## 2 Problem Model
### Environment Model
#### 2.2.2 Situation Information  
The target occurrence is modeled as a binomial distribution event, where each grid cell in the mission area is assigned a target detection probability \(p_{i}\). The collection of all \(p_{i}\) values constitutes the target detection probability map (TDPM). When grid cell \(g_{i}\) lies within the UAV detection range, its TDPM is updated according to the Bayesian criterion:  
\[p_{i}(k+1)=\left\{ \begin{array}{l}\frac{p_{i}p_{i}(k)}{p_{d}p_{i}(k)+p_{f}(1 -p_{i}(k))}s_{i}^{d}(k)=1\\ \frac{(1-p_{d})p_{i}(k)}{(1-p_{d})p_{i}(k)+\big{(}1-p_{f}\big{)}[1-p_{i}(k)]}, s_{i}^{d}(k)=0\end{array} \right.. \tag{3}\]  
where \(k\) is the index value of the time step. The target detection status \(s_{i}^{d}(k)\) is assigned a value of 1 when a UAV detects a target in grid cell \(g_{i}\) and 0 otherwise. The sensor's detection probability \(p_{d}\) and false alarm probability \(p_{f}\) are set as 0.8 and 0.2, respectively. Under the mechanism of Equation (3), if the UAV detects the presence of a target, regardless of whether it is detected or a false alarm, the UAV will consider the probability of the target's existence to be high and will then assign a greater probability of the target's appearance to \(g_{i}\). When this intelligence is broadcasted to other UAVs, they will have a greater chance of accessing \(g_{i}\) for further reconnaissance, preventing missed detections or eliminating false alarms. To reduce redundant search efforts, the digital pheromone map (DPM) method is employed. For any grid cell \(g_{i}\) within the mission area, digital pheromone values are updated according to the following equation [31]:  
\[\eta_{i}(k+1)=(1-V_{a})\big{\{}(1-P_{a})[\eta_{i}(k)+W_{i}o_{i}(k+1)]+I_{i}(k+ 1)\big{\}}, \tag{4}\]  
Figure 2: Grid division of mission area. where \(\eta_{i}(k+1)\) represents the pheromone concentration at grid cell \(g_{i}\) during the \(k\)-th step; \(V_{a}\) and \(P_{a}\) represent the propagation coefficient and volatilization coefficient, respectively; \(W_{i}\) represents the secretion switch, which is typically set to 0 and changes to 1 when a UAV passes through the grid; \(o_{i}(k+1)\) is the pheromone secreted by all UAVs in the grid, including itself, at \(g_{i}\); and \(I_{i}(k+1)\) is the pheromone entering \(g_{i}\) from neighboring grids through propagation, calculated using Equation (5):  
\[I_{i}(k+1)=\frac{P_{a}}{N(\mathbf{G}_{s})}\sum_{j=1}^{N(\mathbf{G}_{s})}\big{[}\eta_{sj }(k)+o_{sj}(k+1)\big{]}, \tag{5}\]  
where \(\mathbf{G}_{s}\) represents the set of grid cells adjacent to \(g_{i}\), and \(N(\mathbf{G}_{s})\) denotes the number of elements in \(\mathbf{G}_{s}\). In Equation (4), for a map stored in a UAV, the pheromones possessed by grid \(g_{i}\) between the \(k\)-th and \(k+1\)-th adjacent time points come from three parts: the pheromones \(\eta_{i}(k)\) retained at the \(k\)-th time point, the pheromones \(W_{i}o_{i}(k+1)\) secreted by \(g_{i}\) by the UAV at the \(k+1\)-th time point, and the pheromones \(I_{i}(k+1)\) propagated to \(g_{i}\) by other grids around \(g_{i}\). The first two terms have already evaporated a part of \(P_{a}\) before the \(k+1\)-th time point, so \((1-P_{a})[\eta_{i}(k)+W_{i}o_{i}(k+1)]+I_{i}(k+1)\) represents all the pheromones passing through \(g_{i}\) between adjacent time points. Finally, these pheromones also propagate to the surrounding grids through a propagation mechanism with a proportion of \(V_{a}\), so the pheromone of \(g_{i}\) at the \(k+1\)-th time point is represented by Equation (4). From the perspective of individual UAVs, the only source of pheromones stored in their maps is the secretion of pheromones caused by UAV access to the grid, which has repulsive properties, and the UAV will prioritize areas with lower concentrations of pheromones. Under this mechanism, the digital pheromone used in this article can reduce the repetitive search of UAV swarms in a certain area and improve search efficiency.