Travelling wave dynamics in the nonlocal dispersal Fisher-KPP equation with distributed delay

• Published in: Nonlinearity Vol. 38; no. 7; pp. 75012 - 75043
• Main Authors: Xu, Zhaoquan, Dong, Luke
• Format: Journal Article
• Language: English
• Published: IOP Publishing 31.07.2025
• Subjects: 34B40; 45E10; 45K05; Fisher-KPP equation; nonlocal diffusion; travelling wave solutions, distributed delay
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/addf0a

We investigate the travelling wave dynamics in the nonlocal dispersal Fisher-KPP equation with distributed delay. The introduction of delay causes the equation to lack the comparison principle, thereby significantly increasing the complexity of studying propagation dynamics. In this paper, we prove that the delayed equation possesses a minimal wave speed, which is identical to that of the classical nonlocal dispersal Fisher-KPP equation without delay studied in previous works. Specifically, the delayed equation admits a travelling wave connecting the trivial steady state and a positive steady state if and only if the wave speed is greater than or equal to this minimal wave speed. This result highlights a significant observation: the delay does not affect the minimal wave speed of travelling waves, but may alter their shape. Additionally, we demonstrate theoretically that the equation can also admit monotone travelling waves when the mean of the delay kernel is small. The uniqueness of such monotone travelling waves, up to translation, is also confirmed.