## 3 Materials and Methods
### [PERSON]  
The biogeography-based optimization (BBO) algorithm was first introduced by [PERSON] in 2008 as an evolutionary optimization technique inspired by the natural distribution of species across different habitats [59]. The fundamental idea of BBO is derived from biogeography, which studies the migration, mutation, and selection processes that govern species distribution in various ecosystems. In an optimization context, each habitat represents a potential solution to a given problem, and the quality of a solution is measured by its habitat suitability index (HSI). The key principle of BBO is that solutions with higher HSI tend to share features with lower-HSI solutions through a migration mechanism, allowing knowledge transfer and convergence toward optimal solutions. In BBO, migration plays a crucial role in the exchange of information among candidate solutions. Each habitat has an immigration rate (\(\lambda\)) and an emigration rate (\(\mu\)), which determine how solutions share information. Migration rates are typically modeled as linear functions of HSI, meaning that high-HSI habitats (better solutions) have a higher emigration rate, while low-HSI habitats (worse solutions) have a higher immigration rate. This is mathematically expressed as:  
\[\mu_{i}=\frac{i}{N}, \tag{35}\]  
\[\lambda_{i}=1-\frac{i}{N} \tag{36}\]  
where \(i\) represents the rank of the habitat in terms of suitability, and \(N\) is the total number of habitats. The migration process involves selecting high-HSI solutions as donors and low-HSI solutions as recipients, enabling the exploration of new promising regions in the search space. In addition to migration, mutation is another critical operator in BBO, ensuring diversity and preventing premature convergence. Mutation is inspired by sudden environmental changes or random genetic variations in species. The mutation probability is often inversely proportional to the HSI of a habitat, meaning that worse solutions have a higher probability of undergoing mutation. The mutation rate (\(m_{i}\)) for a habitat is given by:  
\[m_{i}=m_{\max}\bigg{(}1-\frac{p_{j}}{p_{\max}}\bigg{)} \tag{37}\]  
where \(m_{\max}\) is the maximum mutation rate; \(p_{j}\) represents the probability of species count, and \(p_{\max}\) is the highest value of \(p_{j}\). This mechanism introduces randomness to the optimization process, enhancing global search capability and allowing exploration beyond the solutions obtained through migration. The interaction between host and guest habitats in the BBO is based on the principle that high-HSI habitats act as knowledge sources for lower-HSI habitats. When migration occurs, characteristics of a host (high-HSI habitat) replace some characteristics of a guest (low-HSI habitat), allowing for gradual improvement of weaker solutions. This biogeographical interaction mechanism allows the algorithm to balance exploration (through mutation) and exploitation (through migration), leading to efficient optimization in complex search spaces. The standard linear migration model in BBO suffers from several limitations when dealing with complex optimization landscapes. Linear migration assumes a fixed and simple relationship between immigration and emigration rates, which does not adequately capture the dynamic and nonlinear nature of real-world species migration. This rigid structure limits the adaptability of the algorithm, leading to premature convergence in multimodal search spaces and reducing its ability to explore and exploit solutions effectively. Moreover, in highly non-convex problems, a static migration model fails to maintain diversity across different generations, which is crucial for avoiding stagnation in local optima. To overcome these challenges, we propose a six-stage adaptive migration model, which dynamically adjusts migration rates based on different optimization phases.