Series solution for continuous population models for single and interacting species by the homotopy analysis method

The homotopy analysis method (HAM) is used to find approximate analytical solutions of continuous population models for single and interacting species. The homotopy analysis method contains the auxiliary parameter $hbar,$ which provides us with a simple way to adjust and control the convergence region of series solution. the solutions are compared with the numerical results obtained using NDSolve, an ordinary differential equation solver found in the Mathematica package and a good agreement is found. Also the solutions are compared with the available analytic results obtained by other methods and more accurate and convergent series solution found. The convergence region is also computed which shows the validity of the HAM solution. This method is reliable and manageable.