On the Interplay Among Multiple Factors: E ects of Factor Configuration in a Proof-of-Concept Migration Agent-Based Model

Many researchers haveaddressedwhat factors shouldbe included in theirmodels of couplednaturalhuman systems (CNHSs). However, few studies have explored how these factors should be incorporated (factor configuration). Theoretical underpinning of the factor configurationmay lead to a better understanding of systematic patterns and sustainable CNHSmanagement. In particular, we ask: (1) can factor configuration explain CNHS behaviors based on its theoretical implications? and (2) when disturbed by shocks, do CNHSs respond di erently under varying factor configurations? A proof-of-concept migration agent-based model (ABM) was developed and used as a platform to investigate the e ects of factor configuration on system dynamics and outcomes. Here, two factors, social ties andwater availability, were assumed to have alternative substitutable, complementary, or adaptable relationships in influencing migration decisions. We analyzed how populations are distributed over di erent regions along a water availability gradient and how regions are culturally mixed under di erent factor configurations. We also subjected the system to a shock scenario of dropping 50% of water availability in one region. We found that substitutability acted as a bu er against the e ect of water deficiency and prevented cultural mixing of the population by keeping residents in their home regions and slowing down residential responses against the shock. Complementarity led to the sensitivemigration behavior of residents, accelerating regionalmigration and cultural mixing. Adaptability caused residents to stay longer in new regions, which gradually led to a well-mixed cultural condition. All together, substitutability, complementarity, and adaptability gave rise to di erent emergent patterns. Our findings highlight the importance of how, not just what, factors are included in a CNHS ABM, a lesson that is particularly applicable tomodels of interdisciplinary problems where factors of diverse nature must be incorporated.


Introduction
. Recent studies of coupled natural-human systems (CNHSs) have drawn much attention to interplays between social and natural factors in a wide variety of situations (Liu et al. ). Many scholars have focused on "what" factors should be included in CNHS models (An ; Liu et al. ), but the question of "how" these factors should be incorporated has largely been overlooked; herein, we will refer to how factors are incorporated as "factor configuration." This issue was recently raised by CNHS researchers (Boas et al. ; Kramer et al. ; Metson et al. ; Morzillo et al. ). What if factor configuration itself is more significant to outputs than the mere inclusions of these factors? Factor configuration may even build unique patterns, e.g., flickering patterns, yet existing studies were restricted to examining this aspect. Factor configuration reflects a theoretical implication on how the factors interact and thus leads to a better understanding of decision-making processes for CNHSs. .
Agent-based models (ABMs) take a bottom-up CNHS modeling approach that captures how individuals make their own decisions, follow di erent rules, and interact with each other in heterogeneous environments (Farmer & Foley ; Grimm et al. ). ABMs' strengths make them a great modeling tool to study the e ect of factor configuration. First, ABMs are capable of simultaneously handling both natural and social factors (Bell et al. ; Bousquet & Le Page ; Loomis et al. ). Second, ABMs allow us to include factor configuration into an individual's decision-making processes and generate system-level patterns (An ; Gimblett ). Third, we can test di erent shock scenarios and capture emergent CNHS responses against shocks (Waldrop ). .
The objective of this study is to explore the e ects of factor configuration on system dynamics and outcomes of a simple CNHS ABM. To provide the context and narrative for interpreting the results, we use a highly simplified ABM of migration processes. It is important to note here that the simplified migration ABM is used as a platform for investigating the e ects of factor configuration and is not meant to capture all the complexity of migration process. Indeed, migration is a complex CNHS process, involving a large array of natural and social drivers interacting at multiple temporal and spatial scales and producing multi-dimensional outcomes (Abel et al. ; Adger et al. ; Black et al. ; Czaika & De Haas ). Within the context of this simple ABM, we first delved into how di erent configurations of multiple factors influence individual-level decisions and systematic patterns. We especially looked into the flickering-like behavior-agents in two regions collectively switch their locations back and forth-in a certain configuration (in Figures c and d). This study then tested the e ect of factor configurations in a shock scenario. We considered both stationary values and time series of system outcomes a er the shock for this analysis.

