2016, 13(4): 857-885. doi: 10.3934/mbe.2016021

Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack

1. 

School of Mathematics and Statistics, Southwest University, Chongqing, 400715

Received  August 2015 Revised  February 2016 Published  May 2016

This paper deals with the spatial, temporal and spatiotemporal dynamics of a spatial plant-wrack model. The parameter regions for the stability and instability of the unique positive constant steady state solution are derived, and the existence of time-periodic orbits and non-constant steady state solutions are proved by bifurcation method. The nonexistence of positive nonconstant steady state solutions are studied by energy method and Implicit Function Theorem. Numerical simulations are presented to verify and illustrate the theoretical results.
Citation: Jun Zhou. Bifurcation analysis of a diffusive plant-wrack model with tide effect on the wrack. Mathematical Biosciences & Engineering, 2016, 13 (4) : 857-885. doi: 10.3934/mbe.2016021
References:
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[26]

W. M. Ni and M. X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction,, Trans. Amer. Math. Soc., 357 (2005), 3953.  doi: 10.1090/S0002-9947-05-04010-9.  Google Scholar

[27]

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[28]

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[29]

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[30]

R. Peng and M. X. Wang, Some nonexistence results for nonconstant stationary solutions to the Gray-Scott model in a bounded domain,, Appl. Math. Lett., 22 (2009), 569.  doi: 10.1016/j.aml.2008.06.032.  Google Scholar

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[44]

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L. Xu, G. Zhang and J. F. Ren, Turing instability for a two dimensional semi-discrete oregonator model,, WSEAS Transac. Math, 10 (2011), 201.   Google Scholar

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C. Xu and J. J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model,, Nonlinear Anal. Real World Appl., 13 (2012), 1961.  doi: 10.1016/j.nonrwa.2012.01.001.  Google Scholar

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F. Q. Yi, J. J. Wei and J. J. Shi, Diffusion-driven instability and bifurcation in the lengyel-epstein system,, Nonlinear Anal.: Real World Applications, 9 (2008), 1038.  doi: 10.1016/j.nonrwa.2007.02.005.  Google Scholar

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Y. C. You, Dynamics of two-compartment Gray-Scott equations,, Nonlinear Anal., 74 (2011), 1969.  doi: 10.1016/j.na.2010.11.004.  Google Scholar

[56]

Y. C. You, Global dynamics of the Oregonator system,, Math. Methods Appl. Sci., 35 (2012), 398.  doi: 10.1002/mma.1591.  Google Scholar

[57]

Y. C. You, Robustness of Global Attractors for Reversible Gray-Scott Systems,, J. Dynam. Differential Equations, 24 (2012), 495.  doi: 10.1007/s10884-012-9252-7.  Google Scholar

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J. Zhou and C. L. Mu, Pattern formation of a coupled two-cell Brusselator model,, J. Math. Anal. Appl., 366 (2010), 679.  doi: 10.1016/j.jmaa.2009.12.021.  Google Scholar

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show all references

References:
[1]

Q. Y. Bie, Pattern formation in a general two-cell Brusselator model,, J. Math. Anal. Appl., 376 (2011), 551.  doi: 10.1016/j.jmaa.2010.10.066.  Google Scholar

[2]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations,, John Wiley & Sons, (2003).  doi: 10.1002/0470871296.  Google Scholar

[3]

A. J. Catllá, A. McNamara and C. M. Topaz, Instabilities and patterns in coupled reaction-diffusion layers,, Phy. Rev. E Stat. Nonlinear & Soft Matter Physics, 85 (2012), 489.   Google Scholar

[4]

W. Chen, Localized Patterns in the Gray-scott Model: An Asymptotic and Numerical Study of Dynamics and Stability, PhD thesis, University of British Columbia,, 2009., ().   Google Scholar

[5]

F. A. Davidson and B. P. Rynne, A priori bounds and global existence of solutions of the steady-state Sel'kov model,, Proc. Roy. Soc. Edinburgh Sect. A, 130 (2000), 507.  doi: 10.1017/S0308210500000275.  Google Scholar

[6]

A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model,, Nonlinearity, 10 (1997), 523.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar

[7]

L. L. Du and M. X. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model,, J. Math. Anal. Appl., 366 (2010), 473.  doi: 10.1016/j.jmaa.2010.02.002.  Google Scholar

[8]

J. E. Furter and J. C. Eilbeck, Analysis of bifurcations in reaction-diffusion systems with no-flux boundary conditions: The Sel'kov model,, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 413.  doi: 10.1017/S0308210500028109.  Google Scholar

[9]

M. Ghergu, Steady-state solutions for a general Brusselator system,, In Modern Aspects of the Theory of Partial Differential Equations, (2011), 153.  doi: 10.1007/978-3-0348-0069-3_9.  Google Scholar

[10]

M. Ghergu and V. Rădulescu, Turing patterns in general reaction-diffusion systems of Brusselator type,, Commun. Contemp. Math., 12 (2010), 661.  doi: 10.1142/S0219199710003968.  Google Scholar

