Simulating activity-pressure formation and adaptive redistribution in a historic district: a physarum-inspired multi-agent approach

Abstract

Historic districts face increasing challenges due to the coexistence of intensive tourism, residents' everyday life, and sensitive ecological environments within constrained urban spaces. To address this issue, using the Shichahai historic district in Beijing as a case study, this study develops a Physarum-inspired multi-agent simulation framework to examine activity-pressure patterns under different temporal and behavioral scenarios. Multi-source spatial data were translated into GAMA model inputs representing movement constraints, agent origins, temporal attraction fields, and redistribution targets. Resident, tourist, mixed-activity, and spatial attraction redistribution scenarios were constructed and repeatedly simulated to evaluate spatial coverage, path concentration, mean visit intensity, resident-tourist overlap, activity-pressure coupling, and model stability. The results show that the tourist scenario produced a more concentrated path, with active units ranging from 9,056 to 9,267 across the four observation dates, compared with approximately 10,639 -10,660 in the resident scenario and 10,721 -10,766 in the mixed scenario. The mixed scenario showed the highest overall mean visit intensity, with a mean-of-means visit value of 228.80, followed by the tourist scenario at 217.09 and the resident scenario at 212.82. Resident-tourist overlap remained low at the district scale, with a mean overlap index of approximately 0.023, while localized maximum values reached 0.958 -0.997, indicating that the co-concentration of simulated resident- and tourist-related movement was limited to specific corridors and nodes. Under the spatial attraction redistribution scenario, path concentration decreased, with HHI declining from approximately 5.45 -5.56 × 10^-4 to 4.19 × 10^-4 and the high-density visit share decreasing from 2.45% -2.76% to 0.95%. These findings indicate that adjusting high-density attraction clusters can reshape simulated path concentration and support a balanced distribution of activity pressure without changing the physical structure of the historic district. The proposed framework provides a dynamic modeling approach for identifying localized pressure concentration and supporting low-intervention environmental adaptive regulation in high-density historic districts.

1 Introduction

Historic districts are spatially identifiable urban areas shaped by the long-term accumulation of historic buildings, traditional street patterns, cultural practices, community relations, and everyday urban functions. Rather than static collections of protected objects, they are evolving urban environments in which tangible heritage, social life, functional activities, and diverse stakeholder interests coexist and interact (J. Huang et al., 2026; Xu et al., 2026; S. Wang et al., 2023). The combination of historic spatial structures and continuing urban use distinguishes historic buildings, making their conservation inseparable from the maintenance of contemporary urban life.

The development and governance of historic districts are shaped by the interaction of inherited spatial morphology, contemporary urban functions, and changing patterns of human activity. Their fine-grained street networks and mixed-use environments accommodate residential life, visitor movement, commercial services, and cultural and leisure activities within spatial systems that are often difficult to modify substantially because of heritage conservation requirements (Othmani et al., 2026).

Within this context, how heterogeneous activity demands are accommodated and distributed within constrained historic street-space systems has become a central governance issue for historic districts. Previous research has indicated that, although tourism growth can support economic regeneration and heritage utilization, excessive spatial concentration may transform residential functions, intensify competition for public space, and increase pressure on residents’ daily environments and sense of place (Chahardowli and Sajadzadeh, 2022; Rêgo and Almeida, 2022; Pourbahador and Brinkhuijsen, 2023; Page and Duignan, 2023). Resilience-oriented governance in historic districts has progressively incorporated perspectives related to overtourism and tourism carrying capacity, with particular emphasis on how visitor density interacts with available space, facilities, resident populations, and heritage resources in shaping spatial pressure (Bang and Jang, 2025; Romero-Ojeda et al., 2025). This further indicates that in localized street-space units, heterogeneous activities are spatially coupled, and the capacity of the existing environment to accommodate daily use will be more severely tested. Accordingly, Resilience-oriented governance in historic districts requires the further coordinated protection of physical heritage, improvement of local facilities, and regulation of activity distribution.

The spatial pressure is closely associated with the configuration of the street network and the distribution of urban functions. Street connectivity, accessibility, route continuity, and spatial legibility influence how pedestrian movement is channeled and distributed, while the location and accessibility of destinations shape the direction and concentration of activity across individual streets and nodes (Chakraborty and Ji, 2025; Sevtsuk et al., 2024; Boivin and Tanguay, 2019). In historic districts, these relationships are further conditioned by inherited street morphology, mixed functional organization, and differences between resident- and visitor-oriented behaviors (S. Wang et al., 2023; J. Huang et al., 2026). Narrow streets, limited access points, spatial bottlenecks, and differences in environmental familiarity may consequently lead different user groups to follow distinct routes and concentrate within particular parts of the network (Zhao et al., 2026; Zhang et al., 2026). Understanding activity pressure therefore requires the joint consideration of spatial configuration, functional attraction, and user behaviors, together with attention to how activity is distributed within the street network.

Representing these patterns requires data that capture the physical, functional, and activity-related dimensions of historic districts at compatible spatial and temporal scales. Multi-source urban data provide complementary information on urban form, functional organization, and human activity, allowing complex spatial conditions to be described more comprehensively than through a single data source (Long et al., 2025; Wambugu et al., 2026). Data sources such as OpenStreetMap (OSM) road-network data, polygon-based street-space data, and high-resolution remote-sensing imagery can characterize the physical structures where movement occurs, including street connectivity, accessible space, and the geometry of narrow or fragmented street environments (Song et al., 2026). Point-of-interest (POI) data provide geolocated information regarding the type, density, and distribution of urban functions, which can be utilized to approximate the spatial organization of resident-serving facilities, commercial services, tourism resources, and other potential destinations (Lyu et al., 2025; Boivin and Tanguay, 2019). Location-based service (LBS) data add a temporal dimension by indicating relative variations in activity intensity that are not readily captured by static land-use or morphological datasets (Long et al., 2025; Wang et al., 2025). Integrating these data allows the physical environment, functional organization, and temporal variations in human activity to be examined within a unified spatial framework, providing a more comprehensive representation of existing urban conditions and differentiated activity patterns.

Furthermore, data-driven spatial analysis methods also offer essential diagnostic insights for the planning of historic districts. For example, Space syntax and street-network analysis quantify connectivity, integration, accessibility, and route-choice potential, thereby revealing the structural properties of urban street systems (Lyu et al., 2025; Chakraborty, 2026). Density-based and spatial distribution analyses identify the concentration and differentiation of urban functions and activity patterns (Lyu et al., 2025). Statistical and spatial regression models examine associations between built-environment attributes and human activity, including nonlinear effects and spatially heterogeneous relationships (Wang et al., 2025). Collectively, these methods provide a strong basis for describing existing spatial structures, locating areas of concentration, and identifying relationships among morphology, function, and activity. However, their results are generally derived from fixed spatial attributes, aggregated observations, or cross-sectional statistical relationships. They are therefore less capable of explaining how activity paths emerge through repeated movement decisions, how residents and visitors interact within a shared street network, and how spatial pressure is redistributed over time or under alternative attraction conditions. This limitation has encouraged a broader shift in urban spatial research from static morphological description toward dynamic process analysis (Luo et al., 2026).

