scholarly journals Travelling wave and global attractivity in a competition–diffusion system with nonlocal delays

2010 ◽  
Vol 59 (10) ◽  
pp. 3338-3350 ◽  
Author(s):  
Xiping Yang ◽  
Yifu Wang
Keyword(s):  
Global Attractivity ◽  
Travelling Wave ◽  
Diffusion System ◽  
Competition Diffusion

Dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity

2010 ◽  
Vol 140 (5) ◽  
pp. 1081-1109 ◽  
Author(s):  
Zhi-Cheng Wang ◽  
Wan-Tong Li
Keyword(s):  
Diffusion Equation ◽  
Global Attractivity ◽  
Reaction Diffusion ◽  
Single Species ◽  
Travelling Wave ◽  
Reaction Diffusion Equation ◽  
Positive Equilibrium ◽  
Cauchy Type ◽  
Delayed Reaction ◽  
Non Local

AbstractThis paper is concerned with the dynamics of a non-local delayed reaction–diffusion equation without quasi-monotonicity on an infinite n-dimensional domain, which can be derived from the growth of a stage-structured single-species population. We first prove that solutions of the Cauchy-type problem are positively preserving and bounded if the initial value is non-negative and bounded. Then, by establishing a comparison theorem and a series of comparison arguments, we prove the global attractivity of the positive equilibrium. When there exist no positive equilibria, we prove that the zero equilibrium is globally attractive. In particular, these results are still valid for the non-local delayed reaction–diffusion equation on a bounded domain with the Neumann boundary condition. Finally, we establish the existence of new entire solutions by using the travelling-wave solutions of two auxiliary equations and the global attractivity of the positive equilibrium.

A competition-diffusion system approximation to the classical two-phase Stefan problem

2001 ◽  
Vol 18 (2) ◽  
pp. 161-180 ◽  
Author(s):  
Danielle Hilhorst ◽  
Masato Iida ◽  
Masayasu Mimura ◽  
Hirokazu Ninomiya
Keyword(s):  
Stefan Problem ◽  
Diffusion System ◽  
Two Phase ◽  
Competition Diffusion
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