## 2 Materials and Methods
### Insbbo
\begin{aligned} \mu_{i}=E\times\left(\frac{i}{N}\right)^{6}& \text{i}<\frac{3N}{7}\\ \lambda_{i}=I\times\left(1-\left(\frac{i}{N}\right)^{3}\right)& \end{aligned} \right. \tag{14}\]  
\[\left\{ \begin{aligned} \mu_{i}=E\times\text{ln}\left(\frac{i}{N}+1 \right)&\frac{3N}{7}\leq\text{i}\leq\frac{5N}{7}\\ \lambda_{i}=I\times\frac{3}{5}\left(\text{cos}\left(\frac{i\pi}{N }\right)+1\right)&\frac{5N}{7}<\text{i}\end{aligned} \right. \tag{15}\]  
In the first threshold range, where the habitat suitability index (HSI) is low (\(\text{i}<\frac{3N}{7}\)), a high-degree polynomial function is used for the emigration rate. This decision is based on the fact that habitats with lower HSI should not actively emit individuals since they represent poor-quality solutions. A steep polynomial curve grows slowly near zero, ensuring that weak habitats retain their population, which in turn reduces the dispersion of low-quality traits and enhances the algorithm's convergence by minimizing disruptive migrations. In the same low-HSI range (\(\text{i}<\frac{3N}{7}\)), a cubic inverse function is used for immigration. The rationale is that weaker habitats should be more receptive to incoming individuals in order to improve their quality through exploration. The cubic shape offers a controlled but relatively high immigration rate for these low-HSI habitats. This encourages knowledge infusion into underperforming solutions without overwhelming the population dynamics, allowing gradual adaptation without destabilization. For the middle HSI range (\(\frac{3N}{7}\leq\text{i}\leq\frac{5N}{7}\)), a logarithmic function is applied for the emigration rate. In this moderate zone, the habitat quality is neither optimal nor poor. A logarithmic function increases steadily but gently, enabling these average-quality habitats to contribute to the population by exporting individuals at a reasonable rate. This balances exploration and exploitation, avoiding both stagnation and premature convergence while enriching the diversity of the search space. Simultaneously, in the middle HSI range, a cosine-based function is used for immigration. This function introduces sharp variability in the middle segment of its curve, resulting in moderate but dynamically shifting immigration rates. The goal is to ensure that mid-quality habitats are more reactive to migration, adapting rapidly to fluctuations in the global population structure. This enhances the algorithm's flexibility and encourages efficient migration flow during intermediate search phases. When the HSI is high (\(\frac{5N}{7}<i\)), a hyperbolic tangent function is selected for emigration. This function has a steep increase followed by rapid saturation, allowing high-quality habitats to emit individuals early in the optimization process while gradually stabilizing over time. Such behavior is critical for preserving elite solutions. Initial sharing allows beneficial traits to spread, but as convergence progresses, emigration slows down to protect the refined solution from being disrupted by excessive outward flow. Finally, for immigration in the high-HSI range (\(\frac{5N}{7}<i\)), an inverted hyperbolic tangent function is adopted. It starts with low values and flattens further as HSI increases. This ensures that elite habitats accept very few, if any, new individuals, protecting them from being overwritten by inferior genetic material. By limiting the influx into high-performing habitats, the algorithm prevents deterioration of top solutions and accelerates convergence toward the global optimum. To effectively optimize the hyper-parameters and internal weights of the hybrid transformer-GAN, we propose an enhanced multi-objective version of the BBO algorithm, referred to as INSBBO. In this framework, the optimization targets include not only the learning of weights and biases in the fully connected layers of the transformer and GAN, but also the fine-tuning of critical hyper-parameters such as the learning rate, batch size, and dropout rate.