Agent-based simulation can complement static analysis by representing temporal evolution, agent interactions, feedback, and scenario-dependent redistribution(Z. Wang et al., 2026; Zeng et al., 2025; J. Zhang et al., 2013; Wu and Han, 2025). The value of agent-based simulation lies in its capacity to represent spatial patterns as outcomes of iterative interactions rather than as fixed attributes or externally prescribed relationships (Pan et al., 2020). Such models can describe how individual movement responses accumulate over time, how local environmental information influences subsequent decisions, and how repeated interactions generate reinforcement, diffusion, or redistribution at the collective level. The agent-based simulation of bio-inspired and swarm-intelligence have increasingly been used to represent autonomous decision-making, iterative adjustment, and adaptive spatial change under complex urban constraints, thereby extending spatial analysis from pattern diagnosis to process-based and scenario-oriented experimentation (Castrejon-Esparza et al., 2025; Y. Huang et al., 2025).

Among these approaches, Physarum-inspired models provide an alternative perspective by simulating adaptive path formation through local exploration, positive reinforcement, and negative feedback mechanisms. Physarum polycephalum exhibits adaptive foraging behavior through exploration, reinforcement of efficient pathways, and environmental feedback, which has inspired computational models for solving dynamic network problems (Awad et al., 2023). Early computational studies used slime-mould foraging mechanisms to construct routing networks, generate fault-tolerant graphs, support swarm-robot navigation, solve graph-optimization problems, and produce bicycle-path configurations under topographic constraints. The standard Slime Mould Algorithm (SMA) abstracted these biological responses into a population-based stochastic optimizer, using adaptive weights and positive–negative feedback to balance exploration of alternative solutions with reinforcement around higher-quality solutions (Li et al., 2020). Later variants extended the approach to continuous and discrete optimization, engineering design, network configuration, wireless sensor coverage, multi-objective optimization, and mobile robot path planning and obstacle avoidance (Gharehchopogh et al., 2023). Across these applications, the methodological value of slime-mould-inspired computation lies in its capacity to generate and adjust connections through local attraction, iterative search, and adaptive reinforcement (Heeroo and Gukhool, 2024).

Based on this, the study develops a Physarum-inspired multi-agent simulation to compare the relative diffusion, concentration, overlap, and redistribution of activity patterns within a constrained street-space system. During the model’s operation, path structures emerge through repeated interactions among agents, attraction sources, and movement environments. In urban spatial analysis, this method can be used to simulate the relative intensity of path selection and spatial reinforcement under different activity scenarios, which helps reveal how certain street-space units are repeatedly selected, how path concentration occurs, and how potential overlap between different activity agents is spatially formed.

This study selects the Shichahai historic district in Beijing as an empirical case to examine how heterogeneous urban activities interact with constrained historic street-space systems. The study focuses on three interrelated aspects: the formation of simulated activity-pressure patterns under different resident and tourist scenarios, the spatial coupling between resident- and tourist-related activities, and the potential redistribution of concentrated activity patterns through attraction-based spatial adjustments.

The study addresses three research questions:

• How do the spatial patterns of simulated activity pressure vary among resident-oriented, tourist-oriented, and mixed-activity scenarios within the same historic street-space system?

• How do resident- and tourist-related movement patterns overlap spatially, and how do the intensity and localization of this overlap vary across observation periods?

• How does increasing the relative attraction of clustered secondary nodes affect the distribution and concentration of simulated activity pressure within the inherited street-space structure?

In this study, activity pressure is defined as the simulated spatial concentration and co-distribution of resident and tourist-related movement within the same street-space system. It is comparatively evaluated through spatial coverage, visit intensity, path concentration, and resident–tourist overlap under standardized scenario settings. Environmental adaptive regulation refers to low-intervention adjustments to the spatial distribution of attraction and simulated movement pressure within the inherited street-space structure. The resulting outputs represent the relative situation of simulated path reinforcement and spatial coupling rather than direct measurements of pedestrian volume, crowd density, congestion, social conflict, carrying-capacity exceedance, or environmental impact. The overall research framework and analytical procedure are summarized in the Figure 1.

Figure 1

2 Materials and methods

2.1 Study area

This study focuses on the Shichahai historic district in Beijing, China. To capture the influence of the surrounding urban environment on internal movement activities, the study boundary was moderately extended beyond the conventional administrative or conservation boundary of Shichahai. The research area is bounded by Deshengmenwai Street to the north, Di’anmen West Street to the south, Dongjiaochang Hutong, Dongxinkai Hutong, and Zhao Dengyu Road to the west, and Jiugulou Street and Di’anmenwai Street to the east. The total area is approximately 3.92 km² (Figure 2).

Figure 2

Shichahai is in the core area of central Beijing and is closely connected with the northern section of the city’s central axis. It is a representative high-density historic district where traditional urban fabric, tourism activities, commercial services, and residents’ everyday life coexist within a constrained spatial structure. The district contains the connected water system of Qianhai, Houhai, and Xihai, as well as hutongs, courtyard housing, narrow alleys, waterfront spaces, public services, commercial streets, and cultural attractions (Liu et al., 2019). These elements jointly form an integrated urban environment, making Shichahai a suitable case for examining the formation of activity pressure and environmental adaptive regulation in historic districts. On the one hand, the fine-grained street morphology and limited walkable space constrain movement choices and make localized concentration more likely. On the other hand, the mixed activities and complex environment create temporal variation and potential spatial overlap. Therefore, the district provides an appropriate empirical setting for analyzing how activity pressure diffuses, concentrates, overlaps, and redistributes under different temporal and behavioral scenarios.

2.2 Data sources and preprocessing

This study integrated multiple spatial datasets to construct the simulation environment and support the interpretation of activity-pressure patterns in the Shichahai historic district. The datasets included point-of-interest (POI) data, polygon-based street-space data, OpenStreetMap (OSM) road-network data, and hour-time-segmented Baidu Huiyan gridded population activity heatmap data. The heatmap data constitute an aggregated form of location-based service (LBS) data and were used to represent relative spatiotemporal variations in population activity intensity rather than direct pedestrian counts or exact population density. These datasets were used to represent spatial constraints, agent release locations, and attraction sources within the simulation framework. Before being imported into the GAMA platform, all datasets were standardized in terms of coordinate systems, spatial extents, and attribute structures to ensure consistency across spatial analysis, clustering, grid construction, and multi-agent simulation. Detailed information on data sources and preprocessing procedures is summarized in Supplementary Table 1.

Before analysis and simulation, all datasets were projected to WGS 1984/UTM Zone 50N (EPSG:32650), clipped to the study boundary, and standardized for spatial extent and attribute structure. Relevant preprocessing procedures, including category consolidation, spatial extraction, normalization, and format conversion, were performed to meet the requirements of clustering analysis, grid construction, and agent-based simulation. The original POI dataset contained detailed and uneven category labels, some of which were too fragmented for activity-pressure analysis. Therefore, the records were reorganized according to their relevance to residents’ daily activities, visitor movement, transportation accessibility, commercial services, cultural attractions, and ecological or open-space resources. Records that served mainly as nominal spatial labels rather than as activity-related facilities were removed, including place names, building names, road names, address markers, administrative labels, residential building numbers, and other points without clear activity-generating or attraction-related functions. 3,851 classified POI records were retained within the study area. All these processed datasets were subsequently imported into the GAMA platform as model-ready inputs for spatial constraints, attraction nodes, and agent generation.

