.The area of the modeling unit is noted as A. The subscript "U" in the notation represents the upper reach, and the subscript "L" represents the lower reach.

Water balance of the hydrological sub-system
At the annual time scale, we represent the dominant hydrological process of the system in terms of the water balance equation.The water balance equation for the upper reach is given as where P is the annual precipitation, E t is the annual evapotranspiration from natural vegetation, E c is the annual evapotranspiration from irrigated crop, and E b is the annual evaporation from the bare desert (all expressed in mm yr −1 ).We assume that the precipitation falling on the bare desert is completely evaporated.The evapotranspiration terms are calculated by the following equations if there is sufficient water supply: where E p (mm yr −1 ) is the annual potential evaporation, and k t and k c are the empirical coefficients to calculate the actual evaporation from natural vegetation and crop, respectively.Q inU (m 3 yr −1 ) is the inflow to the upper reach, which is taken as the observed discharge at Aral.Q outU (in m 3 yr −1 ) is the outflow from the upper reach and is determined by Q inU , W U , V CL and other variables.In principle, Q outU could be calculated as where q 1 , q 2 , and q 3 are the corresponding release functions of Q inU , W U , and V CL , of which q 1 increases with upper reach inflow (Q inU ), q 2 increases with W , i.e., allocatable water resource, and q 3 increases with vegetation cover of the lower reach, accounting for the government's environment protection policy.Q outU is calculated using the following procedure according to present water allocation practices in Tarim, which could be generalized in a future study.
If there is sufficient inflow from the headwaters, the streamflow will be released to the lower reach after the water requirements of agriculture and natural vegetation are satisfied (see Eq. 2).Otherwise, the outflow will be equal to the minimum outflow, i.e., k Q Q inU , in line with the water allocation policy adopted in this region, and therefore the water requirement of agriculture and natural vegetation will not be fully satisfied.The outflow , water consumptions from natural vegetation and irrigation are determined in the following procedure.Q outU is then given as After the Q outU is determined, natural vegetation water requirement may not be fully met.If this is the case, the annual evapotranspiration of the natural vegetation, i.e., E tU , is given as (5) Finally, the annual evapotranspiration of the irrigated crop area, E cU , is In a similar way as above, the water balance equation of the lower reach is given as where the symbols are similar to those in the upper reach Eq. ( 1), and Q inL is equal to outflow from the upper reach.
If the water in the lower reach is sufficient, all of the allocatable water will be evaporated in the desert after the water requirements of irrigated agriculture and natural vegetation are satisfied, which can be similarly calculated by Eq. ( 2).Otherwise, the water consumptions in the lower reach are determined in the following procedure and the evaporation on the bare soil only comes from the precipitation.E bL is thus calculated as If the inflow cannot fully meet the water requirement, the natural vegetation water requirement will not be fully satisfied and the minimum water consumption of natural vegetation will be the local precipitation.The annual evapotranspiration of the natural vegetation, i.e., E tL , is given as (10) Finally, the annual evapotranspiration of the irrigated crop area, E cL , is given as

Natural vegetation dynamics of ecological sub-systems
The dynamics of natural vegetation cover is described by the Levins model (Levins and Culver, 1971;Tilman, 1994), which is a logistic growth curve equation (Baudena et al., 2007).This vegetation dynamical model has been applied and validated in the Tarim River in Liu et al. (2012a, b).The dynamical equation of vegetation cover of the upper reach is given as where g V represents the colonization rate and m V represents the mortality rate.V CM represents the maximum of V C .It could be determined by human planning or by the following equation, in which V CMU is the ratio of the vegetated area that the available environmental water could feed to the modeling unit area.
V CMU = available environmental water/water requirement per unit area A U (13) Adopted from Liu et al. (2012a, b), the colonization and mortality rates of natural vegetation depend on the environmental water supply, which basically comes from groundwater reservoirs recharged by streamflow in the river.The readers are referred to Liu et al. (2012a) for details about the hydrological situation in the main stream of the Tarim River.The dependent relationships are shown in Fig. 3, which are described by the following equations.
where g VU , m VU1 , m VU2 and r EWSUC are the empirical parameters.r EWS can be considered as the environmental water supply ratio, i.e., the ratio of available environmental water to the environmental water requirement, and as dimensionless, in [0, 1].r EWSUC is the threshold value of r EWSU , where r EWSU is defined as where T is 1 year and is equal to the time step of the environmental water requirement, and W ERU is the environmental water requirement.
Similarly, the dynamical equations of vegetation cover of the lower reach are V CML = available environmental water/water requirement per unit area The dependent relationships of g VL and m VL are similar and presented in Appendix A for readability.The meanings of all the symbols used above are reported in the separate nomenclature (Table A1) presented at the end of the paper.

