Skip to main content
SpringerLink
Log in

Reaction-Diffusion Models with Delay: Some Properties, Equations, Problems, and Solutions

  • Published:
Theoretical Foundations of Chemical Engineering Aims and scope Submit manuscript

Abstract

The delay reaction-diffusion models used in thermal physics, chemistry, biochemistry, biology, ecology, biomedicine, and control theory were reviewed. New exact solutions were obtained for several classes of one- and three-dimensional nonlinear equations with distributed parameters, in which the kinetic functions involve a delay. The qualitative features of these equations related to nonsmoothness and potential instability of solutions (these features should be taken into account in the mathematical modeling of the corresponding processes) were discussed. The properties of delay reaction-diffusion equations were described, which allow exact solutions to be obtained and multiplied. The key principles of construction, selection, and use of the test problems of the reaction-diffusion type were formulated, which can be used for evaluating the accuracy of rough analytical and numerical methods for solving the delay equations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. CRC Handbook of Lie Group Analysis of Differential Equations, vol. 1: Symmetries, Exact Solutions and Conservation Laws, Ibragimov, N.H., Ed., Boca Raton, Fla.: CRC, 1994.

  2. Galaktionov, V.A. and Svirshchevskii, S.R., Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Boca Raton, Fla.: Chapman & Hall/CRC, 2006.

    Google Scholar 

  3. Polyanin, A.D. and Zaitsev, V.F., Handbook of Nonlinear Partial Differential Equations, Boca Raton, Fla.: Chapman & Hall/CRC, 2012, 2nd ed.

    Google Scholar 

  4. Jackiewicz, Z. and Zubik-Kowal, B., Spectral collocation and waveform relaxation methods for nonlinear delay partial differential equations, Appl. Numer. Math., 2006, vol. 56, p.433.

    Article  Google Scholar 

  5. Marzban, H.R. and Tabrizidooz, H.R., A hybrid approximation method for solving Hutchinson’s equation, Commun. Nonlinear Sci. Numer. Simul., 2012, vol. 17, p.100.

    Article  Google Scholar 

  6. Busenberg, S. and Huang, W., Stability and Hopf bifurcation for a population delay model with diffusion effects, J. Differ. Equations, 1996, vol. 124, p.80.

    Article  Google Scholar 

  7. Su, Y., Wei, J., and Shi, J., Hopf bifurcation in a reaction-diffusion population model with delay effect, J. Differ. Equations, 2009, vol. 247, p. 1156.

    Article  Google Scholar 

  8. Bonnefon, O., Garnier, J., Hamel, F., and Roques, L., Inside dynamics of delayed travelling waves, Math. Model. Nat. Phenom., 2013, vol. 8, no. 3, p.42.

    Article  Google Scholar 

  9. Tian, H., Zhang, D., and Sun, Y., Delay-independent stability of Euler method for nonlinear one-dimensional diffusion equation with constant delay, Front. Math. China, 2009, vol. 4, no. 1, p.169.

    Article  Google Scholar 

  10. Huang, J. and Zou, X., Traveling wave fronts in diffusive and cooperative Lotka–Volterra system with delays, J. Math. Anal. Appl., 2002, vol. 271, p.455.

    Article  Google Scholar 

  11. Faria, T., Stability and bifurcation for a delayed predator-prey model and the effect of diffusion, J. Math. Anal. Appl., 2001, vol. 254, p.433.

    Article  Google Scholar 

  12. Chen, S., Shi, J., and Wei, J., A note on Hopf bifurcations in a delayed diffusive Lotka–Volterra predatorprey system, Comput. Math. Appl., 2011, vol. 62, p. 2240.

    Article  Google Scholar 

  13. Gourley, S.A., Wave front solutions of a diffusive delay model for populations of Daphnia magna, Comput. Math. Appl., 2001, vol. 42, p. 1421.

    Article  Google Scholar 

  14. Smith, F.E., Population dynamics in Daphnia magna, Ecology, 1963, vol. 44, p.651.

    Article  Google Scholar 

  15. So, J.W.-H. and Yang, Y., Dirichlet problem for the diffusive Nicholson’s blowflies equation, J. Differ. Equations, 1998, vol. 150, p.317.

    Article  Google Scholar 

  16. Mei, M., So, J.W.-H., List, M.Y., and Shen, S.S.P., Asymptotic stability of travelling waves for Nicholson’s blowflies equation with diffusion, Proc. R. Soc. Edinburgh, 2004, vol. 134A, p.579.

    Article  Google Scholar 

  17. Wang, X. and Li, Z., Dynamics for a type of general reaction-diffusion model, Nonlinear Anal., 2007, vol. 67, p. 2699.