2.3 Model input construction

The Physarum-inspired multi-agent simulation required a set of spatially explicit model inputs, including the movement environment, simulation units, agent release points, attraction nodes, and attraction redistribution targets. Therefore, the processed spatial datasets were further transformed into model-ready inputs before simulation. The model input construction included five steps: constructing the walkable movement environment, generating grid-based simulation units, defining agent release points, assigning time-segmented attraction weights, and identifying high-density spatial attraction clusters for the spatial attraction redistribution scenario. These steps converted heterogeneous spatial data into a unified simulation structure in which agents could move, respond to attraction sources, deposit pheromone, and generate simulation-unit-level activity-pressure outputs.

2.3.1 Walkable movement environment construction

The first step was to construct a walkable movement environment, which defined the spatial boundary within which agents were allowed to move. In the Shichahai historic district, road centerlines could not adequately represent the actual pedestrian environment because hutongs, narrow alleys, waterfront paths, and irregular street spaces vary substantially in width, continuity, and morphology. Therefore, the polygonal street-space base was used instead of road centerlines as the movement environment of the simulation model.

The street-space base was derived from high-resolution raster imagery through road-space extraction, polygon vectorization, topology verification, and geometric simplification. Invalid geometries, disconnected fragments, and minor topological errors were corrected to improve computational stability. To compensate for positional uncertainty, boundary generalization, and small discontinuities introduced during raster extraction and polygon vectorization, a 20 m buffer was generated around the polygonal street-space layer. The buffered street-space base was then used as the walkable movement boundary. Resident and tourist agents were required to move only within the effective walkable street-space boundary. If a candidate movement step fell outside this boundary, the movement direction was adjusted until the next position remained within the valid movement space.

Owing to the spatial resolution and classification limitations of currently available multi-source urban data, the model could not completely distinguish narrow public alleys, semi-public frontage spaces, courtyard entrances, walls, and private courtyard interiors at the parcel scale. This localized boundary uncertainty may affect the spatial precision of results at the level of individual hutongs or courtyard parcels. Accordingly, the model outputs should be interpreted as district-scale comparative patterns of relative path selection, spatial concentration, and activity distribution, rather than as precise representations of pedestrian trajectories or parcel-level accessibility.

2.3.2 Construct analytical fishnet cells and centroid-based attraction nodes

The second step was to construct standardized analytical units for representing the spatial attraction field in the research area. 200 m * 200 m fishnet was generated over the study area and geometrically intersected with the effective walkable street-space boundary. Only non-empty intersected geometries were retained as analytical units, and the geometric centroid of each retained geometry was calculated as its representative attraction node. A total of 151 attraction nodes were generated within the study area. The 200 m resolution was selected to support the analysis of district-scale differences in spatial attraction and their effects on path diffusion, concentration, and redistribution. A finer grid would subdivide spatially aggregated activity data into excessively dense attraction nodes and amplify local fluctuations, whereas a coarser grid would obscure spatial differences among attraction areas. During preliminary model calibration, alternative fishnet resolutions were compared through trial simulations. The 200 m fishnet provided the clearer balance between retaining meaningful attraction differences and maintaining an interpretable district-scale movement structure.

Let denote the complete set of generated analytical fishnet cells. For each fishnet cell, the effective intersected geometry was defined using Equation 1.

The retained geometries constituted an effective analytical set Q = qi. For each qi ∈ Q, the corresponding geometric centroid was determined using Equation 2.

where c_i denotes the centroid-based attraction node associated with q_i, and x_i and y_i are its projected coordinates in WGS 1984/UTM Zone 50N.

It should be noted that the model used two distinct spatial units. The 200 m * 200 m analytical fishnet cells were used to extract normalized activity-intensity values from the time-segmented Baidu Huiyan heatmap. These values were assigned to the corresponding cell centroids as spatial attraction weights. The finer GAMA simulation units, by contrast, supported agent movement, local sensing, visit recording, and pheromone updating.

2.3.3 Agent release point construction

The third step was to define the release points of different activity agents, because the study aimed to compare activity-pressure patterns under resident-oriented, tourist-oriented, and mixed activity conditions. Resident release points were extracted from residential-related POI records to the spatial distribution of daily living origins within the historic district. Tourist release points were identified from major peripheral roads, access points, gateway spaces, boundary intersections, and transportation interfaces surrounding the study area. These locations represented the primary entry interfaces through which visitors were likely to enter the historic district.

In the model, agent release points define the initial spatial distribution of heterogeneous activity agents. This distinction enabled the simulation to compare how activity pressure emerged from internal residential origins and peripheral tourist entrances. The configuration of agent release points was kept consistent across repeated simulations to ensure comparable scenario evaluation. A total of 107 resident release points were generated from residential-related POI locations, while 57 tourist release points were identified from major access interfaces surrounding the historic district. Resident-oriented and tourist-oriented scenarios activated the corresponding release-point groups separately, whereas the mixed-activity scenario simultaneously activated both groups. These release points represented the spatial origins of simulated agents and were not assigned weights.

2.3.4 Time-segmented attraction weight assignment

The fourth step was to assign time-segmented attraction weights to the centroid-based attraction nodes. Because activity intensity in historic districts varies substantially over time, a single daily average could not adequately represent the differences among resident morning activities, tourist daytime activities, and mixed evening activities. Therefore, hourly urban activity heatmap data were converted into period-specific attraction weights.

Three temporal periods were defined according to dominant activity patterns in the historic district:

Resident morning activity: 07:00-09:00

Tourist daytime activity: 09:00-18:00

Mixed resident-tourist evening activity: 18:00-22:00

For each representative date and observation period, the hourly heatmap values associated with each retained analytical unit were averaged to reduce hour-specific fluctuations and obtain a period-average activity intensity. The resulting averages were then normalized to the interval [0,1]:

In Equation 3, I_i,_h denotes the heatmap intensity of analytical unit i at hourly observation h, T is the number of hourly observations included in the corresponding period,and is the period-average activity intensity of that unit.

In Equation 4, W_i denotes the normalized atraction weight assigned to the corresponding centroid node, while H_min and H_max are the mininum and maximun values of , respectively, among all retained analytical units for the same date and observation period. If , W_i was set to zero. The normalized values were used as attraction weights in the simulation. Nodes with higher W_i values exerted stronger attraction effects on agents, where nodes with lower values exerted weaker effects. A separate attraction-weight layer was generated for each representative date and observation period and applied to the corresponding resident,tourist,or mixed scenario.

The configuration of agent release points and attraction nodes under different simulation scenarios is summarized in Supplementary Table 2. Resident and tourist scenarios differed primarily in activated agent origins, while attraction nodes remained spatially consistent and varied only in their temporal weighting.

