Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

• DOI: 10.1007/s10884-008-9103-8
• Publication Date: 2008-03-10T17:45:24Z
• Authors: Zhi-Cheng Wang; Wan-Tong Li; Shigui Ruan
• Keywords: 0101 mathematics; 01 natural sciences

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay $$\frac{\partial u\left(x, t\right)}{\partial t}= d\Delta u\left(x, t\right)+f\left(u\left(x, t\right),\int\limits_{-\infty }^\infty h\left(x - y\right) u\left(y, t - \tau\right) dy\right)\!.$$ 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*, 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.