Analysis of bifurcation, chaos and pattern formation in a discrete time and space Gierer Meinhardt system

Abstract This paper is concerned with the spatiotemporal behaviors of a Gierer–Meinhardt system in discrete time and space form. Through the linear stability analysis, the parametric conditions are gained to ensure the stability of the homogeneous steady state of the system. Based on the bifurcation theory, as well as center manifold theorem, we derive the critical parameter values of the flip, Neimark–Sacker and Turing bifurcation respectively. Besides, the specific parameter expression to form patterns are also determined. In order to identify chaos among regular behaviors, we calculate the Maximum Lyapunov exponents. The results obtained in this paper are illustrated by numerical simulations. From the simulations, we can see some complex dynamics, such as period doubling cascade, invariant cycles, periodic windows, chaotic behaviors, and some striking Turing patterns, e.g. circle, mosaic, spiral, spatiotemporal chaotic patterns and so on, which can be produced by flip-Turing instability, Neimark–Sacker–Turing instability and chaos.

[1]  S. Petrovskii,et al.  Pattern formation in a space- and time-discrete predator–prey system with a strong Allee effect , 2012, Theoretical Ecology.

[2]  E A Gaffney,et al.  The Influence of Gene Expression Time Delays on Gierer–Meinhardt Pattern Formation Systems , 2010, Bulletin of mathematical biology.

[3]  Wei-Ming Ni,et al.  DIFFUSION, CROSS-DIFFUSION, AND THEIR SPIKE-LAYER STEADY STATES , 1998 .

[4]  Hans Meinhardt,et al.  The Algorithmic Beauty of Sea Shells , 2003, The Virtual Laboratory.

[5]  A. Atangana,et al.  Numerical Simulation of Noninteger Order System in Subdiffusive, Diffusive, and Superdiffusive Scenarios , 2017 .

[6]  Liang Bai,et al.  Nontrivial solutions for a nonlinear discrete elliptic equation with periodic boundary conditions , 2009, Appl. Math. Comput..

[7]  K. Maginu.  Stability of spatially homogeneous periodic solutions of reaction-diffusion equations , 1979 .

[8]  Shigeru Kondo,et al.  Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation , 2010, Science.

[9]  Matthias Winter,et al.  On the two-dimensional Gierer-Meinhardt system with strong coupling , 1999 .

[10]  Matjaz Perc,et al.  Collective behavior and evolutionary games - An introduction , 2013, 1306.2296.

[11]  A. Turing.  The chemical basis of morphogenesis , 1952, Philosophical Transactions of the Royal Society of London. Series B, Biological Sciences.

[12]  Zhujun Jing,et al.  Bifurcation and chaos in discrete-time predator–prey system , 2006 .

[13]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[14]  Xin Li,et al.  Hopf bifurcation and Turing instability in the reaction–diffusion Holling–Tanner predator–prey model , 2013 .

[15]  Jinliang Wang,et al.  Stripe and Spot Patterns for General Gierer-Meinhardt Model with Common Sources , 2017, Int. J. Bifurc. Chaos.

[16]  M. Sokabe,et al.  The application of the Gierer-Meinhardt equations to the development of the retinotectal projection , 1979, Biological Cybernetics.

[17]  M. A. Aziz-Alaoui,et al.  Diffusion driven instability and Hopf bifurcation in spatial predator-prey model on a circular domain , 2015, Appl. Math. Comput..

[18]  Zhen Wang,et al.  Rich dynamics in a spatial predator-prey model with delay , 2015, Appl. Math. Comput..

[19]  Guang Zhang,et al.  Turing instability and wave patterns for a symmetric discrete competitive Lotka-Volterra system , 2011 .

[20]  S. Fortunato,et al.  Statistical physics of social dynamics , 2007, 0710.3256.

[21]  H. Meinhardt,et al.  A theory of biological pattern formation , 1972, Kybernetik.

[22]  Gábor Domokos,et al.  Discrete and continuous state population models in a noisy world. , 2004, Journal of theoretical biology.

[24]  Dongmei Xiao,et al.  Complex dynamic behaviors of a discrete-time predator–prey system , 2007 .

[25]  H. Meinhardt.  Models of biological pattern formation , 1982 .

[26]  Binxiang Dai,et al.  Cross-Diffusion Induced Turing Instability and Amplitude Equation for a Toxic-Phytoplankton-Zooplankton Model with Nonmonotonic Functional Response , 2017, Int. J. Bifurc. Chaos.

[27]  Jinliang Wang,et al.  Stripe and Spot Patterns in a Gierer-Meinhardt Activator-Inhibitor Model with Different Sources , 2015, Int. J. Bifurc. Chaos.

[28]  S. Petrovskii,et al.  Spatiotemporal complexity of biological invasion in a space- and time-discrete predator–prey system with the strong Allee effect , 2012 .

[29]  Huayong Zhang,et al.  Bifurcation, chaos and pattern formation in a space- and time-discrete predator-prey system , 2016 .

[30]  Junjie Wei,et al.  Multiple bifurcation Analysis and Spatiotemporal Patterns in a 1-d Gierer-Meinhardt Model of Morphogenesis , 2010, Int. J. Bifurc. Chaos.

[31]  Robert M. May,et al.  Simple mathematical models with very complicated dynamics , 1976, Nature.

[32]  K. Maginu.  Stability of periodic travelling wave solutions with large spatial periods in reaction-diffusion systems , 1981 .

[33]  A. Nayfeh,et al.  Applied nonlinear dynamics : analytical, computational, and experimental methods , 1995 .

[34]  Shigui Ruan,et al.  DIFFUSION‐DRIVEN INSTABILITY IN THE GIERER‐MEINHARDT MODEL OF MORPHOGENESIS , 1998 .

[35]  Guang Zhang,et al.  Turing instability for a semi-discrete Gierer–Meinhardt system , 2012 .

[36]  Bifurcation and Turing patterns of reaction–diffusion activator–inhibitor model , 2017 .

[37]  Yongli Cai,et al.  Spatiotemporal Complexity of a Leslie-Gower Predator-Prey Model with the Weak Allee Effect , 2013, J. Appl. Math..

[38]  H. Meinhardt,et al.  Biological pattern formation: fmm basic mechanisms ta complex structures , 1994 .

[39]  Canrong Tian,et al.  Turing pattern dynamics in an activator-inhibitor system with superdiffusion. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[40]  Junjie Wei,et al.  Diffusion-driven instability and bifurcation in the Lengyel-Epstein system , 2008 .

[41]  You Li,et al.  Stripe and spot patterns for the Gierer–Meinhardt model with saturated activator production , 2017 .

[42]  Robert I. McKay,et al.  Spatio-temporal dynamics and quantification of daisyworld in two-dimensional coupled map lattices , 2012 .

[43]  Michael J. Ward,et al.  The stability of spike solutions to the one-dimensional Gierer—Meinhardt model , 2001 .

[44]  Juncheng Wei,et al.  Hopf Bifurcations and Oscillatory Instabilities of Spike Solutions for the One-Dimensional Gierer-Meinhardt Model , 2003, J. Nonlinear Sci..