2.3.5 POI-based high-density spatial attraction cluster identification

The fifth step was to identify high-density spatial attraction clusters from the cleaned POI dataset using Density-Based Spatial Clustering of Applications with Noise (DBSCAN). DBSCAN defines clusters as density-connected groups of points and can identify clusters with irregular or non-convex geometries without requiring the number of clusters to be specified in advance. It also explicitly distinguishes sparsely distributed observations as noise, making it particularly suitable for spatial point datasets characterized by uneven density, fragmented distributions, and locally concentrated patterns (Sander et al., 1998; Ester et al., 1996). More recent urban studies have further incorporated DBSCAN-based procedures into the identification and classification of urban functional spaces and district-level spatial patterns (Dong et al., 2025; Zhao et al., 2026).

Because all POI data were projected to WGS 1984/UTM Zone 50N, coordinate values were expressed in meters. Let denote the classified POI dataset, where represents the i-th POI and n is the total number of POIs used for clustering. The distance between two POIs, the ϵ-neighborhood of each POI, and the corresponding core-point condition were defined as follows:

In Equation 5, d(p_i, p_j) denotes the Euclidean distance between POIs p_i and p_j. In Equation 6, N_ϵ (p_i) denotes the set of POIs located within the neighborhood radius ϵ of p_i. Equation 7 gives the core-point condition, where is the number of POIs in the neighborhood and MinPts is the minimum number required for p_i to be identified as a core point. The DBSCAN density definitions followed Ester et al. (1996) and Sander et al. (1998).

DBSCAN requires the specification of two density parameters: the neighborhood radius ϵ and the minimum number of points MinPts. To assess the robustness and analytical suitability of the selected parameters, an additional sensitivity analysis was conducted across combinations of MinPts = 10 − 30 and ϵ = 45 – 150m. For each parameter combination, the number of clusters, noise ratio, silhouette score, and spatial differentiation of the clustering results were compared (Figure 3). Lower values generated a greater number of fragmented and locally incidental clusters, whereas values approaching 20 produced more conservative and spatially continuous aggregation structures. The setting of MinPts=20 retained the principal POI concentration areas while reducing the influence of small and weakly supported point groups, and therefore provided an appropriate representation of the spatial aggregation structure of the Shichahai Historic District at the analytical scale adopted in this study.

Figure 3

The 20-nearest-neighbor distance was defined according to Equation 8, where D₂₀(p_i) denotes the distance from p_i to its 20th nearest neighbor, and represents the 20th nearest neighboring point of p_i. The k-distance curve showed that the mean 20-nearest-neighbor distance was approximately 73.44 m, while a structural inflection appeared at approximately 124.10 m. To avoid excessive fragmentation under small neighborhood radii and over-merging under large neighborhood radii, clustering performance was further evaluated under different ϵ values using the silhouette score, number of clusters, and noise ratio. The evaluation indicated that values around the mean 20-nearest-neighbor distance preserved local spatial differences while suppressing incidental noise. Therefore, the final DBSCAN parameters were set as: MinPts = 20, ϵ = 73.44m. Under this parameter setting, 26 clusters were identified from the 3,851 classified POI samples. A total of 2,755 POIs were assigned to clustered areas, while 1,096 POIs were classified as noise points, corresponding to a noise ratio of 28.46%. The detailed parameter settings and clustering results are summarized in Supplementary Table 3. The largest cluster contained 1,375 POIs, indicating a highly concentrated functional agglomeration within the historic district. (Figure 4).

Figure 4

To incorporate the clustering results into the simulation model, the original analytical fishnet framework was retained for assigning attraction values, while the underlying GAMA units continued to serve as the operational simulation units. In the attraction-reorganization scenario, analytical units whose centroid nodes fell within the identified high-density POI clusters were designated as attraction-redistribution targets, and their normalized attraction weights were set to the maximum value of 1. This scenario represented a hypothetical planning intervention in which spatial attraction was strengthened within existing facility concentrations without modifying the physical street-space structure or agent movement rules. By comparing the resulting tourist-oriented movement pattern with the baseline tourist scenario, the model assessed whether attraction restructuring could reduce path concentration and promote a more spatially distributed movement network within the historic district. .

2.4 Physarum-inspired multi-agent simulation model

After the construction of spatial inputs, a Physarum-inspired multi-agent simulation model was developed in the GAMA platform to simulate the adaptive formation and redistribution of activity-pressure patterns within the Shichahai historic district. The model represents how heterogeneous activity agents interact with spatial attraction inputs and local reinforcement signals within a constrained street-space environment. Resident and tourist agents were used to represent different activity patterns, while centroid-based attraction nodes represented the relative attraction intensity of analytical fishnet units derived from time-specific activity data.

The model was designed to simulate relative path selection, reinforcement, and redistribution processes under different temporal and behavioral scenarios. The simulated outputs describe relative patterns of movement coverage, concentration, and resident-tourist spatial coupling rather than actual pedestrian trajectories, pedestrian volumes, congestion levels, or environmental impacts.

2.4.1 Model framework and adaptive reinforcement mechanism

The model was implemented as a spatially explicit Physarum-inspired multi-agent system in GAMA. Its conceptual framework was derived from the self-organizing characteristics of slime mould systems, where local exploration, positive reinforcement, and negative feedback jointly regulate network formation (Li et al., 2020; Gharehchopogh et al., 2023). In this study, the Physarum-inspired mechanism was translated into an agent-based spatial simulation framework to represent the adaptive evolution of movement patterns within a constrained street-space environment.

The model consisted of four interacting processes: attraction response, agent movement, reinforcement accumulation, and signal diffusion-attenuation. Attraction nodes provided initial spatial preferences by generating localized reinforcement signals according to normalized attraction values. Agents sensed surrounding signal conditions, selected movement directions based on local comparisons, and updated the visited locations during movement. Repeated selection of specific simulation units increased local reinforcement, while diffusion and attenuation mechanisms prevented unlimited accumulation and allowed weaker movement traces to gradually lose influence.

During each simulation step, agents were progressively generated from scenario-specific origins and moved within the effective walkable movement environment. Each agent evaluated nearby candidate simulation units according to their current reinforcement signal values and selected the location with stronger local influence. When multiple candidate locations had equivalent signal values, stochastic selection was applied to maintain movement variability. The selected movement process subsequently updated local visit records and reinforcement signals.

The movement environment was represented by discrete simulation units in GAMA. This discretization was required for stepwise agent movement, neighborhood detection, signal updating, and output recording. The resulting 11,733 simulation units served as the operational units for model execution and output calculation. These simulation units were distinct from the 200 m * 200 m analytical fishnet units used for attraction-value assignment and centroid-based attraction node construction.

2.4.2 Agent configuration and movement rules

Two agent types were defined to represent resident- and tourist-related activities. Resident agents were released from residential origins identified in the study area, whereas tourist agents were released from boundary entrance locations. Both agent types followed identical movement rules, while their differences were introduced through release locations and scenario-specific attraction fields.

The agent generation process was controlled by the maximum fluid particle number, which was set as the reference population size for each simulation run. This parameter regulated the progressive release of agents and was not interpreted as the actual number of residents, tourists, or pedestrians. During each simulation step, newly generated agents were released according to the remaining difference between the current agent population and the predefined reference value.

