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Travelling wave and convergence in stage-structured reaction–diffusion competitive models with nonlocal delays

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  • Xu, Rui
  • Chaplain, M.A.J.
  • Davidson, F.A.

Abstract

In this paper, we first investigate a stage-structured competitive model with time delays, harvesting, and nonlocal spatial effect. By using an iterative technique recently developed by Wu and Zou (Wu J, Zou X. Travelling wave fronts of reaction–diffusion systems with delay. J Dynam Differen Equat 2001;13:651–87), sufficient conditions are established for the existence of travelling front solution connecting the two boundary equilibria in the case when there is no positive equilibrium. The travelling wave front corresponds to an invasion by a stronger species which drives the weaker species to extinction. Secondly, we consider a stage-structured competitive model with time delays and nonlocal spatial effect when the domain is finite. We prove the global stability of each of the nonnegative equilibria and demonstrate that the more complex model studied here admits three possible long term behaviors: coexistence, bistability and dominance as is the case for the standard Lotka–Voltera competitive model.

Suggested Citation

  • Xu, Rui & Chaplain, M.A.J. & Davidson, F.A., 2006. "Travelling wave and convergence in stage-structured reaction–diffusion competitive models with nonlocal delays," Chaos, Solitons & Fractals, Elsevier, vol. 30(4), pages 974-992.
    • Handle: RePEc:eee:chsofr:v:30:y:2006:i:4:p:974-992
    DOI: 10.1016/j.chaos.2005.09.022
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    Cited by:

    1. Li, Wan-Tong & Wu, Shi-Liang, 2008. "Traveling waves in a diffusive predator–prey model with holling type-III functional response," Chaos, Solitons & Fractals, Elsevier, vol. 37(2), pages 476-486.
    2. Upadhyay, Ranjit Kumar & Kumari, Nitu & Rai, Vikas, 2009. "Exploring dynamical complexity in diffusion driven predator–prey systems: Effect of toxin producing phytoplankton and spatial heterogeneities," Chaos, Solitons & Fractals, Elsevier, vol. 42(1), pages 584-594.
    3. Wu, Shi-Liang & Li, Wan-Tong, 2009. "Global asymptotic stability of bistable traveling fronts in reaction-diffusion systems and their applications to biological models," Chaos, Solitons & Fractals, Elsevier, vol. 40(3), pages 1229-1239.
    4. Jin, Chunhua & Yin, Jingxue & Wang, Yifu, 2009. "Travelling wavefronts of a generalized Fisher equation with spatio-temporal delay," Chaos, Solitons & Fractals, Elsevier, vol. 39(2), pages 599-609.

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