## Appendix A Calculation Details of the Marine Predators Algorithm (MPA)
### Population Update  
The following steps are executed in the while loop until the maximum number of iterations is reached:  
Phase 1: In cases of high-velocity ratio or when the predator is moving faster than the prey, it falls under the first scenario for updating the position. Specifically, when the current iteration \(Iter\) is less than one-third of the maximum number of iterations (\(max\_iter\)), the following mathematical model is employed to update the population:  
\[\mathbf{s}_{i}=\mathbf{R}_{B}\otimes\big{(}\mathbf{Elite}_{k}^{j}-\mathbf{R}_{B} \otimes\mathbf{\gamma}_{k}^{i}\big{)},i=1,\ldots,n \tag{1}\] \[\mathbf{\gamma}_{k}^{i}=\mathbf{\gamma}_{k}^{i}+P*\mathbf{R}\otimes\mathbf{s}_{i}\]  
where \(R_{B}\) follows a standard normal distribution. The symbol \(\otimes\) denotes element-wise multiplication of vectors. \(P=0.5\) is a predefined constant. \(R\) is a vector of uniformly distributed numbers in [0, 1]. \(\gamma_{k}^{i}\) represents the \(i\)th individual in the \(\mathbf{Prey}_{k}\). The multiplication of \(R_{B}\) by \(\gamma_{k}^{i}\) simulates the movement of prey. \(\mathbf{s}_{i}\) determines both the distance and direction an individual moves within the search space.  
Phase 2: When both predator and prey are moving at the same pace, or have a unit velocity ratio, the population is divided into two parts. The first half acts as prey and performs Levy flights, while the second half acts as predator and undergoes Brownian motion. Specifically, when the current iteration \(Iter\) is between one-third and two-thirds of the maximum iterations (\(max\_iter\)), the following mathematical model is used to update the population:  
For the first half of the population (prey):  
\[\mathbf{s}_{i}=\mathbf{R}_{L}\otimes\big{(}\mathbf{Elite}_{k}^{i}-\mathbf{R}_{L} \otimes\mathbf{\gamma}_{k}^{i}\big{)},i=1,\ldots,n/2 \tag{2}\] \[\mathbf{\gamma}_{k}^{i}=\mathbf{\gamma}_{k}^{i}+P*\mathbf{R}\otimes\mathbf{s}_{i}\]  
where \(\mathbf{R}_{L}\) is a vector of random numbers following the Levy distribution to simulate the randomness of Levy movement.  
For the second half of the populations (predator):  
\[\mathbf{s}_{i}=\mathbf{R}_{B}\otimes\big{(}\mathbf{Elite}_{k}^{i}-\mathbf{R}_{B} \otimes\mathbf{\gamma}_{k}^{i}\big{)},=\tfrac{n}{2}+1,\ldots,n \tag{3}\] \[\mathbf{\gamma}_{k}^{i}=\mathbf{Elite}_{k}^{i}+P\cdot\mathbf{C}\otimes\mathbf{s}_ {i}\]  
where \(\mathbf{C}F=\left(1-\frac{Iter}{max\_iter}\right)^{2\frac{Iter}{max\_iter}}\), is an adaptive parameter used to regulate the step size for predator movement.  
Phase 3: In cases of low-velocity ratio or when the predator moves faster than the prey, all individuals move in a Levy manner. This situation occurs during the final phase of the optimization process, i.e., when the current iteration \(Iter\) is greater than two-thirds of the maximum iterations (\(max\_iter\)). The following mathematical model is employed to update the population:\[\begin{split}\mathbf{s}_{i}=\mathbf{R}_{L}\otimes\big{(}\mathbf{Elite}_{k}^{i}- \mathbf{R}_{L}\otimes\mathbf{\gamma}_{k}^{i}\big{)},i=1,\ldots,n\\ \mathbf{\gamma}_{k}^{i}=\mathbf{Elite}_{k}^{i}+P*CF\otimes\mathbf{s}_{i}\end{split} \tag{10}\]