Spatiotemporal dynamics in a delayed diffusive predator model

In this paper, we investigate the spatiotemporal dynamics of a delayed reaction-diffusion Leslie-Gower model. Based on the stability analysis, we demonstrate that delayed feedback may generate Hopf and Turing instability under some conditions, resulting in spatial patterns. One of the most interesting findings is that the model exhibits complex pattern replication: Pure Turing instability gives birth to spots, spots-stripes-mixture, stripes, stripes-holes-mixture and holes patterns, pure Hopf instability to spiral wave pattern, and Hopf-Turing instability to chaotic wave pattern. Our results well extend the findings of spatiotemporal dynamics in the delayed reaction-diffusion predator-prey model, and indicate that time delay play an important roles in pattern formation.

[1]  Monika Joanna Piotrowska.  Activator-inhibitor system with delay and pattern formation , 2005, Math. Comput. Model..

[2]  Ranjit Kumar Upadhyay,et al.  Introduction to Mathematical Modeling and Chaotic Dynamics , 2013 .

[3]  M. A. Aziz-Alaoui,et al.  Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with time delay , 2006 .

[4]  Malay Banerjee,et al.  Bifurcation analysis of a ratio-dependent prey-predator model with the Allee effect , 2012 .

[5]  Wenjie Zuo,et al.  Stability and Hopf bifurcation in a diffusive predator–prey system with delay effect , 2011 .

[6]  Shigui Ruan,et al.  On Nonlinear Dynamics of Predator-Prey Models with Discrete Delay ⁄ , 2009 .

[7]  Biao Wang,et al.  Stationary patterns of a predator-prey model with spatial effect , 2010, Appl. Math. Comput..

[8]  Fordyce A. Davidson,et al.  Periodic solutions for a predator-prey model with Holling-type functional response and time delays , 2005, Appl. Math. Comput..

[9]  Y. Kuang.  Delay Differential Equations: With Applications in Population Dynamics , 2012 .

[10]  Ranjit Kumar Upadhyay,et al.  Spatiotemporal Dynamics in a Spatial Plankton System , 2011, 1103.3344.

[11]  Deb Shankar Ray,et al.  Time-delay-induced instabilities in reaction-diffusion systems. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  R. Macarthur.  The Problem of Pattern and Scale in Ecology: The Robert H. MacArthur Award Lecture , 2005 .

[13]  Teresa Faria,et al.  Stability and Bifurcation for a Delayed Predator–Prey Model and the Effect of Diffusion☆ , 2001 .

[14]  Fengde Chen,et al.  On a Leslie―Gower predator―prey model incorporating a prey refuge , 2009 .

[15]  Rui Dilão.  Turing instabilities and patterns near a Hopf bifurcation , 2005, Appl. Math. Comput..

[16]  Canrong Tian.  Delay-driven spatial patterns in a plankton allelopathic system. , 2012, Chaos.

[17]  Lei Zhang,et al.  Pattern formation of a predator-prey system with Ivlev-type functional response , 2008, 0801.0796.

[18]  Bambi Hu,et al.  Pattern formation controlled by time-delayed feedback in bistable media. , 2010, The Journal of chemical physics.

[19]  B. Kendall.  Nonlinear Dynamics and Chaos , 2001 .

[20]  Z. Mei.  Numerical Bifurcation Analysis for Reaction-Diffusion Equations , 2000 .

[21]  Yongkun Li,et al.  Stability and Hopf bifurcation analysis on a delayed Leslie-Gower predator-prey system incorporating a prey refuge , 2013, Appl. Math. Comput..

[22]  Daqing Jiang,et al.  Analysis of a predator-prey model with modified Leslie-Gower and Holling-type II schemes with stochastic perturbation , 2009 .

[23]  J. Gower,et al.  The properties of a stochastic model for the predator-prey type of interaction between two species , 1960 .

