Stability analysis of reactiondiffusion models on evolving domains: The effects of crossdiffusion
Pages: 2133  2170,
Issue 4,
April
2016
doi:10.3934/dcds.2016.36.2133 Abstract
References
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Anotida Madzvamuse  University of Sussex, School of Mathematical and Physical Sciences, Pevensey III, 5C15, Brighton, BN1 9QH, United Kingdom (email)
Hussaini Ndakwo  School of Mathematical and Physical Sciences, Department of Mathematics, University of Sussex, Falmer, Brighton, BN1 9QH, England, United Kingdom (email)
Raquel Barreira  Polytechnic Institute of Setubal, Barreiro School of Technology, Rua Américo da Silva MarinhoLavradio, 2839001 Barreiro, Portugal (email)
1 
D. Acheson, Elementary Fluid Dynamics, Oxford University Press, New York, 1990. 

2 
M. Baines, Moving Finite Elements, Oxford University Press, New York, 1994. 

3 
J. Bard and I. Lauder, How well does Turing's Theory of morphogenesis work?, J. Theor. Bio., 45 (1974), 501531. 

4 
R. Barreira, C. M. Elliott and A. Madzvamuse, The surface finite element method for pattern formation on evolving biological surfaces, J. Math. Bio., 63 (2011), 10951119. 

5 
V. Capasso and D. Liddo, Asymptotic behaviour of reactiondiffusion systems in population and epidemic models. The role of crossdiffusion, J. Math. Biol., 32 (1994), 453463. 

6 
V. Capasso and D. Liddo, Global attractivity for reactiondiffusion systems. The case of nondiagonal diffusion matrices, J. Math. Anal. and App., 177 (1993), 510529. 

7 
E. J. Crampin, W. W. Hackborn and P. K. Maini, Pattern formation in reactiondiffusion models with nonuniform domain growth, Bull. Math. Biol., 64 (2002), 747769. 

8 
G. Gambino, M. C. Lombardo and M. Sammartino, Turing instability and traveling fronts for nonlinear reactiondiffusion system with crossdiffusion, Maths. Comp. in Sim., 82 (2012), 11121132. 

9 
G. Gambino, M. C. Lombardo and M. Sammartino, Pattern formation driven by crossdiffusion in 2D domain, Non. Anal. Real World Applications, 14 (2013), 17551779. 

10 
A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 3039. 

11 
G. Hetzer, A. Madzvamuse and W. Shen, Characterization of Turing diffusiondriven instability on evolving domains, Disc. Con. Dyn. Sys., 32 (2012), 39754000. 

12 
M. Iida and M. Mimura, Diffusion, crossdiffusion an competitive interaction, J. Math. Biol., 53 (2006), 617641. 

13 
K. Korvasova, E. A. Gaffney, M. P. Maini, M. A. Ferreira and V. Klika, Investigating the Turing conditions for diffusiondriven instability in the presence of binding immobile substrate, J. Theor. Biol., 367 (2015), 286295. 

14 
S. Kovács, Turing bifurcation in a system with crossdiffusion, Nonlinear Analysis, 59 (2004), 567581. 

15 
O. Lakkis, A. Madzvamuse and C. Venkataraman, Implicitexplicit timestepping with finite element approximation of reactiondiffusion systems on evolving domains, SIAM JNA, 51 (2013), 23092330. 

16 
C. B. Macdonald, B. Merriman and S. J. Ruuth, Simple computation of reaction diffusion processes on point clouds, Proc. Nat. Acad. Sci. USA., 110 (2013), 92099214. 

17 
C. B. Macdonald and S. J. Ruuth, The implicit closest point method for the numerical solution of partial differential equations on surfaces, SIAM J. Sci. Comput., 31 (2010), 43304350. 

18 
A. Madzvamuse, R. D. K. Thomas, P. K. Maini and A. J. Wathen, A numerical approach to the study of spatial pattern formation in the ligaments of arcoid bivalves, Bulletin of Mathematical Biology, 64 (2002), 501530. 

19 
A. Madzvamuse, P. K. Maini and A. J. Wathen, A moving grid finite element method applied to a model biological pattern generator, J. Comp. Phys., 190 (2003), 478500. 

20 
A. Madzvamuse, A. J. Wathen and P. K. Maini, A moving grid finite element method for the simulation of pattern generation by Turing models on growing domains, J. Sci. Comp., 24 (2005), 247262. 

