Stationary solutions and spreading speeds of nonlocal monostable equations in space periodic habitats

By Wenxian Shen, Aijun Zhang. 2012

84 citations recorded by the source corpus.

This paper deals with positive stationary solutions and spreading speeds of monostable equations with nonlocal dispersal in spatially periodic habitats. The existence and uniqueness of positive stationary solutions and the existence and characterization of spreading speeds of such equations with symmetric convolution kernels are established in the authors’ earlier work [41] for following cases: the nonlocal dispersal is nearly local; the periodic habitat is nearly globally homogeneous or it is nearly homogeneous in a region where it is most conducive to population growth. The above conditions guarantee the existence of principal eigenvalues of nonlocal dispersal operators associated to linearized equations at the trivial solution. In general, a nonlocal dispersal operator may not have a principal eigenvalue. In this paper, we extend the results in [41] to general spatially periodic nonlocal monostable equations. As a consequence, it is seen that the spatial spreading feature is generic for monostable equations with nonlocal dispersal.

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

View Paper: https://www.semanticscholar.org/paper/1e51de99f7c27c896356f0ba771ef89628855e47

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Citations in the corpus

Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations (2012, background): “ satisﬁed. As it is mentioned above, a spatially periodic monostable equation with random dispersal has a spreading speed in every direction. This important feature has been well extended in [54] and [55] to spatially periodic monostable equations with nonlocal dispersal. For given function h : R×RN”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “For the operator L Ω + a, much less is known and only partial results have been obtained when Ω is bounded [25, 28, 32, 37, 40, 41] or in a periodic media [29, 30, 60, 59].”
Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats (2013, background): “See also [20], [35], and [37] for the study of spreading properties of (1.”
On Principal Spectrum Points/Principal Eigenvalues of Nonlocal Dispersal Operators and Applications (2013, background): “1 (4) holds for periodic case (see [36]).”
Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications (2012, background): “In [33], the authors proved that time independent KPP equations with nonlocal dispersal also exhibit monostable feature (see also [2,9] for the study of positive stationary solutions of time independent KPP equations with nonlocal dispersal).”