Traveling Waves And Their Properties Of Stochastic Fisher-KPP Equations

Since the 1970 s, there have been intensive developments in the theory of traveling waves of reaction-diffusion equations. It was found that traveling waves can well model the oscillatory phenomenon and propagation with finite speed of nature, hence the existence,uniqueness and stability of traveling waves have been widely studied. In real world,perturbations of random are common. As the developments of random theory, especially the stochastic analysis and stochastic differential equations, it is necessary to introduce random factors into deterministic models. This thesis is mainly concerned with traveling waves of the stochastic Fisher-KPP equations. We mainly focus on the effects of noise on the traveling wave which develops from deterministic Fisher-KPP equation.In Chapter 1, we mainly introduce the background, research developments, and achievements of research objects. In the end, we give a simply introduction of the main content of this paper.In Chapter 2 and Chapter 3, we prove the existence of stochastic traveling wave by applying two different methods respectively. One is the stochastic order method,the other is the Kolmogorov tightness criterion. The first result has been published by”Journal of Differential Equations”. The second result has been published by ”Nonlinear Differential Equations and Applications(NoDEA)”.In Chapter 4, focusing on the effects of noise on the traveling wave of the deterministic Fisher-KPP equation, we study the asymptotic properties of a stochastic Fisher-KPP equation. The noise is suppose to be Stratonovich type and It?o type respectively. We find sufficient conditions on the strength of noise, under which the traveling wave still persists. Under these conditions, the noise does not affect the wave speed in the case of Stratonovich type. However, the wave speed in the case of It?o type depends on the strength of noise.Following the motivation and results of Chapter 4, in Chapter 5, we study the asymptotic properties of Fisher-KPP equation driven by dually It?o noises. The asymptotic properties of solution under effect of dual noises are more complicated. Similarly, we also find sufficient conditions on the strength of noises, under which the traveling wave still persists. Moreover, we present bifurcations of asymptotic properties of solution under different conditions on strength of noises. This result has been published by ”Journal of Differential Equations”.In Chapter 6, we consider the stochastic Fisher-KPP equation driven by dually Stratonovich noises, which is different form Chapter 5. We establish the existence of random traveling wave solution and estimate the precise value of wave speed.