Each agent was assigned an initial random heading. During movement, agents sensed three candidate locations within the local neighborhood, including the forward direction and two lateral alternatives. The candidate with the highest reinforcement signal value was selected as the movement target. When multiple candidates showed identical signal values, one candidate was randomly selected. A stochastic turbulence parameter was introduced to generate local directional variation and avoid deterministic movement trajectories.

The movement distance was determined according to the reinforcement signal intensity of the current simulation unit through an internal movement coefficient. This coefficient controlled relative movement speed within the model coordinate system and was not interpreted as a physical walking speed. When the forward movement direction exceeded the valid movement boundary, the agent adjusted its direction and continued searching within the available movement environment.

After each valid movement, the selected simulation unit updated its cumulative visit record. In mixed-activity scenarios, resident and tourist visits were recorded separately to support subsequent analysis of spatial overlaps between the two activity patterns.

2.4.3 Attraction-driven signal transmission and reinforcement process

The attraction response mechanism linked external spatial inputs with agent movement behavior. Each attraction node continuously provided a localized reinforcement signal according to its normalized attraction value. The signal influenced nearby simulation units within a predefined search range and established the initial spatial preference for agent movement. When an agent entered the influence range of an attraction node, it acquired a temporary attraction-related signal. The acquired signal was carried by the agent during subsequent movement and gradually decreased according to its remaining signal lifetime. During this process, the agent transferred the carried signal to the simulation units through which it moved, allowing attraction information to extend beyond the original node location.

The reinforcement process therefore emerged from the interaction between persistent attraction inputs and agent-mediated signal transmission. The attraction source was maintained throughout each simulation run, and no progressive depletion of attraction strength was introduced. Path redistribution resulted from the interaction among initial attraction patterns, repeated movement reinforcement, stochastic exploration, signal diffusion, and attenuation. The core parameters controlling attraction response and signal transmission, including attraction search range, signal enhancement coefficient, and signal lifetime, are summarized in Supplementary Table 4.

2.4.4 Reinforcement diffusion, attenuation, and model adaptation

The evolution of simulated paths was regulated by the dynamic updating of the reinforcement signal field. Repeated movement through the same simulation units strengthened local signals, increasing the probability that subsequent agents selected these locations. This positive feedback mechanism enabled frequently used routes to gradually emerge within the constrained street-space system.

To avoid excessive reinforcement of incidental movement traces, the model incorporated signal diffusion and attenuation mechanisms. Diffusion extended reinforcement information to neighboring simulation units and supported the formation of continuous movement structures. Attenuation gradually reduced weakly supported signals, allowing the model to maintain a balance between path persistence and spatial adjustment.

Through the combined effects of reinforcement, diffusion, and attenuation, the model generated adaptive changes in relative movement structures under different activity scenarios. These processes enabled the comparison of how different attraction configurations and agent behaviors influenced the distribution and redistribution of simulated activity pressure.

2.4.5 Simulation termination and stochastic control

Model convergence was evaluated based on the stability of the overall reinforcement signal field. The convergence assessment started after 200 simulation steps to allow sufficient time for agent movement and reinforcement development. A simulation run was considered converged when the relative change in total reinforcement signal remained below 0.001 for 30 consecutive steps. If the convergence condition was not reached, the simulation terminated after a maximum of 1,000 steps.

Because agent initialization, directional perturbation, and movement selection involve stochastic processes, repeated simulations were conducted to reduce the influence of individual random realizations. Each date-scenario combination was repeated 50 times using different random seeds while maintaining identical spatial inputs and baseline model parameters.

The core parameters controlling agent generation, movement behavior, attraction response, signal transmission, convergence criteria, and stochastic repetition are summarized in Table 4. The repeated outputs were subsequently aggregated through statistical analysis to evaluate the consistency of simulated activity-pressure patterns across different scenarios.

2.5 Simulation output data and indicator calculation

The simulation outputs were organized according to their processing sequence. The model first recorded a set of basic variables for each valid GAMA simulation unit during every run. These variables were subsequently used to calculate primary indicators of spatial coverage, visit intensity, path concentration, movement reinforcement, and resident-tourist overlap. Because the model included stochastic movement processes, the indicators obtained from individual runs were further summarized across 50 repetitions using measures of central tendency, variability, and uncertainty. The following subsections describe the direct model outputs, indicator-calculation procedures, and repeated-run statistical processing.

2.5.1 Basic output variables

During each simulation run, four basic output variables were recorded for each valid GAMA simulation unit: total visit count, resident visit count, tourist visit count, and pheromone concentration. Basic output variables recorded during simulation are recorded in Supplementary Table 5.

Total visit count represented the cumulative number of agent visits recorded in each simulation unit during a single run. It provided the source variable for the subsequent calculation of spatial coverage, mean visit intensity, path concentration, and high-density indicators. Resident visit count and tourist visit count recorded the visits generated by the two agent types separately. These variables were used to identify simulation units jointly visited by resident and tourist agents and to calculate resident–tourist overlap indicators in the mixed scenario. Pheromone concentration represented the reinforcement-signal value recorded in each simulation unit and supported the analysis of the spatial distribution and strength of simulated path reinforcement.

2.5.2 Indicators calculated within individual runs

The basic variables recorded for each effective simulation unit were used to calculate indicators addressing four aspects of the model outputs: the spatial extent reached by simulated movement, the overall intensity and concentration of simulated visits, the spatial extent shared by resident and tourist movement, and the state of the pheromone-based reinforcement process. All indicators in this subsection were calculated separately for each individual simulation run.

The spatial extent reached by simulated movement was evaluated using active unit count. An effective simulation unit was considered active when it recorded at least one modeled visit during a simulation run. Let I(·) denote an indicator function that equals 1 when the specified condition is satisfied and 0 otherwise. Active unit count was calculated as:

Active unit count was calculated using Equation 9, where denotes the unit-level visit count of simulation unit i in run r, and denotes the corresponding active unit count. This indicator determines how widely simulated movement extended through the effective movement environment. A larger value indicates that modeled movement reached a greater number of simulation units, whereas a smaller value indicates that movement remained within a more spatially restricted range.

The overall level and spatial allocation of simulated visits were evaluated using mean visit intensity and HHI-based path concentration. Mean visit intensity was calculated as the average unit-level visit count across all effective simulation units:

Mean visit intensity was calculated using Equation 10, where denotes the number of effective simulation units. Mean visit intensity indicates the average magnitude of simulated path reinforcement across the effective movement environment. It provides a measure of the overall level of simulated visitation within a run.

To determine whether the recorded visits were broadly distributed or concentrated within a limited part of the movement environment, the proportional visit share of each simulation unit was first calculated as:

The proportional visit share of each simulation unit was calculated using Equation 11, where denotes the total visit count recorded during run r. HHI-based path concentration was then calculated using Equation 12 (Matsumoto et al., 2012):

A higher HHI value indicates that a greater proportion of simulated visits was allocated to a smaller number of simulation units. In contrast, a lower value indicates a more spatially dispersed distribution(Y. Wang and Liu, 2026). Considered together, active unit count, mean visit intensity, and HHI-based path concentration distinguish the spatial extent reached by simulated movement, the overall intensity of modeled visitation, and the degree to which that intensity was concentrated within particular parts of the movement environment.