Methods
A proof-of-concept migration ABM . In this study, a simple, proof-of-concept migration ABM is used as a platform to investigate the e ects of factor configuration on system dynamics and outcomes. Our ABM incorporates a two-staged migration decisionmaking process (Champion et al. ; Rees et al. ; Stillwell ), focusing on a few key drivers of migration, namely distance, social ties, and the environment (e.g., water availability). We included water availability di erences as a natural factor, based on migration case studies caused by water scarcity (Jägerskog et al. ) and previous examples of ABMs of migration (Krol & Bronstert ; Magallanes et al. ; Rigaud et al. ; Smajgl et al. ). The second factor was cultural a inity-social ties connecting people from the same home region. Social network ties are essential in migration modeling as they are closely related to migration decisions (Klabunde & Willekens ), e.g., information transmission through social connections pulls more people with the same culture to the ethnic enclaves. Lastly, we also used region-to-region distance as a physical factor because the e ect of distance is evident in choosing a destination to migrate (Ravenstein ; Schwartz ).

Model structure .
In the model, five regions are placed at regular intervals along a vector ( Figure ). Each region is assumed to have a di erent total water availability (w i at Region i) that is constant over time, though per capita water availability changes with the population (w i /n i at Region i, where n i is the population at Region i). Water availability of the five regions linearly decreased from le to right (imagine a river or a lake to the le of these five regions), based on the following relationship, w i = s(i − 3) + m, where s represents the water availability gradient and m the average water availability across the five regions. Initially, residents lived in each region and considered it their "homeland." When people in Region i decide to choose a migration destination, they consider the number of people from their homeland in potential destination, Region j (n i,j ), as a beneficial factor (e.g., cultural enclave). Figure : An overview of the proof-of-concept ABM. (A) Five regions (large circles) are located in the -D space with the same distance d to their neighboring regions. Residents (small circles; the color of a resident represents his/her homeland) prefer people from the same homeland (cultural a inity). Empty circles within each region indicate that a resident has migrated to another region. Region has the highest water availability due to its proximity to the water source (e.g., river, lake, etc.; le blue area). Water availability in each region linearly decreases with the distance from the water source (heights of the blue bars). (B) A closer look at Region illustrating di erent migration probabilities.

.
In this model, the decision-making process of a resident is divided into two stages. In the first stage, a resident (small circle in Figure a) decides whether to stay or to leave. Low water availability and/or weak social ties "push" people from the current region, while an aggregate e ect of the two factors is dependent on the factor configuration. In the second stage ("pull"), he/she chooses which region to migrate to. The residents then consider the destination's water availability and stronger social ties as well as the distance from their current region; as in the first stage, the total "pull" e ect depends on the factor configuration. Residents in each region equally share the water available in that region at every time step and make decisions according to their social and environmental conditions. Each resident has a probability of staying µ o (t) (o = origin) (first-stage decision). For those who decide to leave, they have a probability θ j←o (t) (j = destination) to go to Region j (second stage decision). Social ties and water allocated to a person a ect both stages, while distance is only considered in the second stage. Depending on the type of factor configuration described in the next section, the migration probabilities of the two stages take di erent forms. An ODD protocol (Grimm et al. ) of the ABM provides a more detailed explanation in Appendix C. The model code can be found at: https://www.comses.net/codebase-release/43c48c8c-a654-4c64-94ef-f6a477c08702/.

Factor configuration .
Three possible configurations between natural and social factors are explored: ADD, AND, and OR (Figure and  Table ).
. Substitutable factor configuration (ADD) linearly combines the two factors, following the constant elasticity of substitution (CES) production function in neoclassical economics (e.g., Fenichel & Zhao ; Markandya & Pedroso-Galinato ). The relation between factors can be compared to drinking soda products. You could drink, for example, one of two alternative brands (factors). If one is insu icient, another can be an alternative.
The two factors have the same functionality so that one can substitute for the other. Despite this rather strong implicit assumption of substitutability among factors, this configuration is the most widely used in many migration models (e.g., An et al. ; Cai & Oppenheimer ; Li et al. ).
Figure : First stage staying probability at an origin o, µ o (t), with respect to di erent conditions of water availability and social ties. Each plot corresponds to a di erent factor configuration: (A) ADD configuration, (B) AND configuration, and (C) OR configuration (see Table for details).

