Lie Symmetry Analysis and Invariant Solutions of the (1+1)-Dimensional Fisher Equation

Reaction–diffusion equations provide a fundamental framework for modelling spatial population dynamics and invasion processes in mathematical biology. Among these, the Fisher equation combines diffusion with logistic growth to describe the spread of an advantageous gene and the formation of travelling population fronts. In this work, we investigate the one-dimensional Fisher equation using Lie symmetry analysis to obtain a deeper analytical understanding of its wave propagation behaviour. The Lie point symmetries of the governing partial differential equation are derived and used to construct similarity variables that reduce the Fisher equation to ordinary differential equations. These reduced equations are then solved by a combination of direct integration and the tanh method, yielding explicit invariant and travelling–wave solutions. Symbolic computations in MAPLE are employed throughout to compute the symmetries, verify the reductions, and generate illustrative plots of the resulting wave profiles. The obtained solutions capture sigmoidal fronts connecting stable and unstable steady states, providing clear information about wave speed and shape. Overall, the study demonstrates that Lie group methods, combined with hyperbolic-function techniques, offer a powerful and systematic approach for analysing Fisher-type reaction–diffusion models and interpreting their biologically relevant invasion dynamics.