Existence, uniqueness and stability of transition fronts of non-local equations in time heterogeneous bistable media

The present paper is devoted to the study of the existence, the uniqueness and the stability of transition fronts of non-local dispersal equations in time heterogeneous media of bistable type under the unbalanced condition. We first study space non-increasing transition fronts and prove various important qualitative properties, including uniform steepness, stability, uniform stability and exponential decaying estimates. Then, we show that any transition front, after certain space shift, coincides with a space non-increasing transition front (if it exists), which implies the uniqueness, up-to-space shifts and monotonicity of transition fronts provided that a space non-increasing transition front exists. Moreover, we show that a transition front must be a periodic travelling front in periodic media and asymptotic speeds of transition fronts exist in uniquely ergodic media. Finally, we prove the existence of space non-increasing transition fronts, whose proof does not need the unbalanced condition.

Stochastic traveling waves of a stochastic Fisher–KPP equation and bifurcations for asymptotic behaviors

In this paper, traveling wave for a Fisher–KPP equation with stochastic advection and stochastic environmental capacity is investigated. Some conditions are imposed on the reaction rate and noise intensities such that the stochastic transition front exists. Following the results on stochastic transition front, the existence of stochastic traveling waves for the equation is established. Explicit relation between the wave speed and noise attributes including noise intensities and correlation is shown, which can realize the noise effects. It is found that noises reduce the wave speed. In addition, the positive correlation of noises may complement this reduction in a way. But the negative correlation of noises will further aggravate this reduction. There exists a threshold value on the noise correlation making the traveling wave wandering. If the correlation is larger than this threshold value, the wave travels with a forward tendency. Otherwise, the wave travels with a backward tendency. Bifurcations for asymptotic behaviors of the equation induced by the noise intensities and correlation are presented.

Application of the Fokker-Planck Kolmogorov equation to determining the mean value of nucleons generated by the direct ejection process

The statistical description of the process of direct nucleon ejection is the subject of this paper. This description is based on the generalized Fokker-Planck-Kolmogorov equation. The basic proposal is this: deterministic equations and their solutions have the mean values of the stochastic model of the ablation problem. The problem of deformation of the phase transition front is considered. The study is carried out by using the introduced stability position for the dispersion of solutions for mean values. The result of the study is the conclusion that the influence of the Markov diffusion coefficient leads to distortion of the original shape of the boundary phase transition front. The effect of the initial aspiration to resist changing the shape of the phase transition front was found.