[11]

M. Ghergu, Non-constant steady-state solutions for Brusselator type systems,, Nonlinearity, 21 (2008), 2331.  doi: 10.1088/0951-7715/21/10/007.  Google Scholar

[12]

M. Ghergu and V. D. Rădulescu, Nonlinear PDEs: Mathematical Models in Biology, Chemistry and Population Genetics,, Springer Verlag, (2012).  doi: 10.1007/978-3-642-22664-9.  Google Scholar

[13]

A. A. Golovin, B. J. Matkowsky and V. A. Volpert, Turing pattern formation in the Brusselator model with superdiffusion,, SIAM J. Appl. Math., 69 (2008), 251.  doi: 10.1137/070703454.  Google Scholar

[14]

J. K. Hale, L. A. Peletier and W. C. Troy, Stability and instability in the Gray-Scott model: The case of equal diffusivities,, Appl. Math. Lett., 12 (1999), 59.  doi: 10.1016/S0893-9659(99)00035-X.  Google Scholar

[15]

B. D. Hassard, N. D. Kazarinoff and Y. H. Wan, Theory and Applications of Hopf Bifurcation, volume 41., CUP Archive, (1981).   Google Scholar

[16]

D. Iron, J. C. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model,, J. Math. Biol., 49 (2004), 358.  doi: 10.1007/s00285-003-0258-y.  Google Scholar

[17]

J. Jang, W. M. Ni and M. X. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model,, J. Dynam. Differential Equations, 16 (2004), 297.  doi: 10.1007/s10884-004-2782-x.  Google Scholar

[18]

J. Y. Jin, J. P. Shi, J. J. Wei and F. Q. Yi, Bifurcations of patterned solutions in diffusive Lengyel-Epstein system of cima chemical reaction,, Roc. Mount.J. Math., 43 (2013), 1637.  doi: 10.1216/RMJ-2013-43-5-1637.  Google Scholar

[19]

T. Kolokolnikov, T. Erneux and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability,, Phys. D, 214 (2006), 63.  doi: 10.1016/j.physd.2005.12.005.  Google Scholar

[20]

J. van de Koppel and C. M. Crain, Scale-dependent inhibition drives regular tussock spacing in a freshwater marsh,, Amer. Natu., 168 (2006), 36.  doi: 10.1086/508671.  Google Scholar

[21]

J. López-Gómez, J. C. Eilbeck, M. Molina and K. N. Duncan, Structure of solution manifolds in a strongly coupled elliptic system,, IMA J. Numer. Anal., 12 (1992), 405.  doi: 10.1093/imanum/12.3.405.  Google Scholar

[22]

Y. Lou and W. M. Ni, Diffusion, self-diffusion and cross-diffusion,, Journal of Differential Equations, 131 (1996), 79.  doi: 10.1006/jdeq.1996.0157.  Google Scholar

[23]

W. Mazin, K. E. Rasmussen, E. Mosekilde, P. Borckmans and G. Dewel, Pattern formation in the bistable gray-scott model,, Math. Compu. in Simulation, 40 (1996), 371.  doi: 10.1016/0378-4754(95)00044-5.  Google Scholar

[24]

J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model,, Nonlinear Anal. Real World Appl., 5 (2004), 105.  doi: 10.1016/S1468-1218(03)00020-8.  Google Scholar

[25]

W. M. Ni, Qualitative properties of solutions to elliptic problems,, Handbook of Differential Equations Stationary Partial Differential Equations, 1 (2004), 157.  doi: 10.1016/S1874-5733(04)80005-6.  Google Scholar

[26]

W. M. Ni and M. X. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction,, Trans. Amer. Math. Soc., 357 (2005), 3953.  doi: 10.1090/S0002-9947-05-04010-9.  Google Scholar

[27]

R. Peng and M. X. Wang, Pattern formation in the Brusselator system,, J. Math. Anal. Appl., 309 (2005), 151.  doi: 10.1016/j.jmaa.2004.12.026.  Google Scholar

[28]

R. Peng, M. X. Wang and M. Yang, Positive steady-state solutions of the Sel'kov model,, Math. Comput. Modelling, 44 (2006), 945.  doi: 10.1016/j.mcm.2006.03.001.  Google Scholar

[29]

R. Peng, Qualitative analysis of steady states to the Sel'kov model,, J. Differential Equations, 241 (2007), 386.  doi: 10.1016/j.jde.2007.06.005.  Google Scholar

[30]

R. Peng and M. X. Wang, Some nonexistence results for nonconstant stationary solutions to the Gray-Scott model in a bounded domain,, Appl. Math. Lett., 22 (2009), 569.  doi: 10.1016/j.aml.2008.06.032.  Google Scholar

[31]

R. Peng and F. Q. Sun, Turing pattern of the Oregonator model,, Nonlinear Anal., 72 (2010), 2337.  doi: 10.1016/j.na.2009.10.034.  Google Scholar

[32]

