Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays

By Zhi-Cheng Wang, Wan-Tong Li, Shigui Ruan. 2006

252 citations recorded by the source corpus.

This paper deals with the existence of travelling wave fronts in reaction–diffusion systems with spatio-temporal delays. Our approach is to use monotone iterations and a nonstandard ordering for the set of profiles of the corresponding wave system. New iterative techniques are established for a class of integral operators when the reaction term satisfies different monotonicity conditions. Following this, the existence of travelling wave fronts for reaction–diffusion systems with spatio-temporal delays is established. Finally, we apply the main results to a single-species diffusive model with spatio-temporal delay and obtain some existence criteria of travelling wave fronts by choosing different kernels.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

View Paper: https://www.semanticscholar.org/paper/09e1846a62e99fb2e702c3ee4a943d4a0e53eacb

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In Applied Mathematics

In Computational Theory and Mathematics

In Numerical Analysis

Citations in the corpus

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects (2008, background): “Then it is easy to see that g(x, t) satisfies (H0) in [43].”
Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems (2006, methodology): “Other related results can be found in Gourley and Ruan [9], Hosono [11, 12], Kan-on [17], etc.”
On the existence of traveling waves for delayed reaction–diffusion equations☆ (2009, methodology): “[17] Z. Wang, W. Li, S. Ruan, Travelling wave fronts in reaction–diffusion systems with spatio-temporal delays, J. Differential Equations 222 (2006) 185–232.”
Monotone traveling wavefronts of the KPP-Fisher delayed equation (2010, methodology): “In particular, the existence of traveling fronts connecting the trivial and positive steady states in (3) (and its non-local generalizations) was studied in [2,4,7,11,16,27,33,35].”
The non-local Fisher–KPP equation: travelling waves and steady states (2009, methodology): “Spatio-temporal non-local terms were considered in [1, 18], where it was shown that for small delays and short interaction lengths solutions behave qualitatively as in the classical Fisher–KPP case.”
Pattern and Waves for a Model in Population Dynamics with Nonlocal Consumption of Resources (2006, background): “Existence of waves for the exact integro-differential equation is proved in [16] under some conditions on the integral kernel.”