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Applicable Analysis
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Articles

Doubly nonlocal Fisher–KPP equation: front propagation

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Pages 1373-1396 | Received 25 Oct 2018, Accepted 09 Jul 2019, Published online: 18 Jul 2019

References

  • Bolker B, Pacala SW. Using moment equations to understand stochastically driven spatial pattern formation in ecological systems. Theor Popul Biol. 1997;52(3):179–197. doi: 10.1006/tpbi.1997.1331
  • Durrett R. Crabgrass, measles and gypsy moths: an introduction to modern probability. Bull New Ser Am Math Soc. 1988;18(2):117–144. doi: 10.1090/S0273-0979-1988-15625-X
  • Finkelshtein D, Kondratiev Y, Kozitsky Y, et al. The statistical dynamics of a spatial logistic model and the related kinetic equation. Math Models Methods Appl Sci. 2015;25(2):343–370. doi: 10.1142/S0218202515500128
  • Finkelshtein D, Kondratiev Y, Kutoviy O. Semigroup approach to birth-and-death stochastic dynamics in continuum. J Funct Anal. 2012;262(3):1274–1308. doi: 10.1016/j.jfa.2011.11.005
  • Fournier N, Méléard S. A microscopic probabilistic description of a locally regulated population and macroscopic approximations. Ann Appl Probab. 2004;14(4):1880–1919. doi: 10.1214/105051604000000882
  • Mollison D. Possible velocities for a simple epidemic. Adv Appl Probab. 1972;4:233–257. doi: 10.2307/1425997
  • Mollison D. The rate of spatial propagation of simple epidemics. Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability (Univ. California, Berkeley, CA, 1970/1971), Vol. III: Probability Theory. Berkeley (CA): University of California Press; 1972. p. 579–614.
  • Schumacher K. Travelling-front solutions for integro-differential equations. I. J Reine Angew Math. 1980;316:54–70.
  • Andreu-Vaillo F, Mazón JM, Rossi JD, et al. Nonlocal diffusion problems. Vol. 165. Mathematical surveys and monographs. Providence (RI): AMS; 2010. ISBN 978-0-8218-5230-9. xvi+256 pp.
  • Fife PC. Mathematical aspects of reacting and diffusing systems. Vol. 28. Lecture notes in biomathematics. Berlin, New York (NY): Springer-Verlag; 1979. ISBN 3-540-09117-3. iv+185 pp.
  • Murray JD. Mathematical biology. II. Vol. 18. Interdisciplinary applied mathematics. 3rd ed. New York (NY): Springer-Verlag, 2003. ISBN 0-387-95228-4. xxvi+811 pp. Spatial models and biomedical applications.
  • Shigesada N, Kawasaki K. Biological invasions: theory and practice. Oxford: Oxford University Press; 1997.
  • Finkelshtein D, Kondratiev Y, Tkachov P. Existence and properties of traveling waves for doubly nonlocal Fisher–KPP equations. Electron J Differ Equ. 2019;2019(10):1–27.
  • Liang X, Zhao X-Q. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math. 2007;1(60):1–40. doi: 10.1002/cpa.20154
  • Liang X, Zhao X-Q. Spreading speeds and traveling waves for abstract monostable evolution systems. J Func Anal. 2010;259:857–903. doi: 10.1016/j.jfa.2010.04.018
  • Shen W, Zhang A. Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats. J Differ Equ. 2010;249(4):747–795. doi: 10.1016/j.jde.2010.04.012
  • Weinberger H. Long-time behavior of a class of biological models. SIAM J Math Anal. 1982;13(3):353–396. doi: 10.1137/0513028
