Optimal homotopy analysis method for (2+1) time-fractional nonlinear biological population model using $J$-transform

This paper presents a comprehensive study of the (2+1) time-fractional nonlinear generalized biological population model (TFNBPM) using the $J$-transform combined with the optimal homotopy analysis method, a robust semi-analytical technique. The primary focus is to derive analytical solutions for the model and provide a thorough investigation of the convergence properties of these solutions. The proposed method allows for flexibility and accuracy in handling nonlinear fractional differential equations (NFDEs), demonstrating its efficacy through a series of detailed analyses. To validate the results, we present a set of 2D and 3D graphical representations of the solutions, illustrating the dynamic behavior of the biological population over time and space. These visualizations provide insightful perspectives on the population dynamics governed by the model. Additionally, a comparative study is conducted, where our results are juxtaposed with those obtained using other established techniques from the literature. The comparisons underscore the advantages of optimal homotopy analysis $J$-transform method (optimal HA $J$-TM), highlighting its consistency and superior convergence in solving complex fractional models.