The Minimum Speed For Nonlocal Dispersal KPP Equation With Noise

This paper consider the traveling wave solution of nonlocal dispersal KPP equations with small noise,mainly focusing on the minimal wave speed,owning to give out the accurate portrayal of its minimal wave speed.In many cases,we can not get the linear approximation equation of the nonlocal dispersal equations on one point.For that,we select the method of using the minimal speed of Sub-Sup Solutions to approach the minimal speed of traveling wave solutions to solve the problem,in stead of the phase-plane analysis.As we all know,the minimal wave speed of the typical KPP equation without noise is 2,correspondingly,the minimal wave speed of the nonlocal dispersal KPP equation is given out by Coville,relating with the core function J.Adding the noise to the nonlocal dispersal KPP equations,then we can draw lessons from ideas of famous scholar Mueller on Inventional Maths?2012?,to construct a group of Sub-Sup Solutions.By solving the minimal wave speeds of them respectively and the properties of holding orders,we can get the description of the minimal wave speed of traveling wave solutions.Specifically,considering the following problem:u_t = J*u-u + f?u?+???·??u?· W?x,t?.Then we can get listed below results:???when the noise ????0,the minimal wave speed tends to the minimal wave speed without noise?can be solved by the formula which is given by Coville?,signed as c₀+k·?/2·???²?c_??c₀+f'?0?·?/2·???²,where c_?is the minimal wave speed with noise,k staisfying 0<k<f'?0?1,? is a small constant.