Dynamic equations of economic sub-system and social sub-system
The dynamics of the irrigated crop area is balanced by wasteland cultivation and farmland abandonment.The process of its evolution can also be described by the logistic-type equation, whose form is similar to the vegetation dynamical equation (Levins and Culver, 1971;Tilman, 1994).Good andReuveny (2006, 2009) have also presented this kind of conceptualization.In their work they also used a similar equation to describe the resource stock in the ecological-economic model of human-resource interaction.Originally, the logistic growth model is introduced to simulate the growth of biological systems.Subsequently, there have also been several applications of the logistic growth model outside the field of biology.As summarized by Tsoularis and Wallace (2002), the logistic growth model has been used to describe the market penetration of many new products and technologies, world energy usage and source substitution, as an evolutionary process of the industrial revolution.For this case, the evolution of the irrigated crop area is driven by wasteland cultivation and farmland abandonment, which corresponds to the colonization and mortality in our vegetation dynamic equation.We assume that this evolution can be roughly described by the logistic growth model.The dynamical equation of the irrigated crop area ratio of the upper reach is The terms g RU and m RU represent the impact of available water to the area of irrigated field.The terms g R2U and m R2U represent the impact of natural vegetation cover of the upper reach on the area of irrigated field through the environment protection policy.The terms g R3U and m R3U represent the impact of natural vegetation cover of the lower reach on the irrigated field area of the upper reach through the environment protection policy.The dependent relationships adopted are similar to those in Fig. 3 and are also listed in Appendix A.
Similarly, the dynamical equations of irrigated crop area ratio of the lower reach are R IML = available irrigation water/water requirement per unit area Again, the dependent relationships of g RL , m RL , g R2L , and m R2L are described in the Appendix A, and the meanings of symbols are presented in Table A1.
In the socio-economic system, the dynamic evolution of the population is traditionally simulated by the logistic growth model (Tsoularis and Wallace, 2002) although it is usually complicated by human migration and other factors.Both the colonization and mortality terms are dependent on the environment and agriculture.The dynamical equation of the population of the upper reach is where g NU and g N2U represent the colonization and migration rate of the human population.m NU and m N2U represent the mortality and emigration rates.N M represents the maximum of N. It could be assigned depending on the planning arrangement.
The terms g NU and m NU represent the impact of the environment, i.e., V CU to the population of the upper reach.The terms g N2U and m N2U represent the impact of the agriculture, i.e., R IU to the population of the upper reach.The dependent relationships are similar to those in Fig. 3 and are described in Appendix A.
Similarly, the dynamical equation of the population of the lower reach is, The descriptions for the dependent relationships and symbols are referred to in Appendix A and Table A1.

