Patch-generating Land Use Simulation (PLUS)

• PLUS is a raster-based framework that simulates nonlinear, multi-type land use dynamics by linking fine-scale transitions to macro-scale demand.
• The model employs three integrated modules—LEAS, CARS, and MOP—for spatial analysis, patch-based CA dynamics, and scenario-driven planning, respectively.
• PLUS supports sustainable land management by generating scenario-based projections and informing policy through rigorous calibration and validation against observed LULC data.

The Patch-generating Land Use Simulation (PLUS) model is an integrated, raster-based framework for modeling land use and land cover (LULC) dynamics. Designed to resolve the inability of prior Cellular Automata (CA) models to simulate nonlinear multi-type patch evolution, PLUS enables both detailed representation of spatial pattern change and rigorous analysis of its underlying drivers. The architecture comprises three tightly coupled modules: a Land-Expansion Analysis Strategy (LEAS), a CA algorithm with multi-type Random patch Seeds (CARS), and a Multi-Objective Programming (MOP) optimizer for scenario-driven planning. By linking fine-scale transition rules to macro-scale land demand, PLUS underpins scenario-based projections and directly informs policy for sustainable land management (Liang et al., 2020).

1. Methodological Architecture

PLUS operationalizes land use simulation through a sequential workflow synthesized from spatial, statistical, and optimization techniques. The methodological sequence begins with overlaying two temporally separated LULC maps and extracting "growing patches" for each class. The LEAS module trains a binary Random Forest (RF) to estimate the conditional probability, $P_i^k$, of cell $i$ switching to class $k$ under observed biophysical and socioeconomic drivers (e.g., elevation, proximity to infrastructure, population density, GDP). These growth-potential surfaces become the "local attractiveness" layers for the CARS module, which allocates change via patch-seeded CA dynamics, embedding local competition (via neighborhood effects) and steering allocations to meet exogenous demand. Finally, the MOP module generates alternate demand targets for scenario simulation based on optimization over economic, ecological-service, and ecological-capacity criteria subject to area, diversity, and regulatory constraints.

The primary model data inputs are two high-resolution LULC rasters and an array of driver rasters, with outputs comprising probability surfaces, simulated LULC maps, variable-importance rankings, and scenario-based forecasts. The integration of data mining, patch-scale CA, and policy-aligned optimization distinguishes PLUS as a comprehensive, end-to-end modeling platform.

2. Mathematical Formulations and Simulation Algorithms

2.1. Land-Expansion Analysis Strategy (LEAS)

For each target land-use type $k$, the LEAS poses a binary classification: cell $i$ is labeled $y_i^k=1$ if its class transitioned to $k$ between $[T_1, T_2]$, otherwise $0$. Training on a sampled set (typically $5\%$ of all cells), the RF resolves:

$$P_i^k(x) = \frac{1}{M} \sum_{n=1}^{M} I[h_n(x_i) = 1]$$

where $M$ is the number of trees, $h_n(\cdot)$ the $n$th tree vote, and $x_i$ the local driver vector. $P_i^k(x)$ thus encodes the probability surface for expansion, and the RF's $\Delta OOB_j^k$ (variable-importance) quantify each driver's contribution to class $k$'s expansion.

2.2. CA with Multi-type Random Patch Seeds (CARS)

CARS simulates allocation iteratively, contingent on the probability surfaces and macro demand. The dynamic includes:

• Neighborhood Effect:

$$\Theta_{i,k}^t = \frac{1}{n^2-1} \sum_{j \in N_i} I[c_{t-1}(j) = k]\cdot W_k$$

with $N_i$ the Moore neighborhood, typical size $3 \times 3$ in Wuhan calibration.

• Self-adaptive Demand-gap Coefficient:

The iterative correction between current allocation $A_k^t$, target demand $D_k$, and adjustment coefficient $\Delta_k^t$ follows:

$$\Delta_k^t = \begin{cases} \Delta_k^{t-1} & |\!G_k^{t-1}\!| \leq |\!G_k^{t-2}\!| \ \Delta_k^{t-1} \frac{G_k^{t-1}}{G_k^{t-2}} & \text{otherwise} \end{cases}$$

with $G_k^t = A_k^t - D_k$.

• Patch-seeding Mechanism:

Where $\Theta_{i,k}^t=0$, spontaneous seeds are created if $r < P_i^k$ and $r < U_k$ with $r \sim \text{Uniform}(0,1)$; $U_k$ controls the seed probability.

