Asymptotic speed of propagation of wave fronts in a lattice delay differential equation with global interaction

In this paper, we derive a lattice model for a single species in a one-dimensional patchy environment with infinite number of patches connected locally by diffusion. Under the assumption that the death and diffusion rates of the mature population are age independent, we show that the dynamics of the mature population is governed by a lattice delay differential equation with global interactions. We study the well-posedness of the initial-value problem and obtain the existence of monotone travelling waves for wave speeds c > c * . We show that the minimal wave speed c * is also the asymptotic speed of propagation, which depends on the maturation period and the diffusion rate of mature population monotonically.

[1]  Stephen A. Gourley,et al.  A predator-prey reaction-diffusion system with nonlocal effects , 1996 .

[2]  Horst R. Thieme,et al.  Asymptotic estimates of the solutions of nonlinear integral equations and asymptotic speeds for the spread of populations. , 1979 .

[3]  D. Aronson,et al.  Nonlinear diffusion in population genetics, combustion, and nerve pulse propagation , 1975 .

[4]  L. Peletier,et al.  Nonlinear diffusion in population genetics , 1977 .

[5]  O. Diekmann,et al.  Thresholds and travelling waves for the geographical spread of infection , 1978, Journal of mathematical biology.

[6]  A. Offord.  Introduction to the Theory of Fourier Integrals , 1938, Nature.

[7]  O. Diekmann,et al.  The Dynamics of Physiologically Structured Populations , 1986 .

[8]  R. Bracewell.  The Fourier transform. , 1989, Scientific American.

[9]  Xingfu Zou,et al.  A reaction–diffusion model for a single species with age structure. I Travelling wavefronts on unbounded domains , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  Nicholas F. Britton,et al.  Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model , 1990 .

[11]  Odo Diekmann,et al.  Run for your life; a note on the asymptotic speed of propagation of an epidemic : (preprint) , 1979 .

[12]  X. Zou,et al.  Structured population on two patches: modeling dispersal and delay , 2001, Journal of mathematical biology.

[13]  Xingfu Zou,et al.  Traveling Wave Fronts of Reaction-Diffusion Systems with Delay , 2001 .

[14]  Hans F. Weinberger,et al.  Asymptotic behavior of a model in population genetics , 1978 .

[15]  D. Aronson,et al.  Multidimensional nonlinear di u-sion arising in population genetics , 1978 .

[16]  S. A. Gourley.  Travelling front solutions of a nonlocal Fisher equation , 2000, Journal of mathematical biology.

[17]  Hal L. Smith,et al.  Strongly order preserving semiflows generated by functional differential equations , 1991 .

[18]  L. Rass,et al.  The asymptotic speed of propagation of the deterministic non-reducible n-type epidemic , 1986, Journal of mathematical biology.