Feedback loops in the socio-hydrological system
The socio-hydrological processes are coupled by dependent relationships and feedbacks generated in the sociohydrological system.There are four main feedback loops in the socio-hydrological system of the Tarim River, as shown in Fig. 4. The first feedback loop, W U -V CU -R IU -W U , is an inner loop occurring within the upper reach.This is a negative feedback.If the inflow to the upper reach increases, the allocatable water resources (W U ) will increase and then there will be more water to foster natural vegetation (V CU ).With the increase of V CU , the environmental conditions become better and, thus, the irrigated crop area will expand and the irrigation water consumption will increase correspondingly.As a result, the allocatable water resources (W U ) will decrease and receive a negative feedback.The second feedback loop, W L -V CL -R IL -W L , is the corresponding inner loop occurring in the lower reach.It is also a negative feedback.
The third loop, V CL -W U -W L -V CL , is the outer loop linking the upper and lower reaches.If the natural vegetation in the lower reach (V CL ) decreases (degrades), the allocation of the water resources in upper reach (W U ) will be inclined to increase discharge to the lower reach (Q outU ), which depends on water resources management and vegetation protection policy.So the inflow of the lower reach will increase and there will be more water to allocate in the lower reach (W L ).With more water supplied to natural vegetation, the natural vegetation in the lower reach (V CL ) will recover.Obviously, this is also a negative feedback.It is primarily controlled by policies and laws, which are driven by the community awareness discussed in Elshafei et al. (2014).In the baseline model, this feedback is not in effect.Its role will be analyzed later in a subsequent section.
The fourth loop, R IU -W U -W L -V CL -R IU , is a related outer loop linking the two modeling units.This is also a negative feedback.If the irrigated crop area in the upper reach (R IU ) increases, more water (W U ) will be used by irrigation in the upper reach and less water will be released to the lower reach.So the allocatable water resources (W L ) will decrease and there will be less water for the natural vegetation in the lower reach (V CL ).This may lead to a decrease of the natural vegetation (V CL ) and then the irrigated crop area in the upper reach (R IU ) may decrease because of environment protection policy.In the equations, g R3U will decrease and m R3U will increase with the decrease of V CL .As a result, R IU receives a negative feedback.The dependent relationship of the irrigated field area of the upper reach (R IU ) to the natural vegetation coverage of the lower reach (V CL ) is the key chain of the feedback loop.

Socio-hydrologic evolution processes within the Tarim River basin
The conceptual model for socio-hydrologic co-evolution is applied to the main stream of the Tarim River at an annual time step from 1951 to 2010.

Parameters of the model
The parameters of the model are listed in Table A1.The estimation of the parameter values is important for model application.A total of 58 parameters arise from the constitutive relationships presented in the model description above, almost all of which are not measurable directly, at least  A1.

Initial values of the systems states
The initial values of the system states are obtained from the literature and are listed in Table 2.The initial allocatable water volumes of both modeling units are assumed to be zero.The initial population of the upper reach is 17 × 10 4 persons, which refers to the actual population in 1951.The initial population of the lower reach is 21 × 10 4 persons.The vegetation cover and irrigated crop area ratio are assigned reasonable initial values referring to the statistic book and expert knowledge.