Configuration Equation
(1) (3) jo Table : Two-staged migration probability equations of ADD, AND, and OR configurations. µ o (t) is the probability that a resident stays (first stage F 1 ; migration probability θ o (t) = 1 − µ o (t)). x (1) is per capita water availability (natural factor), and x (2) is a proportion of homeland residents in the current region (social factor). θ j←o (t) is the probability that a migrating resident goes to Region j (second stage F 2 ). ∆x (1) jo and ∆x (2) jo are relative x (1) and x (2) values between origin and destination, and ∆x  (1) 2 are parameters used to control the shape of the relationships related to water allocated to each resident (β  (2) 2 are parameters used to control the shape of the relationships related to social ties (β  2 only controls the shape of second-stage probability. In this study, we set all α as . and β as for the simplification. C F represents the normalization constant to satisfy the condition j =o θ j←o (t) = 1 (F = {ADD, AND, OR}).

.
In the complementary factors configuration (AND), the two factors are complementary. For example, both soil moisture and solar radiation are needed for a plant to grow. If either one is insu icient, the plant will wither and die. This case requires both factors for an event to occur. In our model, a resident will stay in the current location (first-stage decision) when the water availability is high and social ties are strong; otherwise, the resident is likely to migrate to other regions. Similarly, in the second stage, the resident prefers a destination with high water availability and strong social ties.
. In the adaptable factors configuration (OR), one factor alone can be enough for an event to happen. For example, on a domestic flight, you may bring either a driver's license or a passport as a form of identification. In our model, this means that a resident is likely to stay when either high water availability or strong social ties. For those who leave, they prefer a destination with either high water availability or strong social ties. .
As discussed in previous paragraphs, di erent factor configurations reflect di erent theoretical reasoning. Our work o ers novel insights into the rules of interactions among multiple factors and their implications in a simple ABM. These lessons can further be extended to other CNHS models. It is also worth noting that the mathematical formulations of the factor configuration capture nonlinear interactions between the factors without the "product" terms (e.g., β (1,2) x (1) x (2) ), used in most models that aim to capture nonlinear interactions. .
We explored some parameter values and in our companion paper (Carmona-Cabrero et al. ), also conducted global sensitivity analysis to determine the importance of the factor configuration on the system outcomes compared to the specific values used for the input parameters. We found that factor configuration had an influential e ect (i.e., explains > % of the outputs variance) on population distributions except for the population in Region , where stochasticity was critical to the outcome.

Systematic patterns: Spatial distribution of population and cultural diversity .
To investigate the e ects of factor configuration, we considered two systematic patterns of populations: Spatial distributions of population and the mixing of cultural groups. We ran ABM realizations under each factor configuration. Selecting the appropriate number of realizations (N = 75) is addressed in Appendix A. .
Initially, each region has residents. Then, regional populations are counted a er the system reaches stable dynamics to understand how populations are distributed over five regions. The level of cultural mixing or diversity is also an important social character of a population, influencing population movements in complex ways (Jiang et al. ; Nathan ). Here, Simpson's index from biodiversity literature (Quétier et al. ; Simpson ) is used to quantify how well cultural groups are mixed in Region j. Previous sociological studies have applied Simpson's index in the context of social diversity (Blau ; Rushton ), and the U.S. Census Bureau has used it to explain degrees of ethnic diversity in the U.S. (U.S. Census Bureau ). This index identifies whether a region is dominated by one cultural group or well-mixed with several groups-such cultural diversity may attract some people or cause social tension; although these consequences and feedbacks have not been explicitly incorporated in this present model yet, we believe that it is important to determine how diversity is a ected by the factor configuration. We used an inverse form of Simpson's index for an easier interpretation: where p j i is the fraction of population in Region j whose homeland is Region i (p j i = n i,j / i n i,j ): p i i represents "locals" from that region i. With D j close to , Region j is dominated by a single cultural group from a certain homeland (unmixed cultural condition). When D j is close to (the total number of cultural groups), cultural groups are well-mixed in Region j. .
We also explored population responses a er a disturbance in the CNHS, namely what would happen to the spatial distributions of population and mixing of cultural groups when water supply in Region reduced to % by an unexpected shock?

