Publisher Summary This chapter discusses large-time asymptotic behavior of solutions to the periodic problem for model nonlinear nonlocal evolution equation. Model equation describes various wave processes in periodic media. It generalizes many well-known equations of modern mathematical physics. The investigation of periodic initial-boundary value problems is studied. The asymptotic stability of stationary periodic solutions for the Fisher equation is proved in the chapter. The study of the periodic problem is in many respects easier than Cauchy problem, and the periodic results are exponential, whereas the problem on the line appears more delicate and has algebraic decay rates.