Travelling Waves and Numerical Approximations in a Reaction Advection Diffusion Equation with Nonlocal Delayed Effects

SummaryIn this paper, we consider the growth dynamics of a single-species population with two age classes and a fixed maturation period living in a spatial transport field. A Reaction Advection Diffusion Equation (RADE) model with time delay and nonlocal effect is derived if the mature death and diffusion rates are age independent. We discuss the existence of travelling waves for the delay model with three birth functions which appeared in the well-known Nicholson's blowflies equation, and we consider and analyze numerical solutions of the travelling wavefronts from the wave equations for the problems with nonlocal temporally delayed effects. In particular, we report our numerical observations about the change of the monotonicity and the possible occurrence of multihump waves. The stability of the travelling wavefront is numerically considered by computing the full time-dependent partial differential equations with nonlocal delay.

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

[2]  Kiyoshi Yoshida,et al.  The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology , 1982 .

[3]  Stephen A. Gourley,et al.  Dynamics of the diffusive Nicholson's blowflies equation with distributed delay , 2000, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[4]  Xin Lu,et al.  On Diffusive Population Models with Toxicants and Time Delays , 1999 .

[5]  S. P. Blythe,et al.  Nicholson's blowflies revisited , 1980, Nature.

[6]  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.

[7]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[8]  Hal L. Smith.  A structured population model and a related functional differential equation: Global attractors and uniform persistence , 1994 .

[9]  Margaret C. Memory.  Bifurcation and asymptotic behavior of solutions of a delay-differential equation with diffusion , 1989 .

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

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

[12]  W. Ricker.  Stock and Recruitment , 1954 .

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

[14]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

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

[16]  Xingfu Zou,et al.  Traveling waves for the diffusive Nicholson's blowflies equation , 2001, Appl. Math. Comput..

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

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