(untitled) (Contemporary Mathematics, 2025)
This paper reveals a nonlinear mathematical Fisher-Kolmogorov-Petrovskii-Piskunov (F-KPP) equation in a biological system. This model has much importance in population genetics as well as nematic liquid crystals, explaining the reaction-diffusion system through traveling waves in population genetics as well as pattern formation in bi-stable systems. We use a fractional derivative to improve accuracy as well as understand the dynamics of the model. By utilizing the modified simplest equation method, we investigate the distinct kinds of exact wave solutions involving trigonometric, hyperbolic, and rational functions. To obtain and verify the solutions, we use the Mathematica software. We demonstrate the obtained solutions through 2-D, 3-D, and contour plots with the use of the Mathematica tool. The main contribution of this paper is to analyze the qualitative analyses, including stability analysis, modulation instability, and asymptotic analysis. At the end, the gained solutions are fruitful in different areas, like ecology, gene mutation, combustion theory, phase transition, and many others.