In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction-diffusion equation with delay ∂ u(x, t)/∂ t= dΔu(x,t)+f(u(x, t),∫ -∞∞ h(x - y) u(y, t - τ) dy). Under the monostable assumption, we show that there exists a minimal wave speed c> 0, such that the equation has no traveling wave front for 0 < c < cand a traveling wave front for each c ≥ c. Furthermore, we show that for c > c, such a traveling wave front is unique up to translation and is globally asymptotically stable. When applied to some population models, these results cover, complement and/or improve a number of existing ones. In particular, our results show that (i) if ∂2 f (0, 0) > 0, then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂2 f (0, 0) = 0, then the delay and nonlocality do not affect the spreading speed.
- Asymptotic stability
- Monostable equation
- Nonlocal reaction-diffusion equation
- Traveling wave front
ASJC Scopus subject areas