Spreading speeds in a lattice differential equation with distributed delay

This paper studies the spreading speed for a lattice differential equation with infinite distributed delay and we find that the spreading speed coincides with the minimal wave speed of traveling waves. Here the model has been proposed to describe a single species living in a 1D patch environment with infinite number of patches connected locally by diffusion.

Spreading speeds in a lattice differential equation with distributed delay

This paper studies the spreading speed for a lattice differential equation with infinite distributed delay and we find that the spreading speed coincides with the minimal wave speed of traveling waves. Here the model has been proposed to describe a single species living in a 1D patch environment with infinite number of patches connected locally by diffusion.

___

  • Aronson DG, Weinberger HF. Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation. Lect Notes Math 1975; 446: 5–49.
  • Aronson DG, Weinberger HF. Multidimensional nonlinear diffusion arising in population genetics. Adv Math 1978; 30: 33–76.
  • Cheng CP, Li WT, Wang ZC. Spreading speeds and traveling waves in a delayed population model with stage structure on a 2D spatial lattice. IMA J Appl Math 2008; 73: 592–618.
  • Cheng CP, Li WT, Wang ZC. Asymptotic stability of traveling wavefronts in a delayed population model with stage structure on a two-dimensional spatial lattice. Discrete Contin Dyn Syst Ser B 2010; 13: 559–575.
  • Cheng CP, Li WT, Wang ZC. Persistence of bistable waves in a delayed population model with stage structure on a two-dimensional spatial lattice. Nonlinear Anal Real World Appl 2012; 13: 1873–1890.
  • Cook HE, Fontaine DD, Hillard JE. A model for diffusion on cubic lattices and its application to the early stages of ordering. Acta Metall 1969; 17: 765–773.
  • Diekmann O. Thresholds and travelling waves for the geographical spread of infection. J Math Biol 1978; 6: 109–130.
  • Diekmann O. Run for your life. A note on the asymptotic speed of propagation of an epidemic. J Differential Equations 1979; 33: 58–73.
  • Fang J, Wei J, Zhao XQ. Spreading speeds and travelling waves for non-monotone time-delayed lattice equations. P Roy Soc Lond A Mat 2010; 466: 1919–1934.
  • Gourley SA, So JWH. Extinction and wavefront propagation in a reaction-diffusion model of a structured population with distributed maturation delay. P Roy Soc Edinb A 2003; 133: 527–548.
  • Gourley SA, Wu J. Extinction and periodic oscillations in an age-structured population model in a patchy environ- ment. J Math Anal Appl 2004; 289: 431–445.
  • Gurney W, Blythe S, Nisbet R. Nicholson’s blowflies revisited. Nature 1980; 287: 17–21.
  • Hsu SB, Zhao XQ. Spreading speeds and traveling waves for non-monotone integrodifference equations. SIAM J Math Anal 2008; 40: 776–789.
  • Kopell N, Ermentrout GB, Williams TL. On chains of oscillators forced at one end. SIAM J Appl Math 1991; 51: 1397–1417.
  • Kyrychko Y, Gourley SA, Bartuccelli MV. Dynamics of a stage-structured population model on an isolated finite lattice. SIAM J Math Anal 2006; 37: 1688–1708.
  • Laplante JP, Erneux T. Propagation failure in arrays of coupled bistable chemical reactors. J Phys Chem 1992; 96: 4931–4934.
  • Li B, Lewis MA, Weinberger HF. Existence of traveling waves for integral recursions with nonmonotone growth functions. J Math Biol 2009; 58: 323–338.
  • Liang X, Yi Y, Zhao XQ. Spreading speeds and traveling waves for periodic evolution systems. J. Differential Equations 2006; 231: 57-77.
  • Liang X, Zhao XQ. Asymptotic speeds of spread and traveling waves for monotone semiflows with applications. Comm Pure Appl Math 2007; 60: 1–40.
  • Lui R. Biological growth and spread modeled by systems of recursions, I. Mathematical theory. Math Biosci 1989; 93: 269–295.
  • Ma S, Weng P, Zou X. Asymptotic speed of propagation and traveling wavefronts in a non-local delayed lattice differential equation. Nonlinear Anal 2006; 65: 1858–1890.
  • Ma S, Zou X. Propagation and its failure in a lattice delayed differential equation with global interaction. J Differential Equations 2005; 212: 129–190.
  • Martin RH, Smith HL. Abstract functional-differential equations and reaction-diffusion systems. T Am Math Soc 1990; 321: 1–44.
  • Niu HL, Wang ZC. Traveling waves in lattice differential equations with distributed maturation delay. Electron J Qual Theo 2013; 44: 1–22.
  • Ruan S, Wu J. Reaction-diffusion equations with infinite delay. Canadian Applied Mathematics Quarterly 1994; 2: 485–550.
  • So JWH, Wu J, Zou X. A reaction-diffusion model for a single species with age structure 1. Traveling wavefronts on unbounded domains. R Soc Lond A Mat 2001; 457: 1841–1853.
  • Taylor JE, Cahn JW, Handwerker CA. Geometric models of crystal growth. Acta Metall Mater 1992; 40: 1443–1474.
  • Thieme HR. Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread. J Math Biol 1979; 8: 173–187.
  • Thieme HR. Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. J Reine Angew Math 1979; 306: 94–121.
  • Thieme HR, Zhao XQ. Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction- diffusion models. J Differential Equations 2003; 195: 430–470.
  • Wang ZC, Li WT. Dynamics of a nonlocal delayed reaction-diffusion equation without quasi-monotonicity. P Roy Soc Edinb A 2010; 140: 1081–1109.
  • Weinberger HF. Long-time behavior of a class of biological models. SIAM J Math Anal 1982; 13: 353–396.
  • Weng P, Huang H, Wu J. Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction. IMA J Appl Math 2003; 68: 409–439.
  • Weng P, Wu J. Wavefronts for a non-local reaction-diffusion population model with general distributive maturity. IMA J Appl Math 2008; 73: 477–495.
  • Weng P, Wu J, Huang H, Ling J. Asymptotic speed of propagation of wave fronts in a 2D lattice delay differential equation with global interaction. Canadian Applied Mathematics Quarterly 2003; 11: 377–414.
  • Wu SL, Liu SY. Asymptotic speed of spread and traveling fronts for a nonlocal reaction-diffusion model with distributed delay. Appl Math Model 2009; 33: 2757–2765.
  • Zhao XQ, Xiao D. The asymptotic speed of spread and traveling waves for a vector disease model. J Dyn Differ Equ 2006; 18: 1001–1019.