Limiting Solutions of Nonlocal Dispersal Problem in Inhomogeneous Media

This paper is concerned with the nonlocal dispersal problem in inhomogeneous media. Our goal is to show the limiting behavior of perturbation equation with parameters. By analyzing the asymptotic behavior of solutions when the parameter is small, we find that convection appears in inhomogeneous media. Moreover, if the effect of inhomogeneous media changes, then we prove a convergence result that convection disappears in nonlocal dispersal problems.

[1]  J. Rossi,et al.  Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions , 2009 .

[2]  Wan-Tong Li,et al.  Approximate the FokkerPlanck equation by a class of nonlocal dispersal problems , 2011 .

[3]  Wenxian Shen,et al.  Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats , 2010 .

[4]  Wenxian Shen,et al.  Criteria for the Existence and Lower Bounds of Principal Eigenvalues of Time Periodic Nonlocal Dispersal Operators and Applications , 2012 .

[5]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[6]  Wenxian Shen,et al.  Spreading Speeds and Traveling Waves of Nonlocal Monostable Equations in Time and Space Periodic Habitats , 2013, 1307.6576.

[7]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[8]  Chufen Wu,et al.  Forced waves and gap formations for a Lotka–Volterra competition model with nonlocal dispersal and shifting habitats , 2021 .

[9]  Wenxian Shen,et al.  Approximations of random dispersal operators/equations by nonlocal dispersal operators/equations , 2014, 1402.6353.

[10]  Guo-Bao Zhang,et al.  Global stability of wavefronts with minimal speeds for nonlocal dispersal equations with degenerate , 2011 .

[11]  Jian-Wen Sun.  Positive solutions for nonlocal dispersal equation with spatial degeneracy , 2018 .

[12]  Grégoire Nadin.  Existence and uniqueness of the solution of a space–time periodic reaction–diffusion equation , 2010 .

[13]  P. Bates,et al.  Traveling Waves in a Convolution Model for Phase Transitions , 1997 .

[14]  Yihong Du,et al.  Order Structure and Topological Methods in Nonlinear Partial Differential Equations: Vol. 1: Maximum Principles and Applications , 2006 .

[15]  W. Shen,et al.  Traveling Wave Solutions of Spatially Periodic Nonlocal Monostable Equations , 2012, 1202.2452.

[16]  J. Roquejoffre,et al.  The effect of a line with nonlocal diffusion on Fisher-KPP propagation , 2015, 1506.09046.

[17]  J. Sethian,et al.  Mathematical Biology , 2006 .

[18]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations: Cantrell/Diffusion , 2004 .

[19]  Yuan Lou,et al.  Random dispersal vs. non-local dispersal , 2009 .

[20]  J. Coville.  On a simple criterion for the existence of a principal eigenfunction of some nonlocal operators , 2010, 1106.5137.

[21]  V. Hutson,et al.  The evolution of dispersal , 2003, Journal of mathematical biology.

[22]  Emmanuel Chasseigne,et al.  Asymptotic behavior for nonlocal diffusion equations , 2006 .

[23]  Andrej Zlatoš.  Propagation of Reactions in Inhomogeneous Media , 2014, 1401.1175.

[24]  Junping Shi,et al.  Spatial pattern formation in activator-inhibitor models with nonlocal dispersal , 2021, Discrete and Continuous Dynamical Systems - Series B.

[25]  Paul C. Fife,et al.  Some Nonclassical Trends in Parabolic and Parabolic-like Evolutions , 2003 .

[26]  H. Weinberger,et al.  Convergence to spatial-temporal clines in the Fisher equation with time-periodic fitnesses , 1990, Journal of mathematical biology.

[27]  J. Rossi,et al.  Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence , 2016 .

[28]  Wan-Tong Li,et al.  Entire solutions in the Fisher-KPP equation with nonlocal dispersal , 2010 .

[29]  Liviu I. Ignat,et al.  A nonlocal convection–diffusion equation , 2007 .

[30]  Noemi Wolanski,et al.  How to Approximate the Heat Equation with Neumann Boundary Conditions by Nonlocal Diffusion Problems , 2006, math/0607058.