Estimating a semi-analytical solution for fish farm model using homotopy analysis method

In this paper, the homotopy analysis method (HAM) is employed to solve the dynamical behavior of nutrient density, fish population and mussel population in a fish farm with the help of a system of nonlinear differential equations. This method enables to compute the solution of a governed system of differential equation as an infinite series. The convergence region of an infinite series solution is found by controlling the auxiliary parameter. In addition, the convergence theorem of the HAM for the fish farm model is proved. A semi-analytical solution of the model has been obtained and then the squared residual errors have been calculated to determine the convergence of the homotopy series. The optimal values of the convergence control parameters have been acquired with the help of residual errors. The numerical results obtained using HAM are compared with the existing solutions of Laplace Adomian decomposition method (LADM) and Runge-Kutta fourth-order method (RK4M). Error comparisons have been tabulated to demonstrate the close agreement between HAM, RK4M and LADM solutions that confirm the effectiveness of HAM. Graphical illustrations show that the study showcases the impact of varying essential parameters in the model, providing valuable insights into the variation of nutrient density, fish population and mussel population and also they help to prove that the HAM is a reliable and efficient approach for analyzing the dynamics of fish farm model.