Stability analysis and finite volume element discretization for delay-driven spatio-temporal patterns in a predator-prey model

[1]  Yang Kuang,et al.  Global existence of periodic solutions in a class of delayed Gause-type predator-prey systems , 1997 .

[2]  Ricardo Ruiz-Baier,et al.  Turing pattern dynamics and adaptive discretization for a super-diffusive Lotka-Volterra model , 2015, Journal of Mathematical Biology.

[3]  B. Hassard,et al.  Theory and applications of Hopf bifurcation , 1981 .

[4]  S. Petrovskii,et al.  Pattern Formation, Long-Term Transients, and the Turing–Hopf Bifurcation in a Space- and Time-Discrete Predator–Prey System , 2011, Bulletin of mathematical biology.

[5]  Ricardo Ruiz-Baier,et al.  Analysis of a finite volume method for a cross-diffusion model in population dynamics , 2011 .

[6]  Wendi Wang.  Epidemic Models with Time Delays , 2009 .

[7]  J. Murray.  Spatial structures in predator-prey communities— a nonlinear time delay diffusional model , 1976 .

[8]  Stefan Vandewalle,et al.  Unconditionally stable difference methods for delay partial differential equations , 2012, Numerische Mathematik.

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

[10]  S. Ruan,et al.  On the zeros of transcendental functions with applications to stability of delay differential equations with two delays , 2003 .

[11]  Mauricio Sepúlveda,et al.  Mathematical and numerical analysis for Predator-prey system in a polluted environment , 2010, Networks Heterog. Media.

[12]  R. D. Driver.  Introduction to Delay Differential Equations , 1977 .

[13]  Hal L. Smith,et al.  An introduction to delay differential equations with applications to the life sciences / Hal Smith , 2010 .

[14]  Mark A. McKibben.  Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations , 2010 .

[15]  Zhen Jin,et al.  Spatial patterns of a predator-prey model with cross diffusion , 2012, Nonlinear Dynamics.

[16]  Ricardo Ruiz-Baier,et al.  Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion , 2014, J. Comput. Phys..

[17]  Wan-Tong Li,et al.  Hopf bifurcation and Turing instability in spatial homogeneous and inhomogeneous predator-prey models , 2011, Appl. Math. Comput..

[18]  Kunal Chakraborty,et al.  Global stability and bifurcation of time delayed prey-predator system incorporating prey refuge , 2012, Math. Comput. Simul..

[19]  Yuan-Ming Wang,et al.  Time-Delayed finite difference reaction-diffusion systems with nonquasimonotone functions , 2006, Numerische Mathematik.

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

[21]  Chia-Ven Pao,et al.  Nonlinear parabolic and elliptic equations , 1993 .

[22]  P. Dechaumphai,et al.  Finite volume element method for analysis of unsteady reaction–diffusion problems , 2009 .

[23]  A. Bellen,et al.  Numerical methods for delay differential equations , 2003 .

[24]  Karline Soetaert,et al.  Solving Ordinary Differential Equations in R , 2012 .

[25]  S. Petrovskii,et al.  Spatiotemporal patterns in ecology and epidemiology : theory, models, and simulation , 2007 .

[26]  H. I. Freedman,et al.  The trade-off between mutual interference and time lags in predator-prey systems , 1983 .

[27]  Vito Volterra,et al.  Leçons sur la théorie mathématique de la lutte pour la vie , 1931 .

[28]  Chengming Huang,et al.  Delay-dependent stability of high order Runge–Kutta methods , 2009, Numerische Mathematik.

[29]  Horst Malchow,et al.  Spatiotemporal Complexity of Plankton and Fish Dynamics , 2002, SIAM Rev..

[30]  Alexandre Ern,et al.  Implicit-Explicit Runge-Kutta schemes and finite elements with symmetric stabilization for advection-diffusion equations , 2012 .

[31]  Marcus R. Garvie,et al.  A three level finite element approximation of a pattern formation model in developmental biology , 2014, Numerische Mathematik.

[32]  So-Hsiang Chou,et al.  Analysis and convergence of a covolume method for the generalized Stokes problem , 1997, Math. Comput..

[33]  Zhiqiang Cai,et al.  On the finite volume element method , 1990 .

[34]  Toshiyuki Koto,et al.  Stability of IMEX Runge-Kutta methods for delay differential equations , 2008 .

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

[36]  Gergana Bencheva,et al.  Comparative Analysis of Solution Methods for Delay Differential Equations in Hematology , 2009, LSSC.

[37]  Peter J. Wangersky,et al.  Time Lag in Prey‐Predator Population Models , 1957 .

[38]  Alfio Quarteroni,et al.  Analysis of a finite volume element method for the Stokes problem , 2011, Numerische Mathematik.

[39]  R. Ruiz-Baier,et al.  Turing pattern dynamics and adaptive discretization for a superdiffusive Lotka-Volterra system , 2018 .

[40]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[41]  J. Cushing,et al.  On the behavior of solutions of predator-prey equations with hereditary terms , 1975 .

[42]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[43]  Paul Nicholas,et al.  Pattern in(formation) , 2012 .

[44]  R. May,et al.  Stability and Complexity in Model Ecosystems , 1976, IEEE Transactions on Systems, Man, and Cybernetics.

[45]  Zhangxin Chen,et al.  A stabilized multi-level method for non-singular finite volume solutions of the stationary 3D Navier–Stokes equations , 2012, Numerische Mathematik.

[46]  Mehdi Dehghan,et al.  The finite volume spectral element method to solve Turing models in the biological pattern formation , 2011, Comput. Math. Appl..

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

[48]  Jianhong Wu,et al.  Nonlocality of Reaction-Diffusion Equations Induced by Delay: Biological Modeling and Nonlinear Dynamics , 2004 .

[49]  F. Brauer,et al.  Mathematical Models in Population Biology and Epidemiology , 2001 .

[50]  J. Hale,et al.  Dynamics and Bifurcations , 1991 .

[51]  Henry S. Greenside,et al.  Pattern Formation and Dynamics in Nonequilibrium Systems , 2004 .

[52]  Aiguo Xiao,et al.  Implicit-explicit time discretization coupled with finite element methods for delayed predator-prey competition reaction-diffusion system , 2016, Comput. Math. Appl..

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

[54]  Raimund Bürger,et al.  Discontinuous finite volume element discretization for coupled flow-transport problems arising in models of sedimentation , 2015, J. Comput. Phys..

[55]  K. Gopalsamy.  Pursuit-evasion wave trains in prey-predator systems with diffusionally coupled delays , 1980 .

[56]  K. Teo,et al.  Existence and uniqueness of weak solutions of the Cauchy problem for parabolic delay-differential equations , 1980, Bulletin of the Australian Mathematical Society.

[57]  A. Quarteroni,et al.  Finite element and finite volume‐element simulation of pseudo‐ECGs and cardiac alternans , 2015 .

[58]  R. Lazarov,et al.  Finite volume element approximations of nonlocal reactive flows in porous media , 2000 .

[59]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[60]  Wendi Wang,et al.  A predator-prey system with stage-structure for predator , 1997 .

[61]  Jianhong Wu,et al.  Spatiotemporal Patterns of Disease Spread: Interaction of Physiological Structure, Spatial Movements, Disease Progression and Human Intervention , 2008 .

[62]  Raimund Bürger,et al.  A Stabilized Finite Volume Element Formulation for Sedimentation-Consolidation Processes , 2012, SIAM J. Sci. Comput..