Entire solutions in the Fisher-KPP equation with nonlocal dispersal

By Wan-Tong Li, Yu-Juan Sun, Zhi-Cheng Wang. 2010

161 citations recorded by the source corpus.

This paper is concerned with entire solutions of the Fisher-KPP equation with nonlocal dispersal, i.e., u(t) =J * u - u + f(u), which is a one-dimensional nonlocal version of the Fisher-KPP equation describing the spatial spread of a mutant in a given population and the dispersion of the genetic characters is assumed to follow a nonlocal diffusion law modeled by a convolution operator. Here the entire solutions are defined in the whole space and for all time t is an element of R. A comparison principle is employed to establish the existence of entire solutions by combining two traveling wave solutions with different speeds and coming from both ends of the real axis and some spatially independent solutions. The main difficulty is that a lack of regularizing effect occurs. This is probably the first time the existence of entire solutions of reaction equations with nonlocal dispersal has been studied. (C) 2009 Elsevier Ltd. All rights reserved.

Subfields: Applied Mathematics, Modeling and Simulation, Public Health, Environmental and Occupational Health

View Paper: https://www.semanticscholar.org/paper/d3446153985c02cb7b38895820adaff1282b40c2

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In Applied Mathematics

In Modeling and Simulation

In Public Health, Environmental and Occupational Health

Citations in the corpus

Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “See, for example, [11], [13], [15], [36], [41], [45], [57], [58], for the study of entire solutions and the existence of spreading speeds and traveling wave solutions connecting the trivial solution u = 0 and a nontrivial positive stationary solution for some special cases of (1.”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, unknown): “The reader is referred to [1], [2], [4], [5], [6], [7], [8], [10], [11], [16], [22], [23], [24], [26], [28], [30], [32], etc.”
Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats (2012, background): “See also [31], [33], [36] and the references therein for the study of entire solutions and traveling wave solutions of certain (delayed) nonlocal monostable systems.”
Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications (2012, unknown): “The reader is referred to [5,8,10,11,13,15,16,19,20,22,24–26,28,32], etc.”