Exponential stability of nonmonotone traveling waves for Nicholson's blowflies equation

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 p/d > e² with a small time-delay r < [π - arctan √ln p/d (ln p/d - 2)]/d √ln p/d (ln p/d - 2), all noncritical traveling waves φ(x + ct) with c > c _* > 0 are exponentially stable, too. As a corollary, we also prove the uniqueness of traveling waves in the case of p/d > e², which to the best of our knowledge was open. Finally, some numerical simulations are carried out. When e < p/d ≤ e², we demonstrate numerically that after a long time the solution behaves like a monotone traveling wave for a small time-delay, and behaves like an oscillatory traveling wave for a big time-delay. When p/d > e², if the time-delay is small, then the solution numerically behaves like a monotone/nonmonotone traveling wave, but if the time-delay is big, then the solution is numerically demonstrated to be chaotically oscillatory but not an oscillatory traveling wave. These either confirm and support our theoretical results or open up some new phenomena for future research.