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Traveling waves in a diffusive predator–prey model with holling type-III functional response

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  • Li, Wan-Tong
  • Wu, Shi-Liang

Abstract

We establish the existence of traveling wave solutions and small amplitude traveling wave train solutions for a reaction-diffusion system based on a predator–prey model with Holling type-III functional response. The analysis is in the three-dimensional phase space of the nonlinear ordinary differential equation system given by the diffusive predator–prey system in the traveling wave variable. The methods used to prove the results are the shooting argument, invariant manifold theory and the Hopf bifurcation theorem.

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  • 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.
    • Handle: RePEc:eee:chsofr:v:37:y:2008:i:2:p:476-486
    DOI: 10.1016/j.chaos.2006.09.039
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    1. 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.
    2. Jiang, Guirong & Lu, Qishao & Qian, Linning, 2007. "Complex dynamics of a Holling type II prey–predator system with state feedback control," Chaos, Solitons & Fractals, Elsevier, vol. 31(2), pages 448-461.
    3. Lin-Lin Wang & Wan-Tong Li, 2004. "Periodic solutions and stability for a delayed discrete ratio-dependent predator-prey system with Holling-type functional response," Discrete Dynamics in Nature and Society, Hindawi, vol. 2004, pages 1-19, January.
    4. Zhang, Shuwen & Tan, Dejun & Chen, Lansun, 2006. "Chaotic behavior of a chemostat model with Beddington–DeAngelis functional response and periodically impulsive invasion," Chaos, Solitons & Fractals, Elsevier, vol. 29(2), pages 474-482.
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    Cited by:

    1. Wu, Chufen & Yang, Yong & Weng, Peixuan, 2013. "Traveling waves in a diffusive predator–prey system of Holling type: Point-to-point and point-to-periodic heteroclinic orbits," Chaos, Solitons & Fractals, Elsevier, vol. 48(C), pages 43-53.
    2. Hu, Guang-Ping & Li, Wan-Tong & Yan, Xiang-Ping, 2009. "Hopf bifurcations in a predator–prey system with multiple delays," Chaos, Solitons & Fractals, Elsevier, vol. 42(2), pages 1273-1285.
    3. Wu, Chufen & Yang, Yong & Zhao, Qianyi & Tian, Yanling & Xu, Zhiting, 2017. "Epidemic waves of a spatial SIR model in combination with random dispersal and non-local dispersal," Applied Mathematics and Computation, Elsevier, vol. 313(C), pages 122-143.
    4. 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.
    5. 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.

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