Novel convergence to forced traveling waves for a nonlocal dispersal population model in a shifting environment

Some new types of the nonlinear stability of forced traveling waves for a nonlocal dispersal population model in a shifting environment are given in this paper. Firstly, when the speed of the shifting habitat edge c is larger than the spreading speed of the species c⁎, by applying the comparison principle, the Fourier transform and the L2-weighted energy estimate, we show that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of e−μt, for some μ>0. In order to extend the stability results to c=c⁎, we take another weight function and construct the different weighted Sobolev space. Then by using the Fourier transform and the L1-weighted energy estimate, we prove that all forced traveling waves with the speed c>c⁎ are exponentially stable in the form of t−12e−νt, for some ν>0, while all forced traveling waves with the speed c=c⁎ are algebraically stable in the form of t−12, which are quite different from the previous stability results.