## 3 Methodology
### Proposed CC-MPSO Algorithm
\[S_{k}\vec{y}(t)=\underset{S_{k}y_{i}(t)}{\text{argmax}}F\Big{(}S_{\{1:M\}-\{k\}} \vec{y}(t),S_{k}y_{i}(t)\Big{)} \tag{26}\]  
In addition, the velocity of each particle \(S_{k}x_{i}\) is controlled by a clamping \(S_{k}v_{\text{max}}\) that regulates the maximum velocity update to a defined range of \([-S_{k}v_{\text{max}},S_{k}v_{\text{max}}]\). The clamping constant controls the exploration-exploitation trade-off by modifying the particle's ability to explore a narrow or large search space. \[S_{k}v_{i}(t+1)\Bigg{\{} \begin{array}{l}S_{k}v_{i}(t+1),if\|S_{k}v_{i}(t+1) \|\leq S_{k}v_{\text{max}}\\ \pm S_{k}v_{\text{max}},otherwise\end{array}  \tag{27}\]  
Step 4: Particle fitness evaluation. In CC-MPSO, each sub-population corresponds to a specific part of the solution vector in the optimization process. Therefore, to evaluate the value of individual fitness in each population, the particle needs to be solved in association with the globally optimal particles of the remaining populations. The formula is shown as follows:  
\[Fitness(S_{k}x_{i})=F\Big{(}S_{\{1:M\}-\{k\}}\vec{y},S_{k}x_{i}\Big{)} \tag{28}\]  
where \(S_{\{1:M\}-\{k\}}\vec{y}\) denotes the global best position of each particle in all populations except the swarm \(k\). Step 5: Update coevolution information. Based on the fitness of each subpopulation, the information on coevolution is updated to guide the algorithm to search towards the global optimal solution. To avoid the algorithm trapping in suboptimal solutions, the swarm particles can be reset into new random positions when one of the swarms converges in a small region of the search space while the global best position of the complementary swarm changes. The determination of whether the swarm is locally convergent or not is based on the computation of the sum of the normalized distances of all particles in the present swarm from the global optimal position, using the following formula:  
\[S_{k}d_{norm}(t)=\frac{S_{k}d(t)}{S_{k}d(0)} \tag{29}\]  
\[S_{k}d(t)=\sum_{i=1}^{N}[\|S_{k}x_{i}(t)-S_{k}\vec{y}(t)\|] \tag{30}\]  
where \(S_{k}d(t)\) denotes the sum of the Manhattan distances between the positions of all particles in the swarm \(k\) and the global optimal position. The swarm \(k\) is considered to be trapped in stagnation when \(S_{k}d_{norm}(t)\) is smaller than a convergence constant \(\varepsilon\). Step 6: Repeat the above steps until the maximum number of iterations is reached. Based on the output of the algorithm, the optimal solution to the problem is obtained, including the optimal path for each UAV and the final target location probability distribution. The algorithm presented in this paper offers two benefits for the challenge of dynamically searching for targets using several UAVs. One benefit is that once the update is finished, each set of individual UAV path parameters will be updated with the path parameters of other UAVs. This more detailed updating mechanism circumvents the issue of making progress and then regressing, which is frequently experienced with conventional PSO. An additional benefit is that the algorithm presented in this research effectively integrates many individuals from distinct communities to augment the diversity of global solutions. Pseudo-code for the proposed CC-MPSO algorithm for dynamic target search using multi-UAVs is given in Algorithm 1.