## 2 Material and Methods
### Methods
#### 2.3.8 Patch-Generating Land Use Simulation (PLUS) Model  
The PLUS model developed by [PERSON] et al. [13] is based on traditional methods for simulating land usage, such as the CA-Markov model [40], the CLUE-S model [41], and the FLUS model [42], combined with a random forest model. This incorporates a CA model with a variety of stochastic seed methods and a rule-mining technique for land expansion analysis. The PLUS model outperforms other models in modeling and predicting soil erosion under future scenarios [43].  
The PLUS model primarily comprises two modules: land expansion analysis strategy (LEAS) and CA based on multi-type random patch seeds (CARS).  
In order to determine the probability of development for each land use type, the LEAS module first extracts the initial land use and then applies the random forest classification (RFC) algorithm to examine the relationship between the growth of various land use types and multiple drivers. This probability is determined by the following formula [13]:  
\[P^{d}_{i,k}(x)=\frac{\sum_{n=1}^{M}I|h_{n}(x)=d|}{M} \tag{11}\]  
where \(x\) is a vector made up of several different drivers. The nth decision tree's \(h_{n}(x)\) prediction type is used for the vector \(x\). \(d\) can take a value of 0 or 1, where 1 representsthe transformation of other land use types on grid \(i\) to this k-type land use type and 0 represents other transformations; \(M\) represents the overall number of decision trees; \(I[h_{n}(x)=d]\) represents the decision tree's indicator function; and \(P_{i,k}^{d}(x)\) represents the likelihood that \(k\) land use type will expand into a space unit \(i\).  
The CARS module simulates the spatial diversity of intricate natural geographic phenomena. It is a scenario-driven land use simulation program that simulates land use distribution patterns, primarily by figuring out the likelihood that various land use types will evolve. The overall probability of the conversion of land use type \(k\) can be estimated using the following equation [13].  
\[OP_{i,k}^{d=1,t}=P_{i,k}^{d=1}\times\Omega_{i,k}^{t}\times D_{k}^{t} \tag{12}\]  
Land use type \(P_{i,k}^{d=1}\) is represented by cell \(k\) and cell growth probability \(i\). \(\Omega_{i,k}^{t}\) is the cell \(i\) domain effect. According to the following formula [13], \(D_{k}^{t}\) represents the demand's effect on future land use:  
\[\Omega_{i,k}^{t}=\frac{con\Big{(}c_{i}^{t-1}=k\Big{)}}{n\times n-1}\times w_{k} \tag{13}\]  
\[D_{k}^{t}=\left\{\begin{array}{l}D_{k}^{t-1}\Big{(}\Big{|}G_{k}^{t-1}\Big{|} \leq\Big{|}G_{k}^{t-2}\Big{|}\Big{)}\\ D_{k}^{t-1}\times\frac{G_{k}^{t-2}}{G_{k}^{t-1}}(0)G_{k}^{t-2}\Big{)}G_{k}^{t -1}\Big{)}\\ D_{k}^{t-1}\times\frac{G_{k}^{t-1}}{G_{k}^{t-2}}\Big{(}G_{k}^{t-1}\Big{)}G_{k }^{t-2}>0\end{array}\right. \tag{14}\]  
In Equations (13) and (14), \(con\) is the first \(n\times n\) window in which the first \(k\) land use type occupied in the last iteration is the total number of grid cells. \(w_{k}\) has a default value of 1 and determines how much importance each sort of land use is given. \(G_{k}^{t-1}\) and \(G_{k}^{t-2}\) are the land use type \(k\) in the first \(t-1\) and \(t-2\) iterations of the current demand and the discrepancy between anticipated demand.