## 2 Materials and Methods
### Insbbo
This dynamic ensures that high-quality solutions tend to influence lower-quality ones, guiding the overall search process toward regions of higher fitness while maintaining sufficient diversity across the population. The migration step thus serves as a primary exploitation mechanism within BBO, promoting the propagation of advantageous traits while allowing the algorithm to explore different parts of the search space through controlled information sharing [52]. In addition to migration, BBO employs a mutation mechanism to maintain population diversity and prevent premature convergence to suboptimal solutions. The mutation process is probabilistically controlled by the quality of each habitat: habitats with lower suitability are more likely to mutate, while highly suitable habitats are protected from unnecessary alterations. This behavior ensures that the algorithm continues to explore the search space even when it has identified regions of relatively high fitness, thus balancing exploration and exploitation. The probability of mutation for each habitat is determined by Equation (13):  
\[m_{i}=m_{max}\times\left(1-\frac{p_{i}}{p_{max}}\right) \tag{13}\]  
where \(m_{i}\) denotes the mutation rate, \(m_{max}\) is chosen by the user, \(p_{i}\) reveals the probability of species count and \(p_{max}\) is the highest value of \(p_{i}\). Although the standard BBO model provides an elegant and effective approach for migration through linear immigration and emigration rates, it exhibits certain fundamental limitations when applied to complex real-world problems. One major shortcoming is its reliance on a simple linear relationship between habitat suitability and migration rates. In reality, most natural phenomena and real-world optimization problems are inherently nonlinear, involving complex interdependencies and dynamic variations. As a result, a linear model may not adequately capture the intricate behavior required for robust and flexible search strategies, particularly in high-dimensional and irregular solution landscapes. Another critical drawback of the linear migration model is its inability to adaptively fine-tune the exploration and exploitation processes during optimization. In the standard model, migration rates change uniformly across all habitats based solely on their HSI ranking, without accounting for the nuanced performance variations that different habitats might exhibit. Consequently, the algorithm may either converge prematurely if migration is too aggressive or stagnate if migration is too conservative, leading to suboptimal optimization performance, especially in dynamic or multi-modal environments [46]. Furthermore, using a single, uniform migration model for all habitats within the same generation reduces the algorithm's diversity and adaptability. Since all solutions are governed by the same linear function, habitats with widely differing HSI values still experience similar migration behavior. This homogenization fails to exploit the strengths of high-quality habitats appropriately while limiting the capacity for weaker habitats to undergo necessary structural changes. Ideally, habitats with different HSIs should utilize distinct migration dynamics, enhancing the algorithm's ability to adjust locally to the specific quality and characteristics of individual solutions. To overcome these limitations, we propose the use of nonlinear migration models that introduce diverse, adaptive migration behaviors across habitats. Specifically, as depicted in Equations (14)-(16), different nonlinear models are employed based on the rank of the habitat within the population. By using polynomial, logarithmic, and hyperbolic tangent functions to define the immigration and emigration rates, the improved migration scheme enables more accurate modeling of natural diversity and dynamic adaptation across different solution qualities. This enhancement is expected to significantly improve the balance between exploration and exploitation, thereby enhancing the optimization performance of the proposed framework. \[\left\{