Results and Discussion
Spatial distributions of population and mixing of cultural groups . We first considered patterns under the pre-shock condition. When the factor configuration was ADD, populations were distributed in a quadratic-like form from Regions -, peaking at Region and dropping at Region (blue boxplots in Figure a); Region had slightly fewer residents, and the population decreased from Region to Region . In water-rich regions (Regions and ), high water availability and strong social ties kept locals in (high µ o (t) in Figures a and b of Appendix B). These regions accommodated more "foreigners" from other regions because weak social ties could be substituted by high water availability. However, foreigners were likely to return to their homelands because when many of these foreigners moved into water-rich regions, their per capita water availability reduced and could no longer substitute weak social ties. In the water-poor region, i.e., Region , low water availability cannot make up for weak social ties, so foreigners avoid the region. Comparing with other regions, Region 's locals also tended to out-migrate more due to low water availability, leading to the lowest population in Region . Region was less populated than Region , despite its higher water availability. Here, distance played an important role in deciding which region to move to in the second-stage decision. Residents in water-poor regions were more likely to migrate to Region simply because Region was closer. Under the ADD configuration, regions were relatively culturally segregated (low values of D j ; blue boxplots in Figure d), with water-rich regions being more culturally mixed. High water availability can substitute for weak social ties of foreigners so that these regions could attract more foreigners. .
A similar trend was observed with a more pronounced gradient in the OR configuration (blue boxplots in Figure   JASSS, ( ) , http://jasss.soc.surrey.ac.uk/ / / .html Doi: . /jasss. c). In this case, either one factor alone was enough to keep a resident. That is, residents were "more tolerant" to changes in either social ties or water availability. It turned out that this adaptability diluted the e ects of social ties, leading to more culturally well-mixed conditions (blue boxplots in Figure f). With greater diversity, all regions had similar strengths of social ties. Residents' decisions now became more dependent on water availability relative to other configurations; more people migrate from water-poor to water-rich regions. Note that the more pronounced distribution of populations implies greater equality in per capita water availability. .
Under the AND factor configuration, spatial distributions of the population had more irregular patterns than ADD and OR configurations (blue boxplots in Figure b). Under this configuration, natural and social factors are complementary: A resident needs both to stay. The results suggested that residents were sensitive to changes in either water availability or social ties, showing highly mobile behaviors. Highly mobile and uncertain characters of residents led to irregular population distributions. Also, all regions became culturally well-mixed conditions and weakened social ties (blue boxplots in Figure e).