In the mixed scenario, the spatial extent jointly occupied by resident and tourist movement was evaluated using overlap unit count. An effective simulation unit was classified as an overlap unit when it recorded at least one resident visit and at least one tourist visit during the same simulation run:

Overlap unit count was calculated using Equation 13, where denotes the overlap unit count. It records the spatial condition of resident-tourist overlap without evaluating the relative intensity of the two movement patterns. The calculation of resident-tourist overlap intensity is described separately in Section 2.5.3.

The state of the pheromone-based reinforcement process at the end of each run was summarized using total pheromone concentration and mean pheromone concentration:

Total and mean pheromone concentrations were calculated using Equation 14, where denotes the unit-level pheromone concentration retained in simulation unit i at the end of run r. Total pheromone concentration represents the overall amount of reinforcement retained within the effective movement environment, while mean pheromone concentration represents its average level across effective simulation units. These indicators were used primarily to examine whether the pheromone-based reinforcement process produced comparable output states across repeated runs. They describe the internal reinforcement state of the model and were not interpreted as measures of actual pedestrian volume or observed environmental pressure. To further identify localized activity concentration and resident-tourist spatial coupling, indicators were calculated through normalization, threshold-based classification, and composite processing. This step aimed to transform unit-level visit records into comparable relative measures and was used to identify high-intensity movement areas and quantify the spatial relationship between resident- and tourist-related activities.

2.5.2.1 High-density simulation unit identification

To identify simulation units with relatively high simulated activity intensity, the visit count of each simulation unit was normalized by the maximum unit-level visit count within the same simulation run:

Unit-level visit intensity was normalized using Equation 15, where denotes the normalized visit intensity of simulation unit i in run r. This normalization converts the unit-level visit intensity into a relative scale ranging from 0 to 1, allowing activity concentration patterns to be compared across different simulation scenarios and repeated runs.

A simulation unit was classified as a high-density simulation unit when its normalized visit intensity exceeded a predefined relative threshold:

High-density simulation units were identified according to the classification rule defined in Equation 16, where represents the high-density state of simulation unit i, and τ represents the threshold value. In this study, τ was set to 0.60. Therefore, simulation units with visit intensity reaching at least 60% of the maximum unit-level visit intensity within the same run were identified as high-density simulation units.

Based on this classification, the high-density simulation unit count was calculated in Equation 17:

where epresents the number of high-density simulation units in run r. This indicator describes the spatial extent occupied by relatively concentrated simulated activity.

To further measure the proportion of simulated movement accumulated within these high-density areas, the high-density visit share was calculated in Equation 18:

where represents the proportion of total simulated visits occurring within high-density simulation units. Together, and distinguish the spatial extent of concentrated activity from the degree to which overall simulated movement was accumulated within these areas.

2.5.2.2 Resident-tourist overlap intensity

In addition to identifying the spatial extent of shared resident and tourist movement through overlap unit count, the mixed scenario required an intensity-based indicator to describe the degree of spatial co-occurrence between the two activity patterns. Resident and tourist visit counts were separately normalized using their maximum unit-level visit counts within the same simulation run:

Resident and tourist visit counts were separately normalized using Equation 19 and Equation 20, respectively, where and respectively denote the normalized resident and tourist activity intensity of simulation unit i.

The resident-tourist overlap in intensity of each simulation unit was then calculated as:

The resident-tourist overlap intensity of each simulation unit was then calculated using Equation 21, where represents the relative degree of spatial co-occurrence between resident- and tourist-related simulated activities within simulation unit i. Higher values indicate that resident and tourist movements were simultaneously concentrated within the same spatial locations, whereas lower values indicate weaker spatial coupling.

This indicator describes the relative spatial coupling of simulated activity patterns within the modeled movement environment. It represents potential areas of resident-tourist activity overlap and does not indicate actual pedestrian congestion, social conflict, or environmental impact.

2.5.3 Stochastic simulation repetition and stability assessment

The Physarum-inspired multi-agent model involves stochastic movement processes, including agent initialization, movement direction adjustment, and local path selection. Therefore, repeated simulations were conducted to evaluate whether the observed differences among scenarios were robust to stochastic variation. Each combination of observation date and activity scenario was independently simulated 50 times using different random seeds.

For each repeated simulation, the model exported scenario-level summary indicators, including active unit count, mean visit intensity, path concentration, high-density unit count, high-density visit share, overlap unit count, and resident-tourist overlap intensity. The mean value of each indicator across repeated runs was calculated to represent the overall scenario performance. The standard deviation SD, coefficient of variation CV, and 95% confidence interval CI were further calculated to quantify the variability of simulation outputs.

For an indicator X obtained from n repeated simulations, the mean value was calculated in Equation 22:

Where X_r represents the value of indicator X in the r-th simulation run.

The standard deviation was calculated using Equation 23:

The coefficient of variation was used to describe the relative variability among repeated simulations in Equation 25:

The coefficient of variation was calculated using Equation 24, where lower CV values indicate smaller relative variation across stochastic realizations. The 95% confidence interval was calculated as:

These statistical measures were used to evaluate the consistency of scenario-level simulation outputs rather than to define an absolute performance threshold for model validity. Indicators with relatively larger variability were interpreted cautiously, particularly threshold-dependent indicators such as high-density unit count and high-density visit share, which may be sensitive to localized changes in simulated activity distribution.

The stability assessment results were summarized using the mean values and 95% confidence intervals of the main output indicators. The stochastic variability analysis ensured that comparisons among residents, tourist, mixed-activity, and spatial attraction redistribution scenarios reflected systematic differences in simulated activity-pressure patterns rather than random fluctuations caused by individual simulation runs.

This study did not establish a fixed coefficient-of-variation threshold for determining the acceptability of simulation outputs. This is because the exported indicators represent different aspects of simulated activity patterns and have distinct sources of variability. Continuous indicators describing overall movement structure, such as active unit count and path concentration, reflect aggregated spatial patterns, whereas threshold-dependent indicators, including high-density simulation unit count and high-density visit share, are more sensitive to localized changes in simulated intensity distribution. Therefore, the stability of simulation outputs was assessed by jointly considering the coefficient of variation, 95% confidence intervals, and the consistency of scenario-level differences across repeated simulations.

3 Result

3.1 Stability assessment of stochastic simulation outputs

The coefficient of variation was calculated to assess the relative variability of simulation outputs across repeated runs. The main structural indicators included active simulation unit count, HHI-based path concentration, mean pheromone concentration, total pheromone concentration, overlap unit count, and resident–tourist overlap intensity. Figure 5 presents the coefficients of variation for the main model output indicators across repeated simulation runs.

Figure 5

Overall, the results indicate that the main structural outputs were sufficiently stable for scenario comparison. Threshold-based hotspot indicators showed higher variability. High-density simulation unit count generally ranged from approximately 23% to 45%, while high-density visit share ranged from approximately 16% to 33%. This indicates that hotspot-related outputs were more sensitive to local changes around the high-density threshold. The instant stable delta showed the highest variability, with many values reaching approximately 70% to 90%. Instant stable delta was used only as a convergence diagnostic to determine whether the model had reached a relatively stable running state. Not used as an output indicator for interpreting spatial patterns or comparing scenario results. The subsequent analysis focuses primarily on active simulation unit count, mean visit density, overlap unit count, and resident–tourist overlap intensity, with hotspot-related indicators only used as supplementary references.

