Existence and nonexistence of monotone traveling waves for the delayed Fisher equation

Abstract In this paper a new approach based on a shooting method in a half line coupled with the technique of upper–lower solution pair is used to study the existence and nonexistence of monotone waves for one form of the delayed Fisher equation that does not have the quasimonotonicity property. A necessary and sufficient condition is provided. This new method can be extended to investigate many other nonlocal and non–monotone delayed reaction–diffusion equations.

[1]  G. E. Hutchinson,et al.  CIRCULAR CAUSAL SYSTEMS IN ECOLOGY , 1948, Annals of the New York Academy of Sciences.

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

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

[4]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[5]  Xiao-Qiang Zhao,et al.  Asymptotic speeds of spread and traveling waves for monotone semiflows with applications , 2007 .

[6]  Gennady Bocharov,et al.  Structured Population Models, Conservation Laws, and Delay Equations , 2000 .

[7]  Wan-Tong Li,et al.  Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays , 2006 .

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

[9]  R. Fisher.  THE WAVE OF ADVANCE OF ADVANTAGEOUS GENES , 1937 .

[10]  K. Hadeler.  Neutral Delay Equations from and for Population Dynamics , 2007 .

[11]  Wenzhang Huang,et al.  Travelling waves for delayed reaction–diffusion equations with global response , 2005, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[12]  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.

[13]  Chunhua Ou,et al.  Persistence of wavefronts in delayed nonlocal reaction-diffusion equations , 2007 .

[14]  Chunhua Ou,et al.  Traveling Wavefronts in a Delayed Food-limited Population Model , 2007, SIAM J. Math. Anal..

[15]  Klaus W. Schaaf.  Asymptotic behavior and traveling wave solutions for parabolic functional-differential equations , 1987 .