Spatial distributions of population and mixing of cultural groups a er shocks
.
In response to a shock of water availability ( % reduction) in Region , residents in Region out-migrated to other regions (red boxplots in Figure ). Median populations in Region reduced to similar amounts ( and ) a er the shock in ADD and OR configurations (red boxplots in Figures a and c), yet post-shock transition forms were di erent between the two (Figrues a and c). Populations in the disturbed region gradually decreased in the ADD configuration. Substitutability absorbed shocks on water availability, preventing a sudden population decrease (Figure a). Moreover, residents in Region moved to other regions at di erent rates. Out-migrating residents in Region were mostly locals under the ADD configuration. As in the pre-shock case, the distance factor was the most critical in choosing which region to move. Thus, the closer a region was to Region , the faster new migrants moved to the region.  Figure c). This behavior appeared inconsistent with the OR configuration's characteristic of being more tolerant, as discussed in the previous section. This seeming inconsistency came from di erent strengths of social ties in ADD and OR configurations. In the ADD configuration, each region became dominated by locals who had strong social ties (Figure d). Regions were culturally mixed in the OR configuration, and residents had weaker social ties. The strength of "pull" became weaker given that regions before the shock were culturally well-mixed, and social ties were not important anymore. .
In the AND configuration, populations in the disturbed region did not change greatly like the other two cases (Figures b and b). Populations in Region were already low under the AND configuration before the shock. This result indicates that per capita water availability was relatively high. Thus, Region was less influenced by the shock than in other configurations. Residents still abruptly out-migrated from Region right a er the shock. It is because staying probability (µ o (t)) drops immediately to deficiency in any one of the factors under the AND configuration (see how µ o (t) changes according to one factor when another factor is su icient in Figure b). This behavior reflects AND's sensitive nature.
. The mixing of cultural groups a er the shock varied across factor configurations. AND and OR configurations did not exhibit significant changes in cultural diversity a er the shock (Figures e and f). Both locals and foreigners in Region had weak social ties before the shock due to the culturally well-mixed condition in these configurations. A er the shock, Region residents su ered from low water availability and weak social ties. No matter which cultural group a resident belonged to, he/she moved out of the region. Cultural diversity was maintained at similar levels. ADD configuration exhibited more significant changes in cultural mixing in the shock scenario. When Region was disturbed, both locals and foreigners out-migrated (population movements of locals were much greater). Foreigners could previously stay in Region as high water availability replaced weak cultural a inity. Now that Region could not make up for weak cultural a inity, most of the foreigners in Region moved to other regions, leading to the almost homogeneous cultural condition (D j ≈ 1). Out-migration from Region increased cultural diversity in other close regions (red boxplots in Figure d). .
Our model results highlighted the importance of how factors are put together in a model. In the context of this proof-of-concept ABM, how agents took water availability and social ties into account-that is, how they configured these two factors-led to di erences in resulting population patterns and a ershock responses. Many of these di erent patterns and changes could be explained by the types of relationships between the two factors encapsulated by our three factor configurations, namely substitutability (ADD), complementarity (AND), and adaptability (OR). When one of the factors became unfavorable at the origin, substitutability provided some bu er to so en the e ects, complementarity led agents to become quite sensitive and incentivized to leave, and adaptability enabled people to stay longer in a new environment a er they migrated.
. Factor configurations were useful, but using them to di erentiate underlying mechanisms of population movements was not always straightforward. For example, the population transition in Region a er the shock was not more tolerant in the OR configuration than in the ADD configuration ( Figure c). Underlying mechanisms fit well into the theoretical implications of factor configuration if all factors are at similar levels. Nonetheless, factor configuration still o ered a way for us to interpret some of these di erent situations in a coherent way.
. Although the population time series reached stationary ranges in the current parameter set, this is not always the case. With larger β values (β  Figure d). Interestingly, such back-and-forth migration patterns have been documented in some migration cases of the real world, e.g., short-term migration in India (Bala ), camps of displaced citizens in Haiti (Fondation Scelles ). Transnationalism literature also discussed back and forth migrations between two locations (Adams & Kasako ; White ). This simple model was rich enough to exhibit diverse patterns according to the changes in β values. β parameters did a ect the model outcomes, yet factor configuration more critically controlled the outcomes, proven by our companion paper (Carmona-Cabrero et al. ). To be clear, the flickering-like behavior was only observed when β values were large in the AND configuration, but not in other cases. Figure : Regional population time series in the AND factor configuration with di erent parameters. Plots represent one realization over realizations. The plots were under pre-shock conditions.

Conclusions
. This study used a proof-of-concept ABM in a stylized migration problem to answer the following research questions: ( ) how does factor configuration between social and natural factors a ect systematic patterns? and ( ) how are post-shock responses distinguished by the factor configuration? In this particular ABM, factor configuration reflected the interactions between cultural a inity and water availability in influencing an agent's decisions. Our results suggested that spatial distributions of the population and the mixing of cultural groups can di er significantly according to the factor configuration. Substitutability, complementarity, and adaptability (ADD, AND, and OR configurations, respectively) exhibited di erent spatial distributions of population and cultural mixing. In the ADD configuration, substitutability resulted in quadratic spatial distributions of population and cultural segregation. High mobility and uncertainty of complementarity shaped irregular population distributions and culturally well-mixed conditions in the AND configuration. Adaptability showed linear population distributions proportional to water availability and high cultural diversity in the OR configuration. Furthermore, we observed nonlinear responses in emergent patterns, both population distribution and cultural mixing, under di erent factor configurations in a shock scenario. ADD and OR configurations exhibited great population changes in Region , while the change was more abrupt in the OR case. AND and OR configurations maintained culturally well-mixed conditions to similar levels a er the shock. On the other hand, ADD configuration had di erent post-shock responses. Region became almost culturally unmixed, and Regions -increased their degrees of cultural mixing. These a ershock population patterns were sometimes unexpected because social and natural factor conditions were di erent for all configurations right before the shock. Population transitions, population distributions, and cultural mixing a er the shock could still be explained through theoretical implications of factor configuration considering these di erences. In sum, these results highlight the importance of how, not just what, natural and social factors are incorporated into ABMs. .
In this paper, we aimed to highlight the e ect of factor configuration rather than to build a realistic model of any particular system. We are aware that real-world migration involves other drivers related to politics, demographics, and economics (Black et al. ). Incorporating all these drivers in the model, however, would have hindered us in answering our research questions in a clear fashion. Thus, our ABM was purposefully built upon a minimalistic design in the conceptual environment. We reduced migration decision-making dimensions by selecting cultural a inity (social factor) and water availability (natural factor) as drivers. Such a design facilitated us to understand the primary mechanisms of the problem-the e ect of factor configuration. Our future work includes implementing several factor configurations to real-world cases and verifying applicability in more complex models, using the present model as a benchmark to capture missing migration factors (e.g., disaster, fatality, income, etc.) at di erent stages. .
While our focus here is on how di erent factors are configured in an ABM, ABMs of such complex processes as migration have other challenges: They tend to involve many parameters to codify the many rules that govern their agents, so many that sometimes it is di icult to tell which parameters and their interactions drive the model outcomes. Our ongoing work addresses that challenge by conducting global sensitivity and uncertainty analysis on the model to disentangle the direct and interactive e ects of the model's inputs (Carmona-Cabrero et al. ). This helps to inform management of the system and to evaluate the model.

