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Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay

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DOI: 10.4236/am.2014.56080    3,573 Downloads   4,867 Views   Citations

ABSTRACT

This paper is concerned with the existence of traveling wave solutions in a reaction-diffusion predator-prey system with Beddington-DeAngelis functional response and a discrete time delay. By introducing a partial quasi-monotonicity condition and constructing a pair of upper-lower solutions, we establish the existence of traveling wave solutions. Moreover, a numerical simulation is carried out to illustrate the theoretical results.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Du, P. , Duan, C. and Liao, X. (2014) Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay. Applied Mathematics, 5, 843-851. doi: 10.4236/am.2014.56080.

References

[1] Nindjina, A.F., Aziz-Alaouib, M.A. and Cadivelb, M. (2006) Analysis of a Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes with Time Delay. Nonlinear Analysis: Real World Applications, 7, 1104-1118. http://dx.doi.org/10.1016/j.nonrwa.2005.10.003
[2] Lu, Z. and Liu, X. (2008) Analysis of a Predator-Prey Model with Modified Holling-Tanner Functional Response and Time Delay. Nonlinear Analysis: Real World Applications, 9, 641-650.
http://dx.doi.org/10.1016/j.nonrwa.2006.12.016
[3] Holling, C.S. (1965) The Functional Response of Predator to Prey Density and Its Role in Mimicry and Population Regulation. Entomological Society of Canada, 45, 1-60.
http://dx.doi.org/10.4039/entm9745fv
[4] Braza, P.A. (2003) The Bifurcations Structure for the Holling-Tanner Model for Predator-Prey Interactions Using Two-Timing. SIAM Journal on Applied Mathematics, 63, 889-904.
http://dx.doi.org/10.1137/S0036139901393494
[5] Haquea, M. and Venturino, E. (2006) The Role of Transmissible Diseases in the Holling-Tanner Predator-Prey Model. Theoretical Population Biology, 70, 273-288.
http://dx.doi.org/10.1016/j.tpb.2006.06.007
[6] Saha, T. and Chakrabarti, C. (2009) Dynamical Analysis of a Delayed Ratio-Dependent Holling-Tanner Predator-Prey Model. Journal of Mathematical Analysis and Applications, 358, 389-402.
http://dx.doi.org/10.1016/j.jmaa.2009.03.072
[7] Peng, R. and Wang, M. (2007) Global Stability of the Equilibrium of a Diffusive Holling-Tanner Prey-Predator Model. Applied Mathematics Letters, 20, 664-670.
http://dx.doi.org/10.1016/j.aml.2006.08.020
[8] Yan, X. and Zhang, C. (2010) Asymptotic Stability of Positive Equilibrium Solution for a Delayed Prey-Predator Diffusion System. Applied Mathematical Modelling, 34, 184-199.
http://dx.doi.org/10.1016/j.apm.2009.03.040
[9] Peng, R. and Shi, J. (2009) Non-Existence of Non-Constant Positive Steady States of Two Holling Type-II Predator-Prey Systems: Strong Interaction Case. Journal of Differential Equations, 247, 866-886.
http://dx.doi.org/10.1016/j.jde.2009.03.008
[10] Ma, S. (2001) Traveling Wavefronts for Delayed Reaction—Diffusion Systems via a Fixed Point Theorem. Journal of Differential Equations, 171, 294-314. http://dx.doi.org/10.1006/jdeq.2000.3846
[11] Ge, Z. and He, Y. (2009) Traveling Wavefronts for a Two-Species Predator-Prey System with Diffusion Terms and Stage Structure. Applied Mathematical Modelling, 33, 1356-1365.
http://dx.doi.org/10.1016/j.apm.2007.09.037
[12] Wu, J. and Zou, X. (2001) Traveling Wave Fronts of Reaction—Diffusion Systems with Delay. Journal of Dynamics and Differential Equations, 13, 651-687.
[13] Zou, X. (2002) Delay Induced Traveling Wave Fronts in Reaction Diffusion Equations of KPP-Fisher Type. Journal of Computational and Applied Mathematics, 146, 309-321.
http://dx.doi.org/10.1016/S0377-0427(02)00363-1
[14] Canosa, J. (1973) On a Nonlinear Diffusion Equation Describing Population Growth. Journal of Research and Development, 17, 307-313. http://dx.doi.org/10.1147/rd.174.0307

  
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