The variability analysis indicated that the main structural indicators maintained relatively consistent patterns across repeated simulations. In contrast, high-density simulation unit indicators showed larger fluctuations due to their sensitivity to local intensity changes. Therefore, they should be interpreted as supplementary measures rather than standalone evidence of spatial concentration.

3.2 Differences in simulated movement coverage and path concentration

Active simulation unit count, HHI, and high-density visit share were used to examine the spatial footprint and concentration patterns of simulated paths across the resident, tourist, and mixed activity scenarios. Across the four observation dates.

Active simulation unit count represents the number of simulation units visited at least once during the simulation and is used to indicate the spatial footprint of simulated paths. Panels correspond to the four observation dates. Figure 6 presents the distribution of active simulation unit count across 50 repeated simulation runs for each observation date and scenario.

Figure 6

The active simulation unit count results indicate that the Tourist 9:00-18:00 scenario consistently covered fewer simulation units than the Resident 7:00-9:00 and Mixed 18:00-22:00 scenarios. Across the four dates, the mean active simulation unit count of the Tourist scenario ranged from approximately 9056 to 9267. In contrast, the Resident scenario remained around 10,639 to 10,660, and the Mixed scenario around 10,721 to 10,766. Among the three baseline scenarios, the Tourist scenario had the smallest active-unit range, while the Mixed scenario had the largest active-unit range.

HHI measures the concentration of simulated path intensity, with higher values indicating stronger concentration within fewer simulation units. Panels correspond to the four observation dates. Figure 7 presents the distribution of HHI path concentration values across repeated simulation runs.

Figure 7

The HHI results show a complementary pattern to active simulation unit count. The Tourist 9:00-18:00 scenario consistently presented the highest path concentration across all four dates. The Resident 7:00-9:00 scenario showed the lowest concentration, while the Mixed 18:00-22:00 scenario remained at an intermediate level. This indicates that although tourist paths covered fewer units, their simulated intensity was much more concentrated. In comparison, residents and mixed scenarios showed broader but more dispersed path structures.

High-density visit share depends on threshold-based identification of high-density simulation unit counts. Small stochastic variations around the threshold may cause individual simulation units to move into or out of the high-density category. For this reason, the indicator is more sensitive to local threshold effects than active simulation unit count or HHI. Its main value here is to confirm the directional consistency of the concentration pattern. It was interpreted as supplementary evidence rather than as a core estimate of hotspot intensity. Figure 8 presents the share of simulated path intensity carried by high-density simulation unit counts across repeated simulation runs.

Figure 8

High-density visit share showed the tourist 9:00-18:00 scenario generally had the highest values, ranging from approximately 2.4% to 2.8% across the four dates. The Mixed 18:00-22:00 scenario remained around 1.1%, while the Resident 7:00-9:00 scenario remained lower, around 0.4% to 0.5%. This indicates that simulated tourist path intensity was more likely driven by locally reinforced high-density simulation unit counts.

Overall, these results suggest that the three activity scenarios differ not only in the extent of spatial coverage but also in the degree of path concentration. The tourist scenario was characterized by a smaller spatial footprint and stronger concentration, implying more selective use of the street-space grid. In contrast, residents and mixed scenarios generated broader coverage with lower concentration, reflecting more diffuse use of the internal network.

3.3 Resident–tourist spatial coupling patterns

To further examine the spatial patterns underlying the quantitative differences identified above, mean visit-intensity maps were generated for the three baseline scenarios across the four observation dates. Each map was based on the mean visit intensity calculated for every valid simulation unit across the 50 repeated runs. The color intensity therefore represents the relative strength of simulated path reinforcement after repeated runs, rather than actual pedestrian volume. Figure 9 presents the spatial distribution of mean visit intensity under the Resident 7:00-9:00, Tourist 9:00-18:00, and Mixed 18:00-22:00 scenarios across the four observation dates. Darker units indicate stronger repeated path reinforcement in the simulation. The high-overlap areas represent scenario-specific spatial coupling patterns within the mixed activity period and should not be interpreted as permanent pressure locations or confirmed conflict zones.

Figure 9

The heatmaps indicate clear differences were observed among the three scenarios. The Resident and Mixed scenarios produced relatively continuous internal path networks, whereas the Tourist scenario showed a more selective pattern, with stronger reinforcement concentrated along several corridors and local nodes. In the Resident 7:00-9:00 scenario, path reinforcement was distributed across a relatively wide internal street-space network. In the Tourist 9:00-18:00 scenario, the heatmaps displayed a more concentrated morphology. Several corridors and local nodes showed high mean visit intensity, while other internal areas remained weakly reinforced. The maximum mean visit values were consistently higher in the Tourist scenario. In the Mixed 18:00-22:00 scenario, the overall mean path density was the highest among the three scenarios. Across the four dates, the average mean-of-mean visit value was 228.80, compared with 217.09 for the Tourist scenario and 212.82 for the Resident scenario.

Date differences were also present, but they were less pronounced than the scenario-level contrasts. 3 May recorded the highest average mean-of-mean visit value across scenarios, at 225.98, whereas 27 April recorded the lowest value, at 212.80. This suggests that date-specific activity conditions may influence the overall level of simulated path reinforcement. Nevertheless, the dominant distinction remained scenario-based: tourist paths were more locally concentrated, while residents and mixed paths were more broadly distributed across the internal street-space network.

3.4 Spatial attraction redistribution scenario and activity-pattern adjustment

To further examine the spatial relationship between daily-life movement and tourism-related movement, resident-tourist bivariate overlap maps were generated for the Mixed 18:00-22:00 scenario across the four observation dates. Figure 10 presents the bivariate spatial distribution of resident and tourist simulated visit intensity under the Mixed 18:00-22:00 scenario. Blue indicates stronger resident-oriented simulated path intensity, red indicates stronger tourist-oriented simulated path intensity, and purple indicates simulation units where both resident and tourist intensities are high.

Figure 10

The results show that the overall resident-tourist overlap level remained low but stable across the four dates. The mean overlap index was approximately 0.023 for all observation dates. Although the district-wide average overlap was low, localized high-intensity overlap was observed in specific corridors and nodes. The maximum overlap index reached 0.977 on 23 April, 0.989 on 27 April, 0.958 on 28 April, and 0.997 on 3 May. These values indicate that both residents and tourist agents simultaneously reinforced some simulation units. Therefore, localized areas of high resident-tourist spatial coupling were not evenly distributed across the historic district but were concentrated in a limited number of shared corridors and nodes where both simulated activity patterns showed stronger reinforcement.

3.5 Spatial attraction redistribution scenario

In the spatial attraction redistribution scenario, the baseline daytime attraction field was replaced by attraction sources derived from DBSCAN-identified clusters. This scenario is set to assess how spatial attraction redistribution can change the organization of tourist routes. Supplementary Table 6 compares the main simulation output indicators before and after spatial attraction redistribution.

