Spectral stability of travelling waves in the Fisher-Kolmogorov-Petrovsky-Piskunov and Fisher-Stefan models

The reaction-diffusion equations including the Fisher-Kolmogorov-Petrovsky-Piskunov (Fisher-KPP) and Fisher-Stefan model have been used to study biological invasion. In this thesis, we first consider the standing waves and travelling waves of the Fisher- KPP equation and the behaviour of the dynamical system near the equilibrium points. Next, we examine the existence of solutions to the Fisher-KPP model by constructing a trapping region in the plane for the wave speed 𝑐 ≠ 0. Then, we study stability of standing waves and travelling waves of this model by looking at the point spectrum, the asymptotic operator and the essential spectrum. However, a limitation of the Fisher-KPP equation is that it cannot be used to model the extinction of invasive populations in practical situations. Hence, the Fisher-Stefan equation is introduced to simulate both biological invasion for 𝑐 > 0 and recession for 𝑐 < 0. This model is reformulated from the Fisher-KPP equation and includes a moving boundary, which gives rise to a spreading-vanishing dichotomy. Moreover, we look at the existence and uniqueness of solutions to the Fisher-Stefan equation. By analysing the phase portrait, numerical results and the perturbation solutions, we can construct approximate solutions of the Fisher-Stefan equation. This phase portrait of this model also suggests that the solutions of the Fisher-KPP model exist for 𝑐 < 2 but they are unstable. Therefore, these solutions are disregarded when 𝑐 < 2 and considered as not exist. This unstable solution has value in describing solution of Fisher- Stefan problem. Thus, the stable solutions exist in the Fisher-Stefan problem. Last, we consider the spectral stability of the Fisher-Stefan equation by analysing its asymptotic operator, continuous spectrum and point spectrum. The results indicate that there is no point spectrum of the Fisher-Stefan asymptotic solution for the eigenvalue parameter 𝜆 > 1 at 𝑐 ≥ 0.