  • Zhang G-B, Li W-T, Wang Z-C. Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity. J Differ Equ. 2012;252(9):5096–5124. doi: 10.1016/j.jde.2012.01.014
  • Perthame B, Souganidis PE. Front propagation for a jump process model arising in spatial ecology. Discrete Contin Dyn Syst. 2005;13(5):1235–1246. doi: 10.3934/dcds.2005.13.1235
  • Xin J. An introduction to fronts in random media. Vol. 5. Surveys and tutorials in the applied mathematical sciences. New York (NY): Springer; 2009. x+159 pp.
  • Finkelshtein D, Tkachov P. The hair-trigger effect for a class of nonlocal nonlinear equations. Nonlinearity. 2018;31(6):2442–2479. doi: 10.1088/1361-6544/aab1cb
  • Kuehn C, Tkachov P. Pattern formation in the doubly-nonlocal Fisher–KPP equation. Discrete Contin Dyn Syst A. 2019;39(4):2077–2100. doi: 10.3934/dcds.2019087
  • Finkelshtein D, Kondratiev Y, Tkachov P. Doubly nonlocal Fisher–KPP equation: speeds and uniqueness of traveling waves. J Math Anal Appl. 2019;475(1):94–122. doi: 10.1016/j.jmaa.2019.02.010
  • Bouin E, Garnier J, Henderson C, et al. Thin front limit of an integro-differential Fisher–KPP equation with fat-tailed kernels. SIAM J Math Anal. 2018;50:3365–3394. doi: 10.1137/17M1132501
  • Finkelshtein D, Tkachov P. Accelerated nonlocal nonsymmetric dispersion for monostable equations on the real line. Appl Anal. 2019;98(4):756–780. doi: 10.1080/00036811.2017.1400537
  • Garnier J. Accelerating solutions in integro-differential equations. SIAM J Math Anal. 2011;43(4):1955–1974. doi: 10.1137/10080693X
  • Tkachov P. Front propagation in the non-local Fisher–KPP equation [PhD thesis]. Germany: Bielefeld University; 2017.
  • Tkachov P. On stability of traveling wave solutions for integro-differential equations related to branching Markov processes. arXiv:1808.00411, 2018.
  • Aguerrea M, Gomez C, Trofimchuk S. On uniqueness of semi-wavefronts. Math Ann. 2012;354(1):73–109. doi: 10.1007/s00208-011-0722-8
  • Coville J, Dávila J, Martínez S. Nonlocal anisotropic dispersal with monostable nonlinearity. J Differ Equ. 2008;244(12):3080–3118. doi: 10.1016/j.jde.2007.11.002
  • Coville J, Dupaigne L. On a non-local equation arising in population dynamics. Proc Roy Soc Edinburgh Sect A. 2007;137(4):727–755. doi: 10.1017/S0308210504000721
  • Li W-T, Sun Y-J, Wang Z-C. Entire solutions in the Fisher–KPP equation with nonlocal dispersal. Nonlinear Anal Real World Appl. 2010;11(4):2302–2313. doi: 10.1016/j.nonrwa.2009.07.005
  • Sun Y-J, Li W-T, Wang Z-C. Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity. Nonlinear Anal. 2011;74(3):814–826. doi: 10.1016/j.na.2010.09.032
  • Yagisita H. Existence and nonexistence of traveling waves for a nonlocal monostable equation. Publ Res Inst Math Sci. 2009;45(4):925–953. doi: 10.2977/prims/1260476648
  • Finkelshtein D, Kondratiev Y, Molchanov S, et al. Global stability in a nonlocal reaction–diffusion equation. Stoch Dynam. 2018;18(5):1850037 (15 pp.). doi: 10.1142/S0219493718500375
  • Yu Z, Yuan R. Existence, asymptotics and uniqueness of traveling waves for nonlocal diffusion systems with delayed nonlocal response. Taiwanese J Math. 2013;17(6):2163–2190. doi: 10.11650/tjm.17.2013.3794
  • Weng P, Zhao X-Q. Spreading speed and traveling waves for a multi-type SIS epidemic model. J Differ Equ. 2006;229(1):270–296. doi: 10.1016/j.jde.2006.01.020

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