Compared with the baseline tourist scenarios, the spatial attraction redistribution scenario showed a similar overall mean visit intensity but lower path concentration. Its mean visit intensity was 219.76, close to the average value of the baseline tourist scenarios, which was 217.09. The HHI of the spatial attraction redistribution scenario was 4.19×10^-4, lower than the baseline tourist scenarios, whose HHI values ranged from approximately 5.45 × 10–⁴ to 5.56 × 10^-4. The high-density visit share also decreased from approximately 2.45%-2.76% in the baseline tourist scenarios to 0.95% in the spatial attraction redistribution scenario. The comparison shows that the redistribution scenario had a similar mean visit intensity to the baseline tourist scenarios, while its HHI, maximum mean visit, and high-density visit share were lower than those of the baseline tourist scenarios.

4 Discussion

4.1 Identification of simulated activity-pressure patterns and spatial coupling

The comparison among the resident, tourist, and mixed scenarios reveals distinct spatial patterns of simulated activity pressure. Tourist-oriented movement generated a smaller but more concentrated spatial footprint, whereas resident-oriented and mixed movement extended across broader parts of the internal street-space system. These differences indicate that activity pressure within the same historic street network may vary not only in overall intensity but also in spatial coverage and concentration according to the origins, attraction fields, and movement conditions represented in each scenario. Tourist-related activity was more selectively reinforced along particular corridors and nodes, while resident-related and mixed activity produced more spatially diffuse path structures. Additionally, the findings indicate that the spatial relationship between residents’ daily movements and tourist-oriented movements is uneven. While the overall level of overlap remains low at the district scale, localized areas of high-intensity overlap are observed within specific corridors and nodes.

The results show that activity pressure is influenced by the coexistence of tourism, residents’ daily activities, commercial services and the constrained morphology of historic streets. In districts with high historical density, ecological and environmental regulations extend beyond the protection of green and waterfront areas. They also encompass how the built environment accommodates, directs, and redistributes intense human activities without compromising daily liveability, environmental quality, and the continuity of the historic setting.

From a planning perspective, the principal value of these findings lies in distinguishing broad spatial diffusion from localized concentration. In historic districts such as Shichahai, residential access, visitor circulation, commercial activity, waterfront use, and other urban functions operate within the same fine-grained street system. The model identifies spatial units where movement patterns are concentrated or overlap. These locations may represent areas where multiple activity demands are simultaneously concentrated and therefore deserve further empirical investigation.

4.2 Adaptive spatial regulation through attraction redistribution scenarios

The attraction-redistribution scenario remained an exploratory model experiment to examine whether activity concentration could be adaptively adjusted through the reorganization of spatial attraction structures under conditions where the physical street-space system remains unchanged. In this scenario, DBSCAN-identified clusters were represented as spatially concentrated attraction areas derived from existing POI distributions, and the reassignment of maximum normalized attraction values was used as a controlled model experiment to test the potential effects of strengthening selected attraction locations on simulated activity redistribution. It represents a general category of non-physical spatial regulation measures that may influence where activities are attracted and distributed, including potential adjustments in activity organization, visitor guidance, information provision, or attraction management.

Compared with the baseline tourist scenarios, the redistribution scenario maintained a similar overall level of simulated movement but showed lower concentration-related outputs. This means that the scenario did not simply reduce activity intensity. Instead, it changed how simulated activity pressure was distributed across the grid system. This distinction is important for historic district governance: the aim is not necessarily to reduce all activity, but to prevent excessive concentration in limited street-space units and to improve the spatial distribution of use within the existing environment.

This scenario also addresses a limitation in existing historic-district management approaches, which often evaluate accessibility or movement patterns under existing spatial conditions but have limited ability to examine how alternative attraction configurations may influence emergent activity distributions. By treating attraction redistribution as a controlled model intervention, the framework provides a means to evaluate potential adaptive regulation strategies before physical changes are introduced. Such simulation-based exploration does not predict the outcome of a specific planning action but helps identify how modifications in attraction structure may influence the spatial concentration and redistribution of activities.

While the proposed framework provides a computational approach for diagnosing activity-pressure patterns and exploring spatial redistribution mechanisms, several interpretive boundaries should be acknowledged. First, the simulated activity intensity represents relative path reinforcement and spatial coupling rather than observed pedestrian volume, congestion, social conflict, or environmental impact. Therefore, the identified high-overlap areas should be regarded as spatial hypotheses for further investigation rather than confirmed conflict locations. Second, the temporal scenarios simplify complex user compositions by representing dominant activity regimes, while future studies could integrate individual trajectory data, field observations, or perception surveys for empirical triangulation. Third, although the simulation framework enables comparative analysis of spatial redistribution, its outputs depend on the selected spatial resolution and attraction representation. Future research could examine multi-scale modelling strategies and integrate additional behavioral evidence.

5 Conclusion

The methodological contribution of this study lies in translating multi-agent movement processes into measurable indicators for spatial diagnosis. The Physarum-inspired model generated relative measures of simulated path reinforcement, spatial concentration, resident–tourist overlap, and attraction-redistribution effects.

Density, network, and spatial statistical methods are effective for describing existing spatial structures, locating functional concentrations, and identifying relationships between the built environment and human activity. However, because their results are generally based on fixed spatial attributes, aggregated observations, or cross-sectional relationships, they are less able to represent how movement patterns emerge and change through repeated interactions over time (Lyu et al., 2025; Wang et al., 2025; Luo et al., 2026). Multi-agent simulation can extend such analyses by representing temporal evolution, local behavioural responses, environmental feedback, and scenario-dependent redistribution(Z. Wang et al., 2026; Zeng et al., 2025).

In the present model, simulated path structures emerged from the interaction of movement constraints, spatial attraction weights, agent release locations, stochastic movement, reinforcement-signal updating, diffusion, and attenuation. This design draws on the adaptive-feedback logic of slime-mould-inspired computation, in which locally reinforced connections retain greater influence while weakly supported alternatives gradually lose influence (Li et al., 2020; Gharehchopogh et al., 2023). By converting these processes into comparable indicators, the framework supports the identification of broad movement coverage, localized visitor concentration, and resident-tourist spatial overlap within a constrained historic street-space environment.

Several limitations should nevertheless be recognized. First, simulated visit intensity and overlap represent relative path reinforcement and spatial co-occurrence rather than actual pedestrian volume, crowd density, congestion, social conflict, carrying-capacity exceedance, or environmental impact. The results should therefore be interpreted as comparative model outputs rather than direct measurements of real pedestrian movement. Second, the attraction weights were derived from aggregated heatmap data. Although these data describe relative temporal differences in spatial activity, they do not distinguish individual identities, travel purposes, behavioural preferences, or real-time route decisions. The resident, tourist, and mixed scenarios consequently represent controlled analytical settings rather than complete reproductions of the actual population composition during each period. Third, the framework was examined in a single historic district, and its transferability to urban environments with different street morphologies, functional structures, and activity patterns requires further testing.

Future research should combine the simulation framework with observed pedestrian trajectories, field counts, mobile signalling data, and survey-derived behavioural information to calibrate agent rules and independently validate the simulated spatial patterns. Comparative applications across different historic-district types would also help determine which model mechanisms and parameter settings are transferable and which require context-specific adjustment.

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