Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects
By Zhi-Cheng Wang, Wan-Tong Li, Shigui Ruan. 2008
194 citations recorded by the source corpus.
AbstractIn 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 < c* and 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 ∂2f (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 ∂2f (0, 0) = 0, then the delay and nonlocality do not affect the spreading speed.
Subfields: Genetics, Modeling and Simulation, Public Health, Environmental and Occupational Health
View Paper: https://www.semanticscholar.org/paper/3238f7381a0a7624faa6610d6aa95b923870c8ef
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Genetics
- Guo Lin
- Sergei I. Trofimchuk
- Maitere Aguerrea
- Carlos Gómez
- Wan-Tong Li
- Fei-Ying Yang
- Ming Ma
- King-Yeung Lam
In Modeling and Simulation
In Public Health, Environmental and Occupational Health
- Wan-Tong Li
- Guo Lin
- Ming Ma
- Fei-Ying Yang
- Maitere Aguerrea
- Carlos Gómez
- Sergei I. Trofimchuk
- Zhi-Cheng Wang
Citations in the corpus
- Traveling Wave Solutions for Delayed Reaction–Diffusion Systems and Applications to Diffusive Lotka–Volterra Competition Models with Distributed Delays (2014, unknown): “[34–36], Wu and Zou [38], and Yi et al.”
- Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models (2009, background): “Notice the study on the stability and the uniqueness of traveling wave solutions (see [31, 41, 52, 58]), we think the asymptotic behavior is very important in investigating more properties of traveling wave solutions.”
- On uniqueness of semi-wavefronts (2010, methodology): “It would be interesting to consider further generalizations of (1) in order to include more applications (for example, equations with distributed delays considered in [16,17], see also [25,33,39]).”
- Traveling waves for a nonlocal dispersal SIR model with standard incidence (2014, background): “The proof is mainly based on that of Wang and Wu [38] and several earlier studies, see [13, 20, 23, 36, 37].”
- Asymptotic spreading of competition diffusion systems: The role of interspecific competitions (2012, background): “1) and essentially the combination of the theory of abstract functional differential equations (see [29]) with the monotone dynamical systems (see [40]); see also [25, 30, 41, 47] for some related topics.”