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BIFURCATION ANALYSIS OF A SINGLE SPECIES REACTION-DIFFUSION MODEL WITH NONLOCAL DELAY

  • Zhou, Jun (School of Mathematics and Statistics Southwest University)
  • Received : 2019.01.12
  • Accepted : 2019.07.08
  • Published : 2019.12.30

Abstract

A reaction-diffusion model with spatiotemporal delay modeling the dynamical behavior of a single species is investigated. The parameter regions for the local stability, global stability and instability of the unique positive constant steady state solution are derived. The conditions of the occurrence of Turing (diffusion-driven) instability are obtained. The existence of time-periodic solutions, the existence and nonexistence of nonconstant positive steady state solutions are proved by bifurcation method and energy method. Numerical simulations are presented to verify and illustrate the theoretical results.

References

  1. J. F. G. Auchmuty and G. Nicolis, Bifurcation analysis of nonlinear reaction-diffusion equations. I. Evolution equations and the steady state solutions, Bull. Math. Biology 37 (1975), no. 4, 323-365. https://doi.org/10.1007/bf02459519 https://doi.org/10.1007/BF02459519
  2. N. F. Britton, Aggregation and the competitive exclusion principle, J. Theoret. Biol. 136 (1989), no. 1, 57-66. https://doi.org/10.1016/S0022-5193(89)80189-4 https://doi.org/10.1016/S0022-5193(89)80189-4
  3. N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math. 50 (1990), no. 6, 1663-1688. https://doi.org/10.1137/0150099 https://doi.org/10.1137/0150099
  4. K. J. Brown and F. A. Davidson, Global bifurcation in the Brusselator system, Nonlinear Anal. 24 (1995), no. 12, 1713-1725. https://doi.org/10.1016/0362-546X(94)00218-7 https://doi.org/10.1016/0362-546X(94)00218-7
  5. A. J. Catlla, A. McNamara, and C. M. Topaz, Instabilities and patterns in coupled reaction-diffusion layers, Phy. Rev. E 85 (2012), no. 2, 026215-1. https://doi.org/10.1103/PhysRevE.85.026215 https://doi.org/10.1103/PhysRevE.85.026215
  6. W. Chen, Localized patterns in the gray-scott model: an asymptotic and numerical study of dynamics and stability, A thesis submitted in partial fulfillment of the requirments for the degree of doctor of philosophy in the faculty of graduate studies (Mathematics), The Uinversity of British Columbia(Vancouver) July, 2009.
  7. C. Chris and C. R. Stephen, Spatial Ecology via Reaction-Diusion Equations, John Wiley & Sons, Ltd., Chichester, 2003.
  8. 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), no. 3, 507-516. https://doi.org/10.1017/S0308210500000275 https://doi.org/10.1017/S0308210500000275
  9. A. Doelman, T. J. Kaper, and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity 10 (1997), no. 2, 523-563. https://doi.org/10.1088/0951-7715/10/2/013 https://doi.org/10.1088/0951-7715/10/2/013
  10. L. Du and M. Wang, Hopf bifurcation analysis in the 1-D Lengyel-Epstein reaction-diffusion model, J. Math. Anal. Appl. 366 (2010), no. 2, 473-485. https://doi.org/10.1016/j.jmaa.2010.02.002 https://doi.org/10.1016/j.jmaa.2010.02.002
  11. 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), no. 2, 413-438. https://doi.org/10.1017/S0308210500028109 https://doi.org/10.1017/S0308210500028109
  12. M. Ghergu, Non-constant steady-state solutions for Brusselator type systems, Nonlinearity 21 (2008), no. 10, 2331-2345. https://doi.org/10.1088/0951-7715/21/10/007 https://doi.org/10.1088/0951-7715/21/10/007
  13. M. Ghergu and V. Radulescu, Turing patterns in general reaction-diffusion systems of Brusselator type, Commun. Contemp. Math. 12 (2010), no. 4, 661-679. https://doi.org/10.1142/S0219199710003968 https://doi.org/10.1142/S0219199710003968
  14. M. Ghergu and V. Radulescu, Nonlinear PDEs, Springer Monographs in Mathematics, Springer, Heidelberg, 2012. https://doi.org/10.1007/978-3-642-22664-9
  15. J. K. Hale, L. A. Peletier, and W. C. Troy, Stability and instability in the Gray-Scott model: the case of equal diusivities, Appl. Math. Lett. 12 (1999), no. 4, 59-65. https://doi.org/10.1016/S0893-9659(99)00035-X https://doi.org/10.1016/S0893-9659(99)00035-X
  16. B. D. Hassard, N. D. Kazarino, and Y. H. Wan, Theory and applications of Hopf bifurcation, London Mathematical Society Lecture Note Series, 41, Cambridge University Press, Cambridge, 1981.