Dynamics of the socio-hydrological system
The simulation results of discharge, vegetation cover, ratio of irrigated area and population of the two modeling units are shown in Figs.5-11, respectively.Although the water balance equation is at the annual scale, the simulated outflow of the upper reach is close to the observed outflow, as shown in Fig. 5.For comparison, the observed inflow of the upper reach is also shown in Fig. 5 and the simulated outflow shows almost the same trend as the inflow.The average annual runoff of the inflow at Aral from 1957 to 2008 is 4.536 × 10 9 m 3 and the average annual runoff of the outflow at Yingbazha is 2.760 × 10 9 m 3 , which is 60.8 % of the inflow.The simulated annual mean runoff of the outflow is 2.312 × 10 9 m 3 , which is 51.0 % of the inflow, and is 16.   and 7, the simulated vegetation cover decreases from 1951 to 2010 and stays at the similar level with the vegetation cover of the vegetation reference point of the upper reach and lower reach, respectively.In future, the vegetation cover evolution should be validated based on more historical data.Irrigated crop area is one of the critical variables of the socio-economic system in the Tarim River basin.The catchment area of the main stream of Tarim River is 4.6 × 10 4 km 2 and the main stream is the typical inland river, where no runoff can be generated and all water comes from the headwaters (Song et al., 2003).In the calculation of the irrigated crop area ratio, the area of the modeling unit represents the area within 10 km of both of the riversides along the river channel.Most of the farmland is located within this area.The simulated values of irrigated crop area ratio are shown in Figs. 8 and 9. From 1951 to 2010, the simulated irrigated crop area ratio (R I ) increases throughout the simulation period and is similar to the observed R I from 1989 to 2010.The average absolute value of relative error of the simulated R I in the upper reach is 5.2 and 12.3 % in the lower reach.The R I of the upper reach is much higher than the R I of lower reach because there is more water in the upper reach than in   the lower reach of the Tarim River, which is an inland river basin.In contrast, there is more runoff in the lower reach than in the upper reach, which is normal in the exorheic rivers.The human population is another important variable in the socio-economic system, especially in this agriculturedominated river basin.The simulated population is shown in Figs. 10 and 11.In both modeling units, the simulated population numbers are very close to the observed values.Although, the simulated population of the lower reach is higher than the observed after 1990, the dynamical equation describes the evolution of the population quite well.The average absolute value of the relative error of the simulated population in the upper reach is 3.9 and 2.7 % in the lower reach.Based on the outcomes of the co-evolution model, it appears that the system has not yet reached a steady state.The inflow of the upper reach and the policy of the river basin management, i.e., water allocation scheme, will influence the future trajectories of the system status.
The evolution of the socio-hydrological system is driven by the interactions of humans and water as governed by the Taiji-Tire Model (Liu et al., 2014), and the productive and restorative actions of humans (van Emmerik et al.,   2014) invoked either actively or passively, intentionally or un-intentionally, are at the core of these interactions.In fact, the observed co-evolution is the consequence of the balancing of water's economic and ecological value.At different stages of the socio-hydrological system, the dominant driving forces may be different.During the study period the dominant driving force was indeed the productive force, i.e., expanding agricultural production within the Tarim River basin.The realization of productive force is the water allocation scheme established as part of the river basin management.From 1951 to 2010, agricultural production increased significantly and contributed to the growth of agricultural productivity.During this period, irrigation water was unconstrained and water that otherwise would have served the ecological system was instead exploited and consumed for agricultural irrigation.The ecological water ratio, i.e., the ratio of ecological water to the total water consumption, decreased from 67.0 % (1951-1990) to 35.1 % (1991-2010).Consequently, vegetation cover decreased, as shown in Figs. 6 and 7.
The degradation of the ecological system since 1990 contributed to a re-evaluation of the original water allocation scheme within the Tarim River basin.A research project by Xi'an University of Technology and the Tarim River basin  Management Bureau, funded by the Ministry of Water Resources of the People's Republic of China, studied a more rational water resources allocation scheme for the Tarim River basin.The results of a scenario analysis suggested that the ratio of ecological water to the total water consumption could reach 50.2 % if the recurrence interval of the annual runoff at Aral in 2020 is 2 years in the commendatory scenario.With the implementation of the new water resources allocation scheme, the dominant driving force may have been switched to the environmental restorative force.For the model, it means that the negative feedback, outer loop V CL -W U -W L -V CL , should be switched on to analyze the long-term evolutionary dynamics.
In order to study the evolution of the socio-hydrological system, precipitation, evapotranspiration and inflow are repeated another 4 times after 1951-2010 to obtain a synthetic time series of 300 years.In the current modeling framework, denoted as the baseline model, a quasi-steady state of the system is reached in the 300-year simulation.The dynamics of the resulting co-evolution are shown in Fig. 12.After 2100, vegetation cover, irrigated crop area ratio and population almost approach quasi-steady states.The average values of system variables in the last 60 years, i.e., from 2191 to 2250, are shown in Table 3.It shows that 34.6 % of the inflow is released into the lower reach.The average vegetation cover of the upper reach is 0.220 and 0.005 in the lower reach (much smaller than that in the upper reach).The average irrigated crop area ratio of the upper reach is 0.299 and 0.115 in the lower reach.The average population of the upper reach is 109.7 × 10 4 and 50.5 × 10 4 in the lower reach.All 6 variables above are much smaller than the maximum values shown in Table A1.
In the baseline model, the outflow of the upper reach (Q outU ) is not connected to the natural vegetation of the lower reach.In fact, after 2000 the outflow of upper reach has been now regulated through changes to the river basin management policy in order to restore the natural vegetation of the lower reach, i.e., the negative feedback V CL -W U -W L - V CL is now in effect.So the parameter k Q in Eq. ( 4) could be calculated as

