Research ArticleOpen AccessGoogle Scholar indexed
Dynamics Behaviors of a Reaction-Diffusion Predator-Prey System with Beddington-DeAngelis Functional Response and Delay
School of Mathematics and Physics, University of South China, Hengyang, China
School of Mathematics and Physics, University of South China, Hengyang, China
School of Mathematics and Physics, University of South China, Hengyang, China
- 1 School of Mathematics and Physics, University of South China, Hengyang, China
- 2 School of Mathematics and Physics, University of South China, Hengyang, China
- 3 School of Mathematics and Physics, University of South China, Hengyang, China
Copy link · social · email
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.
KeywordsTraveling Wave SolutionsReaction-Diffusion SystemUpper-Lower SolutionsLocal Stability
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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
- 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