.
Our lessons and findings about the importance of how, not just what, factors are included in an ABM are applicable to other models of co-evolutionary systems, including CNHS, socio-ecological systems, and socio-technical systems. These models integrate several components-e.g., humans, ecology, hydrology, technology, infrastructure, etc.-and therefore require an understanding of how components are incorporated. Clarifying the effects of the way in which multiple factors are configured will improve the development of these complex models as well as their contributions to the development of a coherent theory.
Since this model is stochastic, we need to run the model several realizations with the same inputs for robust results. The realization size should be large enough to make outputs representative and robust for the same inputs by reducing the the e ect of stochasticity over the outputs. At the same time, it should be computationally e icient without excessive runs to minimize the simulation duration. We used bootstrapping technique to identify the best realization size objectively in this tradeo .
We bootstrapped two outputs at time step (pre-shock condition) in the AND configuration: the population in Region and the degree of cultural mixing in Region (D 3 ) in the AND configuration. AND configuration was the most variable and uncertain from our understanding. The realization size selected from the AND configuration would produce stable outputs in the other two configurations. Also, Region was particularly the most stochastic among the five regions as it o en worked as a transit. Therefore, the optimal realization size selected from these outputs will be used for other outputs and factor configurations. First, the model generated results for iterations with the same inputs. We then resampled N outputs from realizations. Here, N is directly related to the realization size. We tested di erent resampling sizes of N = 5, 10,25,50,75,100,150,200. Coe icient of variation was calculated with N resampled outputs: where µ s is the mean of N resampled outputs, and σ s is the standard deviation of N resampled outputs. c v involves the relative variability of N resampled outputs. Resampling and c v calculation were repeated times for each N .

Purpose
This model is built upon a proof-of-concept condition to verify the e ects of factor configuration in simplified migration decision-making processes. The model particularly focuses on incorporating cultural a inity as a social factor and water availability per resident as a natural factor. The model aims to (i) reveal how di erent factor configurations influence the emergent patterns and (ii) explore how factor configurations are related to changing regimes of system outcomes in response to unexpected shock.

Entities, state variables, and scales
The environment of this stylized model is five patches in a row (in the x-axis direction), representing five independent regions (Figures and ). Neighboring regions are one patch away from each other. Available water in Region j (w j ) is linearly decreasing from Region to : w j = s(j − 3) + m (Table for the explanation of  parameters). Then, w j is equally distributed to each resident in Region j (shown in the le bottom in Figure ). Figure : A NetLogo interface of our ABM. In the le top, you can select how natural and social factors are combined (factor configuration) in the chooser per simulation. Water available in each region is shown in the plot below the chooser. Le bottom part monitors attributes of Region j including population (p j ), water availability (w j ), and water allocated to each resident (w j /p j ). In the center, we display model outcomes-spatial distributions of population and the mixing of cultural groups-in the forms of both distribution graph and time series. Parameters can be controlled in the right column.
Agents of our ABM are residents of five regions (circles in Figure ). Residents take their initial locations (patches at t = 0) as their homelands and form social ties with people from the same homeland due to their cultural background (social factor). Water availability is also an important factor in migration decisions (natural factor). Each resident has di erent levels of social and natural factors in deciding whether to stay or to leave their current locations. A resident calculates a migration probability based on factor levels and parameters for his/her migration decision-making (equations for the migration probability are given in Table and parameters are illustrated in Table ).