  17. M. Herschkowitz-Kaufman, Bifurcation analysis of nonlinear reaction-diffusion equations. II. Steady state solutions and comparison with numerical simulations, Bull. Math. Biology 37 (1975), no. 6, 589-636. https://doi.org/10.1007/BF02459527
  18. D. Iron, J. Wei, and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol. 49 (2004), no. 4, 358-390. https://doi.org/10.1007/s00285-003-0258-y https://doi.org/10.1007/s00285-003-0258-y
  19. J. Jang, W.-M. Ni, and M. Tang, Global bifurcation and structure of Turing patterns in the 1-D Lengyel-Epstein model, J. Dynam. Differential Equations 16 (2004), no. 2, 297-320. https://doi.org/10.1007/s10884-004-2782-x https://doi.org/10.1007/s10884-004-2782-x
  20. J. Y. Jin, J. P. Shi, J. J. Wei, and F. Q. Yi, Bifurcations of patterned solutions in the diffusive Lengyel-Epstein system of CIMA chemical reactions, Rocky Mountain J. Math. 43 (2013), no. 5, 1637-1674. https://doi.org/10.1216/RMJ-2013-43-5-1637 https://doi.org/10.1216/RMJ-2013-43-5-1637
  21. T. Kolokolnikov, T. Erneux, and J. Wei, Mesa-type patterns in the one-dimensional Brusselator and their stability, Phys. D 214 (2006), no. 1, 63-77. https://doi.org/10. 1016/j.physd.2005.12.005 https://doi.org/10.1016/j.physd.2005.12.005
  22. C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), no. 1, 1-27. https://doi.org/10.1016/0022-0396(88)90147-7 https://doi.org/10.1016/0022-0396(88)90147-7
  23. J. Lopez-Gomez, 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), no. 3, 405-428. https://doi.org/10.1093/imanum/12.3.405 https://doi.org/10.1093/imanum/12.3.405
  24. Y. Lou and W.-M. Ni, Diusion, self-diffusion and cross-diffusion, J. Differential Equa- tions 131 (1996), no. 1, 79-131. https://doi.org/10.1006/jdeq.1996.0157 https://doi.org/10.1006/jdeq.1996.0157
  25. W. Mazin, K. Rasmussen, E. Mosekilde, P. Borckmans, and G. Dewel, Pattern formation in the bistable gray-scott model, Math. Comput. Simu. 40 (1996), no. 3, 371-396. https://doi.org/10.1016/0378-4754(95)00044-5
  26. J. S. McGough and K. Riley, Pattern formation in the Gray-Scott model, Nonlinear Anal. Real World Appl. 5 (2004), no. 1, 105-121. https://doi.org/10.1016/S1468-1218(03)00020-8 https://doi.org/10.1016/S1468-1218(03)00020-8
  27. W.-M. Ni, Qualitative properties of solutions to elliptic problems, Handbook of Differential Equations: Stationary Partial Differential Equations, Volume 1, Chapter 3, 157-233, 2004.