Hydrol
where k qa , k qb and k qc are parameters, as shown in Table A1.
The negative feedback V CL -W U -W L -V CL is quantitatively described through the constitutive relationship of V CL and After the decrease of V CL , k Q will increase and then more water will be released into the lower reach.As a result of increase of water in the lower reach, V CL will increase.So the outflow of the upper reach (Q outU ) in Eq. ( 4) is The resulting model is denoted here as the revised model.The dynamics of the Tarim socio-hydrological system governed by the inclusion of Eq. ( 25) and using the 300 years of forcing data are shown in Fig. 13.It shows that the natural vegetation of the lower reach is obviously improved.The average values of system variables in the last 60 years are compared in Table 3.In contrast, the vegetation cover, irrigated crop area ratio and the population in the upper reach in the last 60 years modeled by the revised model are smaller than those in the baseline model, while the vegetation cover, irrigated crop area ratio and population in the lower reach in the last 60 years modeled by the revised model are larger.This behavior is attributed to the equation for Q outU , i.e., Eq. ( 25), which is the driver for water release from the upstream to the downstream.In Eq. ( 25), as the vegetation cover in the lower reach decreases, the environmental feedback forces the system to release more water to the downstream; i.e., the third feedback loop of V CL -W U -W L -V CL , is thus activated in the revised model.In this way, the restorative force is invoked to restore the vegetation in the lower reach.With water flowing into the lower reach, vegetation cover, irrigated crop area and population also effectively "flow" into the lower reach.The runoff flowing into the lower reach increases by 36.7 % and the variable which changes most is the vegetation cover in the lower reach, which increases from 0.005 to 0.017, i.e., an increase of 240.0 %, as shown in Table 3.The state to which the vegetation cover could be restored is determined by the water resources allo-  cation, i.e., the relative preference by humans between the economic value and the ecologic value.It is exhibited as the relative priority given to water resource allocation between the upper and the lower reaches, and between different sectors within one reach.Considering the whole simulation, we can find a costly pendulum swing between a balanced distribution of socio-economic resources and natural ecologic resources for the upper and lower reaches and a centered distribution in the upper reach.This pendulum swing of spatial distribution of resources is very similar to the pendulum swing of values between agricultural socio-economic benefits and ecosystem services found in the Murrumbidgee River basin by Kandasamy et al. (2014).In fact, the first pendulum swing is driven by the second one.
In both the baseline model and the revised model, the socio-hydrological system reaches a quasi-steady state after 2100.The rate at which the quasi-steady state is reached turned out to be faster than our intuition suggested, which could be ascribed to the absence of technology improvement in the co-evolution model.As irrigation technology advances, the crop coefficient of evapotranspiration (k c ) will decrease and irrigation water requirement will decrease.As a result, the quasi-steady state may be attained much later.The importance of technological advances was highlighted by Good and Reuveny (2009), who, however, assumed that technology is static in their work.Alvarez et al. (2011) and Bilancini and D'Alessandro (2012) included the development of technology in their social system model.Based on the results presented here, it is clear that the technological advances should be incorporated in future efforts at the modeling of socio-hydrological systems, also following the example of van Emmerik et al. (2014).