21 
A. Madzvamuse, Timestepping schemes for moving grid finite elements applied to reactiondiffusion systems on fixed and growing domains, J. Sci. Phys., 214 (2006), 239263. 

22 
A. Madzvamuse and M. K. Maini, Velocityinduced numerical solutions of reactiondiffusion systems on fixed and growing domains, J. Comp. Phys., 225 (2007), 100119. 

23 
A. Madzvamuse, Diffusiondriven instability for growing domains with divergence free mesh velocity, Nonlinear Analysis: Theory, Methods and Applications, 17 (2009), e2250e2257. 

24 
A. Madzvamuse, E. A. Gaffney and M. K. Maini, Stability analysis of nonautonomous reactiondiffusion systems: the effects of growing domains, J. Math. Biol., 61 (2010), 133164. 

25 
A. Madzvamuse and R. Barreira, Exhibiting crossdiffusioninduced patterns for reactiondiffusion systems on evolving domains and surfaces, Physical Review E, 90 (2014), 043307, 14pp. 

26 
A. Madzvamuse, H. S. Ndakwo and R. Barreira, Crossdiffusiondriven instability for reactiondiffusion systems: Analysis and simulations, Journal of Math. Bio., 70 (2015), 709743. 

27 
P. K. Maini, E. J. Crampin, A. Madzvamuse, A. J. Wathen and R. D. K. Thomas, Implications of domain growth in morphogenesis, in Mathematical Modelling and Computing in Biology and Medicine, Capaso, V., ed., Proceedings of the 5th European Conference for Mathematics and Theoretical Biology: Conference, Milan, Italy. 1 (2003), 6773. 

28 
M. S. McAfree and O. Annunziata, Crossdiffusion in a colloidpolymer aqueous system, Fluid Phase Equilibria, 356 (2013), 4655. 

29 
C. C. McCluskey, A strategy for constructing Lyapunov functions for nonautonomous linear differential equations, Linear Algebra and its Applications, 409 (2005), 100110. 

30 
J. D. Murray, Mathematical Biology. II, Volume 18 of Interdisciplinary Applied Mathematics. SpringerVerlag, New York. Third edition. Spatial models and biomedical applications, 2003. 

31 
R. G. Plaza, F. SánchezGarduño, P. Padilla, R. A. Barrio and P. K. Maini, The effect of growth and curvature on pattern formation, J. Dynam. and Diff. Eqs., 16 (2004), 10931121. 

32 
I. Prigogine and R. Lefever, Symmetry breaking instabilities in dissipative systems. II, J. Chem. Phys., 48 (1968), 16951700. 

33 
F. Rossi, V. K. Vanag, E. Tiezzi and I. R. Epstein, Quaternary crossdiffusion in waterinoil microemulsions loaded with a component of the BelousovZhabotinsky reaction, J. Phys. Chem. B, 114 (2010), 81408146. 

34 
R. RuizBaier and C. Tian, Mathematical analysis and numerical simulation of pattern formation under crossdiffusion, Non. Anal. Real World Applications, 14 (2013), 601612. 

35 
J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389400. 

36 
L. Z. Tian and M. Pedersen, Instability induced by crossdiffusion in reactiondiffusion systems, Non. Anal.: Real World Applications, 11 (2010), 10361045. 

37 
A. Turing, On the chemical basis of morphogenesis, Phil. Trans. Royal Soc. B, 237 (1952), 3772. 

38 
V. K. Vanag and I. R. Epstein, Crossdiffusion and pattern formation in reaction diffusion systems, Phys. Chem. Chem. Phys., 11 (2009), 897912. 

39 
C. Venkataraman, O. Lakkis and A. Madzvamuse, Global existence for semilinear reactiondiffusion systems on evolving domains, Journal of Mathematical Biology, 64 (2012), 4167. 

40 
A. Vergara. F. Capuano, L. Paduano and R. Sartorio, Lysozyme mutual diffusion in solutions crowded by poly(ethylene glycol), Macromolecules, 39 (2006), 45004506. 

41 
Z. Xie, Crossdiffusion induced Turing instability for a three species food chain model, J. Math. Analy. and Appl., 388 (2012), 539547. 

42 
J. F. Zhang, W. T. Li and Y. X. Wang, Turing patterns of a strongly coupled predatorprey system with diffusion effects, Non. Anal., 74 (2011), 847858. 

43 
E. P. Zemskov, V. K. Vanag and I. R. Epstein, Amplitude equations for reactiondiffusion systems with crossdiffusion, Phys. Rev. E., 84 (2011), 036216. 

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