## 3 Methodology
### Proposed CC-MPSO Algorithm  
Based on Bayesian theory, this paper constructs the multi-UAV dynamic target search problem as an optimization problem that maximizes the target detection probability in finite time. With the increase in the number of UAVs and targets, the complexity of the problem grows exponentially, and the performance of traditional algorithms can no longer meet the requirements. A cooperative coevolution framework (CC framework) is an important tool for solving optimization problems. The idea is to divide the original optimization problem into multiple subproblems, each corresponding to a sub-population. Through the coevolution between sub-problems, the global optimal solution can be searched for more efficiently. [PERSON] et al. applied this framework to genetic algorithms [PERSON] et al. (2016), where they divided the solution vector into multiple chromosomes, and each chromosome belonged to a different population. [PERSON] et al. applied a similar framework to PSO algorithms [PERSON] et al. (2017), where the space of n-dimensional solution vectors is partitioned into one-dimension solution space, and each solution space corresponds to a sub-population that is independent in the update. Considering that the objective function in this paper is determined by the joint target detection probability of multiple UAVs, the CC-MPSO fusion algorithm is proposed based on the CC framework and the MPSO algorithm by drawing on the idea of cooperative evolution. Unlike traditional optimization algorithms, our approach views the search task of multi-UAVs as a co-evolutionary process, an idea that makes the algorithm more adaptive, flexible, and better adapted to complex environments. Another innovation is that we introduce a motion-encoded mechanism. Coding based on UAV motion paths not only optimizes their detection performance but also enables more effective information sharing and collaborative decision-making in collaborative tasks, thus improving overall performance. A schematic overview of numerous swarms cooperating to optimize the search paths of multi-UAVs is shown in Figure 5, and the basic steps of CC-MPSO are shown as follows:Step 1: Problem modeling. The challenge of dynamically searching for targets using multi-UAVs is divided into several sub-problems, with each sub-problem corresponding to a specific UAV. Each sub-problem defines the state representation, search range, and objective function for its own UAV. Step 2: Population initialization. The particle of each population is initialized for each subproblem and motion-encoded. The position vector of each particle reflects the motion path of each UAV. Step 3: Subpopulation update. The standard PSO algorithm is used to update the position and velocity of the particles in each subpopulation with the following equations:  
\[S_{k}v_{i}(t+1)=wS_{k}v_{i}(t)+c_{1}r_{1}[S_{k}y_{i}(t)-S_{k}x_{i}(t)]+c_{2}r_{2 }[S_{k}\widetilde{y}_{i}(t)-S_{k}x_{i}(t)] \tag{23}\]  
\[S_{k}x_{i}(t+1)=S_{k}x_{i}(t)+S_{k}v_{i}(t+1) \tag{24}\]  
where \(S_{k}x_{i}(t)\) and \(S_{k}v_{i}(t)\) denote the position and velocity for particle \(i\) of swarm \(k\), respectively. The personal best position \(S_{k}y_{i}(t)\) for each particle \(i\) of each swarm \(k\) depicts the best result found for this particle up to iteration \(t\):  
\[S_{k}y_{i}(t+1)=\left\{ \begin{array}{ll}S_{k}y_{i}(t),&if\quad F\Big{(}S_{ \{1:M\}-\{k\}}\widetilde{y}(t),S_{k}y_{i}(t)\Big{)}\geq F\Big{(}S_{\{1:M\}-\{ k\}}\widetilde{y}(t),S_{k}x_{i}(t+1)\Big{)}\\ S_{k}x_{i}(t+1),if\quad F\Big{(}S_{\{1:M\}-\{k\}}\widetilde{y}(t),S_{k}y_{i}(t) \Big{)}\leq F\Big{(}S_{\{1:M\}-\{k\}}\widetilde{y}(t),S_{k}x_{i}(t+1)\Big{)} \end{array} \right. \tag{25}\]  
The global best position of each particle \(i\) for each swarm is updated using:  
Figure 5: A schematic overview of numerous swarms cooperating to optimize the search paths of multi-UAVs.