Description
Unit Value m Mean of available water for five regions. This input is directly related to the amount of water supply over five regions.
[W ] s A slope of water availability over five regions. Related to how equally/inequally water is distributed in five regions.
[−] - Weight of natural factor (water availability) in calculating first-stage migration probability. Controls a shape of logistic function for the first-stage probability (flat curve versus steep curve).
Weight of social factor (cultural a inity) in calculating first-stage migration probability.
Controls a shape of logistic function for the first-stage probability (flat curve versus steep curve). [−] α (1) Increase/decrease the probability of a resident to leave the current location (controls degrees of first-stage probability) with respect to water availability.
Increase/decrease the probability of a resident to leave the current location (controls degrees of first-stage probability) with respect to cultural a inity.
Weight of natural factor (water availability) in calculating second-stage migration probability. Controls a shape of logistic function for the second-stage probability (flat curve versus steep curve).
Weight of social factor (cultural a inity) in calculating second-stage migration probability. Controls a shape of logistic function for the second-stage probability (flat curve versus steep curve).
Weight of distance factor in calculating second-stage migration probability. Controls a shape of logistic function for the second-stage probability (flat curve versus steep curve).
1 patch Table : Descriptions, units, and values of parameters used in the model. This ABM is built in the artificial world that the units and scales are not based on real-world measures. In Table , "[W ]" is an artificial unit related to the water amount, "[N ]" is a unit for the number of residents, and "[patch]" represents a distance between regions.

Par
Description Initial value p j Count a population size in Region j D j Identify how well cultural groups are mixed in region j Table : Descriptions and initial values of state variables used in the model.

Process overview and scheduling
Every time step, the followings are done.
. First-stage migration decision making: Each resident decides whether to stay or leave the current location. He/she first identifies the current levels of natural and social factors in the current region.
• Natural factor x (1) : Water availability per resident in the current region j (x (1) = w j /p j ) • Social factor x (2) : Strength of social ties in Region j stemming from the same cultural background. x (2) = p j k /p j ; p j k is the population in region j from homeland k) He/she calculates a staying probability at the current location o (µ o (t)) with these factors and parameters in Table . An equation of staying probability depends on the first-stage factor configuration (F 1 ).
1 ( x (2) −α (2) ) Then, the resident rolls a random dice between and . If the dice value is smaller than µ o (t), he/she stays in the current location. Otherwise, he/she decides to leave the current region and proceeds to the second-stage decision making to select where to go.
. Second-stage migration decision making: Each resident who decided to leave the current region in the first stage chooses which region to migrate to. He/she first identifies the levels of natural, social, geographical factors in the other four regions.
• Natural factor ∆x He/she calculates a second-stage migration probability (θ j←o (t)) with these factors and parameters in Table  . An equation of migration probability depends on the second-stage factor configuration (F 2 ). Geographical factor always has an inverse relationship to natural and social factors.
Then, the resident rolls a random dice between and (independent from the previous one). If the dice value is in the range of cumulative probability for θ j←o , he/she chooses Region j as the destination.
. Move: Residents who decided to leave in the first stage migrate to destinations selected in the second stage. Residents who decided not to leave in the first stage just stay in their current region.
. Update social and environmental properties: A er the migration process is finished for all residents, the model newly updates natural and social factors.

Shock scenario
At t = 2001, we drop % of water availability in Region to explore how the system responds to the shock in di erent factor configurations. This drop is kept until the end of the simulation (t = 3000).

Design Concepts
Basic Principles. This model is a proof-of-concept ABM which simplifies environmentally induced migration to test the e ect of factor configuration. Though many drivers exist in this problem (e.g., economic, political, demographic) (Black et al. ), we focus on natural (related to water availability), social (related to cultural a inity), and geographical (related to distance) factors.

Emergence.
Two key outputs of our ABM are the spatial distribution of populations and the mixing of cultural groups. The former explains how populations are spread over five regions, and the latter illustrates how much each region is culturally homogeneous/heterogeneous. For the cultural mixing, we use Simpson's diversity index (D j ) from ecology to quantify a level of how well Region j is culturally mixed.