## 2 Materials and Methods
### Adaptive DBO for DPE
#### 2.3.2 Adaptive DBO Algorithm
After determining the optimal foraging area, the position of the young dung beetle will be updated by  
\[x_{l}(t+1)=x_{l}(t)+c_{1}\Big{(}x_{l}(t)-Lb^{b}\Big{)}+c_{2}\Big{(}x_{l}(t)-Ub^{ b}\Big{)}, \tag{19}\]  
where \(x_{l}(t)\) denotes the position of the \(l\)-th small dung beetle at the \(t\)-th iteration, \(c_{1}\) denotes a random number obeying normal distribution, and \(c_{2}\) is a random variable in the range of \((0,1)\). Stealing behavior: A subset of dung beetles engage in stealing behavior, stealing the dung balls of other dung beetles. The stealing dung beetles will steal near the best place to steal, i.e., the global optimum, and the positions of the stealing dung beetles will be updated as follows:  
\[x_{l}(t+1)=X^{b}+S\times g\times\Big{(}|x_{l}(t)-X^{*}|+|x_{l}(t)-X^{b}|\Big{)}, \tag{20}\]  
where \(x_{l}(t)\) represents the position of the \(l\)-th stealing dung beetle at the \(t\)-th iteration, \(g\) is a normally distributed random vector of size \(1\times D\), and \(S\) is a constant. In general, optimization algorithms are most effective when their strengths align with the challenging characteristics of the problem. However, the fixed proportions of dung beetles and the single linear convergence factor in the DBO algorithm limit its dynamic adaptability. Moreover, without a mechanism to escape local optima, the standard DBO algorithm struggles with multi-modal problems, making it less suitable for handling the challenges of DPE. Thankfully, FLA provides a data-driven approach to understanding optimization problems. The basic principle is that in the swarm intelligence algorithm, each individual is part of a population, which collectively provides a discrete observation of the local fitness landscape. Several descriptive characteristics, such as FDC and ruggedness, have been proposed to assess algorithm performance and guide improvements. Building upon this concept, we propose an improved ADBO algorithm, featuring two key enhancements. Adjustment of Search Capabilities Based on FDC  
According to the formulation, spawning dung beetles enhance local exploration capability, as their position updates are strictly confined within the optimal spawning boundary. Conversely, foraging dung beetles contribute to global exploration capability since their position updates are not strictly restricted to the optimal foraging area. The convergence factor determines the extent of the optimal spawning and feeding areas in (15) and (18). We aim for the algorithm to dynamically adjust the populations of spawning dung beetles and foraging dung beetles, as well as the value of the convergence factor, according to the characteristics of the optimization problem at hand. FDC is a static measure characterizing the local landscape where a population is present. It sorts individuals by their distances from the best one and examines how fitness changes with distance to infer the landscape type [31]. To effectively measure the extent to which fitness function values correlate with distance to the global optimum, one should examine a problem with known optima, take a sample of individuals, and compute the correlation coefficient for the set of pairs. Given a population \(P=\{x_{1},\cdots,x_{L}\}\) with \(L\) individuals, there is a corresponding fitness values vector \(\mathbf{f}=[f_{1},\cdots,f_{L}]\). Identify the best individual in the population, denoted as \(x_{best}\), which corresponds to the highest fitness value. The distance from each individual to the best individual is represented by \(\mathbf{d}=[d_{1},\cdots,d_{L}]\), where \(d_{l}=|x_{l}-x_{best}|\). The FDC \(C_{f}\) is given by  
\[C_{f}=\frac{\delta_{fd}}{\delta_{f}\delta_{d}}, \tag{21}\]  
in which  
\[\delta_{fd}=\frac{1}{L}\sum_{l=1}^{L}(f_{l}-\bar{f})(d_{l}-\bar{d}) \tag{22}\]  
is the covariance of \(\mathbf{f}\) and \(\mathbf{d}\). \(\delta_{f}\) and \(\delta_{d}\) are the standard deviations, \(\bar{f}\) and \(\bar{d}\) are the means of \(\mathbf{f}\) and \(\mathbf{d}\), respectively.