  28. W.-M. Ni and M. Tang, Turing patterns in the Lengyel-Epstein system for the CIMA reaction, Trans. Amer. Math. Soc. 357 (2005), no. 10, 3953-3969. https://doi.org/10.1090/S0002-9947-05-04010-9 https://doi.org/10.1090/S0002-9947-05-04010-9
  29. R. Peng, Qualitative analysis of steady states to the Sel'kov model, J. Differential Equations 241 (2007), no. 2, 386-398. https://doi.org/10.1016/j.jde.2007.06.005 https://doi.org/10.1016/j.jde.2007.06.005
  30. R. Peng, J. Shi, and M. Wang, On stationary patterns of a reaction-diffusion model with autocatalysis and saturation law, Nonlinearity 21 (2008), no. 7, 1471-1488. https://doi.org/10.1088/0951-7715/21/7/006 https://doi.org/10.1088/0951-7715/21/7/006
  31. R. Peng and F. Sun, Turing pattern of the Oregonator model, Nonlinear Anal. 72 (2010), no. 5, 2337-2345. https://doi.org/10.1016/j.na.2009.10.034 https://doi.org/10.1016/j.na.2009.10.034
  32. R. Peng and M. X. Wang, Pattern formation in the Brusselator system, J. Math. Anal. Appl. 309 (2005), no. 1, 151-166. https://doi.org/10.1016/j.jmaa.2004.12.026 https://doi.org/10.1016/j.jmaa.2004.12.026
  33. 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), no. 4, 569-573. https://doi.org/10.1016/j.aml.2008.06.032 https://doi.org/10.1016/j.aml.2008.06.032
  34. R. Peng, M. Wang, and M. Yang, Positive steady-state solutions of the Sel'kov model, Math. Comput. Modelling 44 (2006), no. 9-10, 945-951. https://doi.org/10.1016/j.mcm.2006.03.001 https://doi.org/10.1016/j.mcm.2006.03.001
  35. J. Schnakenberg, Simple chemical reaction systems with limit cycle behaviour, J. Theoret. Biol. 81 (1979), no. 3, 389-400. https://doi.org/10.1016/0022-5193(79)90042-0 https://doi.org/10.1016/0022-5193(79)90042-0
  36. E. Sel'Kov, Self-oscillations in glycolysis, Eur. J. Bioch. 4 (1968), no. 1, 79-86. https://doi.org/10.1111/j.1432-1033.1968.tb00175.x
  37. J. Shi and X. Wang, On global bifurcation for quasilinear elliptic systems on bounded domains, J. Differential Equations 246 (2009), no. 7, 2788-2812. https://doi.org/10.1016/j.jde.2008.09.009 https://doi.org/10.1016/j.jde.2008.09.009
  38. A. M. Turing, The chemical basis of morphogenesis, Philos. Trans. Roy. Soc. London Ser. B 237 (1952), no. 641, 37-72. https://doi.org/10.1098/rstb.1952.0012
  39. J. Wang, J. Shi, and J. Wei, Dynamics and pattern formation in a diffusive predator-prey system with strong Allee effect in prey, J. Differential Equations 251 (2011), no. 4-5, 1276-1304. https://doi.org/10.1016/j.jde.2011.03.004 https://doi.org/10.1016/j.jde.2011.03.004
  40. M. Wang, Non-constant positive steady states of the Sel'kov model, J. Differential Equa- tions 190 (2003), no. 2, 600-620. https://doi.org/10.1016/S0022-0396(02)00100-6 https://doi.org/10.1016/S0022-0396(02)00100-6
  41. M. Wang and P. Y. H. Pang, Global asymptotic stability of positive steady states of a diffusive ratio-dependent prey-predator model, Appl. Math. Lett. 21 (2008), no. 11, 1215-1220. https://doi.org/10.1016/j.aml.2007.10.026 https://doi.org/10.1016/j.aml.2007.10.026
  42. M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math. 109 (2002), no. 3, 229-264. https://doi.org/10.1111/1467-9590.00223 https://doi.org/10.1111/1467-9590.00223
  43. J. Wei, Pattern formations in two-dimensional Gray-Scott model: existence of single-spot solutions and their stability, Phys. D 148 (2001), no. 1-2, 20-48. https://doi.org/10.1016/S0167-2789(00)00183-4 https://doi.org/10.1016/S0167-2789(00)00183-4
  44. J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol. 57 (2008), no. 1, 53-89. https://doi.org/10.1007/s00285-007-0146-y https://doi.org/10.1007/s00285-007-0146-y
  45. J. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetics, J. Math. Biol. 64 (2012), no. 1-2, 211-254. https://doi.org/10.1007/s00285-011-0412-x https://doi.org/10.1007/s00285-011-0412-x
  46. C. Xu and J.Wei, Hopf bifurcation analysis in a one-dimensional Schnakenberg reaction-diffusion model, Nonlinear Anal. Real World Appl. 13 (2012), no. 4, 1961-1977. https://doi.org/10.1016/j.nonrwa.2012.01.001 https://doi.org/10.1016/j.nonrwa.2012.01.001
  47. L. Xu, G. Zhang, and J. F. Ren, Turing instability for a two dimensional semi-discrete oregonator model, WSEAS Trans. Math. 10 (2011), no. 6, 201-209.