Y. W. Qi, The development of travelling waves in cubic auto-catalysis with different rates of diffusion,, Phys. D, 226 (2007), 129.  doi: 10.1016/j.physd.2006.11.010.  Google Scholar

[33]

E. E. Sel'Kov, Self-oscillations in glycolysis,, European Journal of Biochemistry, 4 (1968), 79.   Google Scholar

[34]

J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour,, J. Theoret. Biol., 81 (1979), 389.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar

[35]

J. P. Shi and X. F. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains,, J. Differential Equations, 246 (2009), 2788.  doi: 10.1016/j.jde.2008.09.009.  Google Scholar

[36]

I. Takagi C. S. Lin and W. M. Ni, Large amplitude stationary solutions to a chemotaxis system,, J. Differential Equations, 72 (1988), 1.  doi: 10.1016/0022-0396(88)90147-7.  Google Scholar

[37]

J. J. Tyson, K. Chen and B. Novak, Network dynamics and cell physiology,, Nature Rev. Molecular Cell Bio., 2 (2001), 908.   Google Scholar

[38]

A. M. Turing, The chemical basis of morphogenesis,, Philosophical Transactions of the Royal Society of London. Series B, 237 (1952), 37.   Google Scholar

[39]

M. X. Wang, Non-constant positive steady states of the Sel'kov model,, J. Differential Equations, 190 (2003), 600.  doi: 10.1016/S0022-0396(02)00100-6.  Google Scholar

[40]

M. X. Wang, R. Peng and J. P. Shi, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law,, Nonlinearity, 21 (2008), 1471.  doi: 10.1088/0951-7715/21/7/006.  Google Scholar

[41]

M. X. Wang and P. Y. H. Pang, Global asymptotic stability of positive steady states of a diffusive ratio-dependent pre-predator model,, Applied Mathematics Letters, 21 (2008), 1215.  doi: 10.1016/j.aml.2007.10.026.  Google Scholar

[42]

J. F. Wang, J. P. Shi and J. J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong allee effect in prey,, J. Differential Equations, 251 (2011), 1276.  doi: 10.1016/j.jde.2011.03.004.  Google Scholar

[43]

M. J. Ward and J. C. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model,, Stud. Appl. Math., 109 (2002), 229.  doi: 10.1111/1467-9590.00223.  Google Scholar

[44]

J. M. Wei, Pattern formations in two-dimensional Gray-Scott model: Existence of single-spot solutions and their stability,, Phys. D, 148 (2001), 20.  doi: 10.1016/S0167-2789(00)00183-4.  Google Scholar

[45]

J. C. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems,, J. Math. Biol., 57 (2008), 53.  doi: 10.1007/s00285-007-0146-y.  Google Scholar

[46]

J. C. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics,, J. Math. Biol., 64 (2012), 211.  doi: 10.1007/s00285-011-0412-x.  Google Scholar

[47]

S. Wiggins and M. Golubitsky, Introduction to Applied Nonlinear Dynamical Systems and Chaos, volume 2., Springer, (1990).  doi: 10.1007/978-1-4757-4067-7.  Google Scholar

[48]

L. Xu, G. Zhang and J. F. Ren, Turing instability for a two dimensional semi-discrete oregonator model,, WSEAS Transac. Math, 10 (2011), 201.   Google Scholar

[49]

C. Xu and J. J. Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model,, Nonlinear Anal. Real World Appl., 13 (2012), 1961.  doi: 10.1016/j.nonrwa.2012.01.001.  Google Scholar

[50]

F. Q. Yi, J. J. Wei and J. J. Shi, Diffusion-driven instability and bifurcation in the lengyel-epstein system,, Nonlinear Anal.: Real World Applications, 9 (2008), 1038.  doi: 10.1016/j.nonrwa.2007.02.005.  Google Scholar

[51]

F. Q. Yi, J. J. Wei and J. P. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predator-prey system,, J. Differential Equations, 246 (2009), 1944.  doi: 10.1016/j.jde.2008.10.024.  Google Scholar

[52]

F. Q. Yi, J. J. Wei and J. P. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system,, Appl. Math. Lett., 22 (2009), 52.  doi: 10.1016/j.aml.2008.02.003.  Google Scholar

[53]

Y. C. You, Global dynamics of the Brusselator equations,, Dyn. Partial Differ. Equ., 4 (2007), 167.  doi: 10.4310/DPDE.2007.v4.n2.a4.  Google Scholar

[54]

Y. C. You, Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems,, Commun. Pure Appl. Anal., 10 (2011), 1415.  doi: 10.3934/cpaa.2011.10.1415.  Google Scholar

[55]

Y. C. You, Dynamics of two-compartment Gray-Scott equations,, Nonlinear Anal., 74 (2011), 1969.  doi: 10.1016/j.na.2010.11.004.  Google Scholar

[56]

Y. C. You, Global dynamics of the Oregonator system,, Math. Methods Appl. Sci., 35 (2012), 398.  doi: 10.1002/mma.1591.  Google Scholar

[57]

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