Sensitivity analysis
In order to assess the effect of the initial values and boundary conditions to system quasi-steady state, the sensitivity of system behaviors to initial values and boundary conditions, i.e., the precipitation, potential evaporation and inflow of the upper reach, was analyzed with the baseline model.By increasing and decreasing the initial values, precipitation, potential evaporation, and inflow of the upper reach by 10 %, the model is re-run under different combinations of initial and boundary conditions; the results are listed in Table 4 and results show that all the tested conditions can alter the system's quasi-steady state except for the initial values.In our tested ranges, the system presents unique quasi-steady states without regard to initial values.The relative change rate of quasi-steady states compared with the baseline results are shown in Table 5, which indicates that the changes of the precipitation have a slight effect with the range from −1 to 1 %, which can be attributed to the absolutely small precipitation amount in this area.Otherwise, the potential evaporation (very high compared to precipitation) and inflow of the upper reach have significant effects, especially, on the vegetation cover of the lower reach (V CL ).Furthermore, our test results show that initial/boundary conditions have only a slight impact on the populations of the upper and that the lower reaches, and the relative change is in the range from −1 to 1 %.Generally, the boundary conditions including potential evaporation and upper reach inflow play an important role in the system co-evolution.
Analyzing the sensitivity of the parameters in the model is an important approach for identifying the critical parameters that affect the performance of the model.In the hydrological sub-system, the crop coefficient of evapotranspiration (k c ) is an important parameter for system behavior.The k c is decreased or increased by 10 %, respectively, to assess its effect on the system quasi-steady state and the results are listed in Tables 4 and 5.The k c has significant effects on the vegetation cover of the lower reach (V CL ) and slight effects on the other variables.In the ecological sub-system, economic sub-system and social sub-system, the state variables are governed by a logistic-type equation, whose characteristics have been investigated by many scholars (Levins and Culver, 1971;Tilman, 1994;Tsoularis and Wallace, 2002).Because the interactions between the sub-systems will affect the colonization rate and mortality rate, the colonization rate only changes in the range from 0 to g VU0 and mortality rate change in the range from m VU1 to m VU2 , taking the vegetation cover of the upper reach as an example.The evolution process and quasi-steady states are determined by the colonization rate, mortality rate and carrying capacity (i.e., the maximum of the variable, such as V CMU ).The parameters of the vegetation cover of the upper reach (V CU ) were decreased or increased by 10 % to assess the effects on the quasi-steady states of the system and the results are shown in Tables 4  and 5. V CMU and g VU0 have a marked effect on the quasisteady states of V CU and V CL because the change of the V CU induces the change of the outflow from the upper reach.The quasi-steady states are not sensitive to the other parameters.The results are consistent with the performance of the parameters in the modeling calibration.These two types of parameters, i.e., growth rate (such as g VU0 ) and maximum of the variable (such as V CMU ), have important effects on the quasi-steady states.

Conclusions
For socio-hydrological systems, their hydrological processes, ecological processes and socio-economic processes are coupled together via water consumption activities and water allocation policies.To explore such interactive processes, a conceptual dynamical model is developed by coupling the water balance equation for hydrological process and logistic growth equations for evolution of vegetation, irrigation, and population.Four state variables, i.e., water storage, vegetation cover, irrigated crop area ratio, and human pop-ulation, are adopted to represent the state of hydrological, ecological, economic, and social sub-systems, respectively.Each growth equation contains several colonization terms and mortality terms, which are jointly determined by the state variables of different sub-systems through the corresponding constitutive relations.We recognize that a few previous studies have proposed the concepts of community sensitivity (Elshafei et al., 2014) or environmental awareness (van Emmerik et al., 2014) to explicitly represent the feedback mechanisms between social and environmental systems.In our model, the feedback mechanisms are implicitly incorporated into the model through constitutive relations and hidden feedback loops.At the current stage of our understanding of complex socio-hydrological processes, the logistic growth model can reduce the need for an explicit representation of human behavior and also benefit from the vast amount of literature on the growth model in biologic and social sciences.
Forced by the annual precipitation, pan evaporation and streamflow of the headwater basins, the co-evolution model reproduces the past trajectories of the human-water system in the Tarim River basin.The simulated evolution processes are consistent with observed patterns such as the outflow of the upper reach, vegetation cover, irrigated crop area ratio and human population, which suggests a reasonably good performance of the model.The long-term simulation results from both baseline and revised model runs show a pendulum swing between a balanced distribution of socio-economic resources and natural ecologic resources for the upper and lower reaches and a highly skewed distribution towards the upper reach.The real history of Tarim River basin discussed in Liu et al. (2014) confirms this simulation result.During the traditional agricultural period with lower levels of productive force, the population was distributed relatively uniformly along the Tarim River.When the time of the industrialized agriculture came, the water consumption in the upper reach saw a tremendous increase.As a result, the natural vegetation in the lower reach deteriorated and thus the agriculture and population shrank.The pendulum swung to the opposite end.However, with the increasing awareness of the environment in human consciousness, the society changed the water allocation policy and more water is now required to be released to the lower reach.Consequently, the natural ecological resources and also socio-economic resources could recover in the lower reach, and the pendulum swung back to the former mode.This costly pendulum swing of the spatial distribution of resources is very similar to the pendulum swing of values between agricultural socio-economic benefits and ecosystem services that was observed in the Murrumbidgee River basin in Australia by Kandasamy et al. (2014).Simulation models of the kind presented here can shed light on the possible future trajectories of similar socio-hydrological systems.
In the current model, we acknowledge that there are many parameters in the dependent relationships, which is favorable for a flexible model.In future, the parameters should be reduced after the test of functional forms of dependent relationships.At the current stage of socio-hydrology, our understanding of the dominant socio-hydrological processes is limited and the social behaviors, such as the interim policy of the government and the development of technology (van Emmerik et al., 2014), are not incorporated in the model.The interim policy of the government to promote the economy may have prominent effects on the irrigated area.The management policy, as a kind of social behavior, should be designed and evaluated further to improve the constitutive relationship of the water flux depending on the human activities.The model can also be improved in both of the dominant socio-hydrological processes and the data used in the model.This study focuses on the modeling framework and feedback network, especially the negative feedback loops which make the socio-hydrological system stable.However, the abrupt jump in the evolution, the multiple steady states of the system, the abrupt change of the steady states (Manfreda and Caylor, 2013), and the tipping point or catastrophic critical transition (Scheffer et al., 2009) are all important characteristics of the system, which need further research in future.

