Novel Exact Solutions for a Biological Population Model Using the Power Index Method

In this paper, we study a nonlinear biological population model that describes the spatiotemporal evolution of population density, incorporating nonlinear diffusion and reaction effects. Using the Power Index Method, we derive exact solutions for this model. In this approach, we select appropriate indexes for the independent variable in the similarity transformation, allowing the unknown functions to take polynomial, rational, or other elementary forms. These transformations reduce the nonlinear partial differential equation (NLPDE) to nonlinear ordinary differential equations (NLODEs). We then solve the nonlinear ordinary differential equations (NLODEs) exactly using Maple. Finally, by applying the similarity transformations and the exact solutions of the nonlinear ordinary differential equations (NLODEs), we obtain the exact solutions of the nonlinear biological population model. The behavior of the solutions is illustrated through $3$D graphs. The results demonstrate that the proposed method is straightforward, powerful, and capable of yielding exact solutions, revealing biological phenomena such as nonlinear diffusion, wave propagation, and decay effects.