Traveling wavefronts of a diffusive competing pioneer and climax system with delays

Abstract In this paper, we study a delayed diffusion–reaction system for competing pioneer and climax species. We establish the existence criterion of traveling wavefronts connecting the pioneer-invasion-only equilibrium and the co-invasion equilibrium. A transformation of variables makes the monotone iteration scheme and the Schauder fixed point theorem is applicable. By constructing a pair of admissible upper and lower solutions, we show that the system does support such co-invasion wavefronts for c ≥ c ∗ , and c ∗ is the minimal wave speed, where c ∗ > 0 can be decided by a single characteristic equation which is a quadratic algebraic equation. We also discuss the asymptotic behavior of the wave tail for the traveling wavefronts.

[1]  Zhixian Yu,et al.  Traveling wave solutions in temporally discrete reaction‐diffusion systems with delays , 2011 .

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

[3]  M. Hassell,et al.  Discrete time models for two-species competition. , 1976, Theoretical population biology.

[4]  Kai Zhou,et al.  Traveling wave solutions in delayed reaction-diffusion systems with mixed monotonicity , 2010, J. Comput. Appl. Math..

[5]  Co-invasion waves in a reaction diffusion model for competing pioneer and climax species , 2010 .

[6]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[7]  Rui Xu,et al.  Travelling waves of a delayed SIRS epidemic model with spatial diffusion , 2011 .

[8]  N. Shigesada,et al.  Biological Invasions: Theory and Practice , 1997 .

[9]  Peixuan Weng,et al.  Wavefronts for a global reaction–diffusion population model with infinite distributed delay☆ , 2008 .

[10]  STABLE PERIODIC BEHAVIOR IN PIONEER‐CLIMAX COMPETING SPECIES MODELS WITH CONSTANT RATE FORCING , 1998 .

[11]  Xingfu Zou,et al.  Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity , 2003 .

[12]  G. Lin,et al.  Traveling waves in delayed lattice dynamical systems with competition interactions , 2010 .

[13]  Van Vuuren,et al.  The existence of travelling plane waves in a general class of competition-diffusion systems , 1995 .

[14]  Hopf bifurcation in pioneer-climax competing species models. , 1996, Mathematical biosciences.

[15]  B. Buonomo,et al.  Linear and nonlinear stability thresholds for a diffusive model of pioneer and climax species interaction , 2009 .

[16]  J. Dockery,et al.  Traveling wave solutions of a reaction diffusion model for competing pioneer and climax species. , 2005, Mathematical biosciences.

[17]  Shiwang Ma,et al.  Traveling Wavefronts for Delayed Reaction-Diffusion Systems via a Fixed Point Theorem , 2001 .

[18]  Jim M Cushing,et al.  NONLINEAR MATRIX MODELS AND POPULATION DYNAMICS , 1988 .

[19]  Wan-Tong Li,et al.  Existence of travelling wave solutions in delayed reaction–diffusion systems with applications to diffusion–competition systems , 2006 .

[20]  J. Roberds,et al.  Lumped-density population models of pioneer-climax type and stability analysis of Hopf bifurcations. , 1996, Mathematical biosciences.