On a Bistable Delayed Nonlinear Reaction–Diffusion Equation for a Two-Phase Free Boundary: Semi-Wave and Its Numerical Simulation

In this paper, we investigate the existence and uniqueness of the semi-wave solution to a bistable delayed reaction–diffusion equation with a double free boundary. This equation captures key features of biological systems and has broad applications in mathematical biology, including population dynamics, gene expression, virus propagation, and tumor growth. In particular, we employ the technique that was developed in the previous study by M. Alfaro et al. [Proceedings of the London Mathematical Society (3), 116, no. 4 (2018): 729–759], and use it to prove the existence and uniqueness of semi-wave solutions. Our results demonstrate that the semi-wave solutions depend on both the delay parameter and the free boundary condition. Additionally, we conduct numerical simulations to validate our theoretical results, and explore the effects of various parameters on the semi-wave solution and spreading speed.