Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems
By Wan-Tong Li, Guo Lin, Shigui Ruan. 2006
221 citations recorded by the source corpus.
This paper is concerned with the existence of travelling wave solutions in a class of delayed reaction–diffusion systems without monotonicity, which concludes two-species diffusion–competition models with delays. Previous methods do not apply in solving these problems because the reaction terms do not satisfy either the so-called quasimonotonicity condition or non-quasimonotonicity condition. By using Schauder's fixed point theorem, a new cross-iteration scheme is given to establish the existence of travelling wave solutions. More precisely, by using such a new cross-iteration, we reduce the existence of travelling wave solutions to the existence of an admissible pair of upper and lower solutions which are easy to construct in practice. To illustrate our main results, we study the existence of travelling wave solutions in two delayed two-species diffusion–competition systems.
Subfields: Applied Mathematics, Computer Networks and Communications, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/67af193a59d0f63949516860e20d0aa098ac3d90
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In Applied Mathematics
In Computer Networks and Communications
In Public Health, Environmental and Occupational Health
Citations in the corpus
- Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models (2009, methodology): “φ 1 (t) ≤ F1(φ1, φ2, · · · , φn)(t), F1 ( φ1, φ2, · · · , φn ) (t) ≤ φ1(t) (14) for all φj(t) ∈ {φj(t), φj(t)}, t ∈ R and j ∈ I1, for which one refers to [21, 28, 61].”
- Traveling wave solutions for a continuous and discrete diffusive predator–prey model (2017, background): “For this aspect, we refer the reader to the works [23, 17, 18, 21, 15, 19, 22] for the continuous case and [16, 20] for the discrete case.”
- Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects (2008, background): “[23] developed a new cross iteration scheme, which is different from that defined in [28,47].”
- Time periodic traveling wave solutions for periodic advection–reaction–diffusion systems (2014, unknown): “[22], Sandstede and Scheel [31], Volpert et al.”
- Asymptotic spreading of competition diffusion systems: The role of interspecific competitions (2012, background): “It is well known that the asymptotic speeds of spreading equal to the minimal wave speeds of travelling wave solutions in many models [26, 44, 48], and we refer to [25] for the role of inter-specific delays in the travelling wave solutions.”