Existence, uniqueness and stability of travelling waves in a discrete reaction–diffusion monostable equation with delay

In this paper, we study the existence, uniqueness and asymptotic stability of travelling wavefronts of the following equation:ut(x,t)=D[u(x+1,t)+u(x-1,t)-2u(x,t)]-du(x,t)+b(u(x,t-r)),where , t>0, D,d>0, r 0, and b(0)=dK-b(K)=0 for some K>0 under monostable assumption. We show that there exists a minimal wave speed c*>0, such that for each c>c* the equation has exactly one travelling wavefront U(x+ct) (up to a translation) satisfying U(-∞)=0,U(+∞)=K and , where λ=Λ1(c) is the smallest solution to the equation cλ-D[eλ+e-λ-2]+d-b′(0)e-λcr=0. Moreover, the travelling wavefront is strictly monotone and asymptotically stable with phase shift in the sense that if an initial data satisfies and for some ρ0 (0,+∞), then the solution u(x,t) of the corresponding initial value problem satisfies for some .