• Overall Change Probability:

$$\Pi_{i,k}^t = P_i^k \cdot \Theta_{i,k}^t \cdot \Delta_k^t$$

• Threshold-descent Competition:

Competing patch seeds undergo a descending admission threshold $t \leftarrow \beta t r'$, with $\beta \approx 0.9$, $r' \sim \text{Uniform}(0,2)$, admitting only cells where $\Pi_{i,k}^t > t$.

2.3. Multi-Objective Programming (MOP) for Scenarios

The MOP defines future area allocations $x = (x_1,...,x_7)'$ for LULC classes (grass, deciduous, cropland, urban, bare, water, evergreen), subject to:

• Three objectives: maximize economic benefit $f_1(x)=\sum e_i x_i$, ecological service value $f_2(x)=\sum s_i x_i$, and ecological capacity $f_3(x)=\sum c_i x_i$, using empirically specified coefficients;
• Constraints on total area, population, diversity, green-equivalence, food security, and class-specific bounds.

Three single-objective and one multi-objective posterior (SD) scenario formulations are solved.

3. Calibration and Model Validation

Calibration leverages a decadal LULC change atlas (e.g., Wuhan 2003–2013) to empirically fit the LEAS and tune CARS parameters. Key settings include: $5\%$ cell random sampling for training, $M=50$ trees, $mtry=14$ drivers/split, $U_k=0.1$, $\beta=0.9$, patch step size $=500$ area cells. Iterative allocation continues until the observed 2013 class areas are matched.

Model validation prioritizes two classes of quantitative metrics:

• Figure-of-Merit (FoM):

$\text{FoM} = \frac{|Sim \cap Obs|}{|Sim \cup Obs|}$

PLUS achieved FoM = $0.2642$, outperforming baseline CA comparators.

• Landscape Pattern Metrics:

Number of patches (NP), largest-patch index (LPI), perimeter–area (PARA), nearest-neighbor statistics (ENN), and like adjacency (PLADJ) are used to assess spatial realism. PLUS achieved the best match in 7 out of 15 metrics and second-best in 6 others. Optional metrics include overall accuracy (OA) and Kappa coefficient, though the study emphasizes FoM and landscape congruence.

4. Transition Rules and Drivers of Land-Use Change

The variable-importance ($\Delta OOB_j^k$) outputs from the LEAS module enable detailed attribution of land-use transitions. In the Wuhan application:

• Grassland expansion is predominantly explained by increased distance to administrative centers, indicating that grassland proliferation occurs where human presence is weakest.
• Deciduous forest increases are closely associated with proximity to arterial roads, consistent with targeted reforestation and corridor plantings.
• Urban expansion is primarily driven by proximity to tertiary roads, reflecting built-up edges advancing along feeder-road networks.

These empirically derived transition rules have direct interpretive and policy uses, such as identifying regions for land-use controls or anticipating the impact of infrastructure expansion.

5. Scenario-Based Projections for 2035

Scenario simulations utilize 2035 demand vectors computed via MOP, with CARS enacting spatial allocation under each scenario:

• Baseline (Markov): Projects continuation of historical trends, predicting rapid cropland and grassland loss.
• ED (Economic Development): Maximizes GDP, yielding compact urban growth and large deciduous patches at the expense of forest continuity.
• EP (Ecological Protection): Favors forest and grass recovery, minimizing urban spread but at a cost to economic output.
• SD (Sustainable Development): Balances objectives, maintaining forest corridors, incorporating evergreen fragments, and constraining urban intrusion into sensitive areas.

Trade-off analyses reveal that the SD scenario achieves 88% of maximal GDP, 93% of ecosystem service value, and 95% of ecological capacity as compared to the maximum obtained in the specialized scenarios.

6. Policy Implications and Software Accessibility

PLUS provides policy-relevant, spatially explicit recommendations. Without intervention (baseline), critical losses in cropland and grassland are projected. Sole focus on economic growth leads to fragmentation of ecological corridors, while ecological targets alone suppress economic output. A sustainable, balanced approach achieves high fractions of both economic and ecological objectives.

The complete PLUS software stack, including documentation and demo data, is available at https://github.com/HPSCIL/Patch-generating_Land_Use_Simulation_Model. Deployment requires Python ($\geq$3.7), scikit-learn, numpy, gdal, and can be accessed via GUI or command line for all standard modeling steps: preparing input rasters, running LEAS, defining scenario MOP targets, executing CARS, and evaluating outputs. This accessibility ensures that researchers and policymakers can replicate, adapt, and extend the modeling framework for other regions and applications, linking rigorous driver analysis, patch-based CA, and scenario-driven optimization in one unified platform (Liang et al., 2020).