[24]  Shanshan Chen,et al.  Global stability and Hopf bifurcation in a delayed diffusive Leslie-Gower predator-prey System , 2012, Int. J. Bifurc. Chaos.

[25]  Dongwoo Sheen,et al.  Turing instability for a ratio-dependent predator-prey model with diffusion , 2009, Appl. Math. Comput..

[26]  Baba Issa Camara,et al.  Complexité de dynamiques de modèles proie-prédateur avec diffusion et applications. (Dynamics omplexity of diffusive predator-prey models and applications) , 2009 .

[27]  Peter V. E. McClintock,et al.  Pattern formation : an introduction to methods. , 2006 .

[28]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[29]  Yongli Cai,et al.  Dynamical Analysis of a Delayed Reaction-Diffusion Predator-Prey System , 2012 .

[30]  S. Levin.  The problem of pattern and scale in ecology , 1992 .

[31]  Wan-Tong Li,et al.  Periodic solutions of delayed Leslie-Gower predator-prey models , 2004, Appl. Math. Comput..

[32]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[33]  Juan Zhang,et al.  Pattern formation of a spatial predator-prey system , 2012, Appl. Math. Comput..

[34]  S. Levin,et al.  Diffusion and Ecological Problems: Modern Perspectives , 2013 .

[35]  Soovoojeet Jana,et al.  Stability and bifurcation analysis of a stage structured predator prey model with time delay , 2012, Appl. Math. Comput..

[36]  Ling Wang,et al.  Hopf bifurcation and Turing instability of 2-D Lengyel-Epstein system with reaction-diffusion terms , 2013, Appl. Math. Comput..

[37]  M. A. Aziz-Alaoui,et al.  Boundedness and global stability for a predator-prey model with modified Leslie-Gower and Holling-type II schemes , 2003, Appl. Math. Lett..

[38]  Marcus R. Garvie.  Finite-Difference Schemes for Reaction–Diffusion Equations Modeling Predator–Prey Interactions in MATLAB , 2007, Bulletin of mathematical biology.

[39]  Hamad Talibi Alaoui,et al.  Limit cycle and numerical similations for small and large delays in a predator-prey model with modified Leslie-Gower and Holling-type II schemes , 2008 .

[40]  Sze-Bi Hsu,et al.  Global Stability for a Class of Predator-Prey Systems , 1995, SIAM J. Appl. Math..

[41]  Eduardo Sáez,et al.  Three Limit Cycles in a Leslie--Gower Predator-Prey Model with Additive Allee Effect , 2009, SIAM J. Appl. Math..

[42]  Sahabuddin Sarwardi,et al.  A Leslie-Gower Holling-type II ecoepidemic model , 2011 .

[43]  Peixuan Weng,et al.  Stability Analysis of Diffusive Predator-Prey Model with Modified Leslie-Gower and Holling-Type II Schemes , 2011, Appl. Math. Comput..

[44]  C. Cosner,et al.  Spatial Ecology via Reaction-Diffusion Equations , 2003 .

[45]  P. H. Leslie.  SOME FURTHER NOTES ON THE USE OF MATRICES IN POPULATION MATHEMATICS , 1948 .

[46]  Ilkka Hanski,et al.  Specialist predators, generalist predators, and the microtine rodent cycle. , 1991 .

[47]  Eduardo Sáez,et al.  Two limit cycles in a Leslie-Gower predator-prey model with additive Allee effect , 2009 .

[48]  Wan-Tong Li,et al.  Hopf bifurcation analysis for a delayed predator–prey system with diffusion effects , 2010 .

[49]  Pushpita Ghosh,et al.  Control of the Hopf-Turing transition by time-delayed global feedback in a reaction-diffusion system. , 2011, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Sanling Yuan,et al.  RETRACTED: Bifurcation analysis in the delayed Leslie–Gower predator–prey system , 2009 .

[51]  P. Cochat,et al.  Et al , 2008, Archives de pediatrie : organe officiel de la Societe francaise de pediatrie.