Figure 1 .
Figure 1.Location of the main stream of the Tarim River.

Figure 2 .
Figure 2. Sketch map of the main stream of the Tarim River.
et al.: A conceptual socio-hydrological model of the co-evolution of humans and water

Figure 3 .
Figure 3. Dependent relationships of the colonization and mortality of natural vegetation depending on the environmental water supply.

Figure 4 .
Figure 4. Feedbacks in the socio-hydrological system of the Tarim River.

Figure 5 .
Figure 5. Discharge of the upper reach of the Tarim River.The outflow (Q out ) of the upper reach is the inflow of the lower reach.

Figure 6 .
Figure 6.Vegetation cover (V c ) of the upper reach of the Tarim River.

Figure 7 .
Figure 7. Vegetation cover (V c ) of the lower reach of the Tarim River.

Figure 8 .
Figure 8. Ratio of irrigated area (R I ) of the upper reach of the Tarim River.

Figure 9 .
Figure 9. Ratio of irrigated area (R I ) of the lower reach of the Tarim River.

Figure 10 .
Figure 10.Population of the upper reach of the Tarim River.

Figure 11 .
Figure 11.Population of the lower reach of the Tarim River.

Figure 12 .
Figure 12.Quasi-steady state of the socio-hydrological system with the 300-year series (baseline model).

Figure 13 .
Figure 13.Quasi-steady state of the socio-hydrological system with the improved Q outU equation (revised model).

Table 1 .
Modeling framework for socio-hydrology co-evolution of the Tarim River.
where g RU , g R2U and g R3U represent the cultivation rate of new irrigated field.m RU , m R2U and m R3U represent the abandonment rate of the current irrigated field.R IMU represents the maximum of R IU .It could be determined by planning or by the available irrigation water as follows:

Table 2 .
Initial values of system state variables.

Table 3 .
Mean values of state variables during the last 60 years of system evolution.

Table 4 .
Mean values of state variables during the last 60 years of system evolution for sensitivity tests.ConditionsQ outU /10 9 m 3 yr −1 V CU /-V CL /-R IU /-R IL /-N U /10 4 persons N L /10 4 persons

Table 5 .
Changing rate of mean values of state variables during the last 60 years of system evolution for sensitivity tests.