  48. G. Yang and J. Xu, Analysis of spatiotemporal patterns in a single species reaction-diffusion model with spatiotemporal delay, Nonlinear Anal. Real World Appl. 22 (2015), 54-65. https://doi.org/10.1016/j.nonrwa.2014.07.013 https://doi.org/10.1016/j.nonrwa.2014.07.013
  49. F. Yi, J. Wei, and J. Shi, Diusion-driven instability and bifurcation in the Lengyel-Epstein system, Nonlinear Anal. Real World Appl. 9 (2008), no. 3, 1038-1051. https://doi.org/10.1016/j.nonrwa.2007.02.005 https://doi.org/10.1016/j.nonrwa.2007.02.005
  50. F. Yi, J. Wei, and J. Shi, Bifurcation and spatiotemporal patterns in a homogeneous diffusive predatorprey system, J. Differential Equations 246 (2009), no. 5, 1944-1977. https://doi.org/10.1016/j.jde.2008.10.024 https://doi.org/10.1016/j.jde.2008.10.024
  51. F. Yi, J. Wei, and J. Shi, Global asymptotical behavior of the Lengyel-Epstein reaction-diffusion system, Appl. Math. Lett. 22 (2009), no. 1, 52-55. https://doi.org/10.1016/j.aml.2008.02.003 https://doi.org/10.1016/j.aml.2008.02.003
  52. Y. You, Global dynamics of the Brusselator equations, Dyn. Partial Differ. Equ. 4 (2007), no. 2, 167-196. https://doi.org/10.4310/DPDE.2007.v4.n2.a4 https://doi.org/10.4310/DPDE.2007.v4.n2.a4
  53. Y. You, Asymptotic dynamics of reversible cubic autocatalytic reaction-diffusion systems, Commun. Pure Appl. Anal. 10 (2011), no. 5, 1415-1445. https://doi.org/10.3934/cpaa.2011.10.1415 https://doi.org/10.3934/cpaa.2011.10.1415
  54. Y. You, Dynamics of two-compartment Gray-Scott equations, Nonlinear Anal. 74 (2011), no. 5, 1969-1986. https://doi.org/10.1016/j.na.2010.11.004 https://doi.org/10.1016/j.na.2010.11.004
  55. Y. You, Global dynamics of the Oregonator system, Math. Methods Appl. Sci. 35 (2012), no. 4, 398-416. https://doi.org/10.1002/mma.1591 https://doi.org/10.1002/mma.1591
  56. Y. You, Robustness of global attractors for reversible Gray-Scott systems, J. Dynam. Differential Equations 24 (2012), no. 3, 495-520. https://doi.org/10.1007/s10884-012-9252-7 https://doi.org/10.1007/s10884-012-9252-7
  57. J. Zhou and C. Mu, Pattern formation of a coupled two-cell Brusselator model, J. Math. Anal. Appl. 366 (2010), no. 2, 679-693. https://doi.org/10.1016/j.jmaa.2009.12.021 https://doi.org/10.1016/j.jmaa.2009.12.021