Exponential Stability of Nonmonotone Traveling Waves for Nicholson's Blowflies Equation

Abstract. This paper is concerned with Nicholson’s blowflies equation, a kind of time-delayed reaction-diffusion equation. It is known that when the ratio of birth rate coefficient and death rate coefficient satisfies 1 p d ≤ e, the equation is monotone and possesses monotone traveling wavefronts, which have been intensively studied in previous research. However, when p d e, the equation losses its monotonicity, and its traveling waves are oscillatory when the time-delay r or the wave speed c is large, which causes the study of stability of these nonmonotone traveling waves to be challenging. In this paper, we use the technical weighted energy method to prove that when e p d ≤ e2, all noncritical traveling waves φ(x+ ct) with c c ∗ 0 are exponentially stable, where c ∗ 0 is the minimum wave speed. Here, we allow the traveling wave to be either monotone or nonmonotone with any speed c c ∗ and any size of the time-delay r 0; however, when pd e 2 with a small time-delay r π−arctan ln p