    Article  Google Scholar 

  18. Ling, Z. and Lin, Z., Traveling wave front in a hematopoiesis model with time delay, Appl. Math. Lett., 2010, vol. 23, p.426.

    Article  Google Scholar 

  19. Mackey, M.C. and Glass, L., Oscillation and chaos in physiological control system, Science, 1977, vol. 197, p.287.

    Article  CAS  Google Scholar 

  20. Mackey, M.C., Unified hypothesis for the origin of aplastic anemia and periodic hematopoiesis, Blood, 1978, vol. 51, p.941.

    CAS  PubMed  Google Scholar 

  21. Hattaf, K. and Yousfi, N., A generalized HBV model with diffusion and two delays, Comput. Math. Appl., 2015, vol. 69, no. 1, p.31.

    Article  Google Scholar 

  22. Wang, K. and Wang, W., Propagation of HBV with spatial dependence, Math. Biosci., 2007, vol. 210, p.78.

    Article  PubMed  Google Scholar 

  23. Wang, K., Wang, W., and Song, S., Dynamics of an HBV model with diffusion and delay, J. Theor. Biol., 2008, vol. 253, p.36.

    Article  PubMed  Google Scholar 

  24. Hattaf, K. and Yousfi, N., Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response, Comput. Appl. Math., 2014. https://doi.org/10.1007/s40314-014-0143-x

    Google Scholar 

  25. Xu, R. and Ma, Z.E., An HBV model with diffusion and time delay, J. Theor. Biol., 2009, vol. 257, p.499.

    Article  PubMed  Google Scholar 

  26. Zhang, Y. and Xu, Z., Dynamics of a diffusive HBV model with delayed Beddington–DeAngelis response, Nonlinear Anal.: Real World Appl., 2014, vol. 15, p.118.

    Article  Google Scholar 

  27. Li, J., Sun, G.-Q., and Jin, Z., Pattern formation of an epidemic model with time delay, Phys. A (Amsterdam), 2014, vol. 403, p.100.

    Article  Google Scholar 

  28. Cai, Y., Yan, Sh., Wang, H., Lian, X., and Wang, W., Spatiotemporal dynamics in a reaction-diffusion epidemic model with a time-delay in transmission, Int. J. Bifurcation Chaos Appl. Sci. Eng., 2015, vol. 25, no. 8, article 1550099. https://doi.org/10.1142/S0218127415500996

    Google Scholar 

  29. Kmet, T., Modelling and simulation of food network, Proc. 22nd Eur. Conf. on Modelling and Simulation, Nicosia, Cyprus, 2008, p.157.

    Google Scholar 

  30. Likhoshvai, V.A., Fadeev, S.I., Demidenko, G.V., and Matushkin, Yu.G., Modeling of multistage synthesis without branching by an equation with the delayed argument, Sib. Zh. Ind. Mat., 2004, vol. 7, no. 1, p.73.

    Google Scholar 

  31. Polyanin, A.D. and Zhurov, A.I., Exact solutions of linear and nonlinear differential-difference heat and diffusion equations with finite relaxation time, Int. J. Non-Linear Mech., 2013, vol. 54, p.115.

    Article  Google Scholar 

  32. Jordan, P.M., Dai, W., and Mickens, R.E., A note on the delayed heat equation: Instability with respect to initial data, Mech. Res. Commun., 2008, vol. 35, p.414.

    Article  Google Scholar 

  33. Polyanin, A.D. and Vyazmin, A.V., Differential-difference heat-conduction and diffusion models and equations with a finite relaxation time, Theor. Found. Chem. Eng., 2013, vol. 47, no. 3, pp. 217–224. https://doi.org/10.1134/S0040579513030081

    Article  CAS  Google Scholar 

  34. Ismagilov, R.S., Rautian, N.A., and Vlasov, V.V., Examples of very unstable linear partial functional differential equations, Cornell University Library. https://doi.org/arxiv.org/abs/1402.4107. Accessed May 11, 2017.

  35. Polyanin, A.D. and Zhurov, A.I., Exact solutions of nonlinear differential-difference equations of a viscous fluid with finite relaxation time, Int. J. Non-Linear Mech., 2013, vol. 57, p.116.

    Article  Google Scholar 

  36. Trofimchuk, E., Pinto, M., and Trofimchuk, S., Traveling waves for a model of the Belousov–Zhabotinsky reaction, J. Differ. Equations, 2013, vol. 254, p. 3690.

    Article  Google Scholar 

  37. Gromov, Yu.Yu., Zemskoi, N.A., Lagutin, A.V., Ivanova, O.G., and Tyutyunnik, V.M., Sistemy avtomaticheskogo upravleniya s zapazdyvaniem: uchebnoe posobie (Automatic Control Systems with Delay: A Textbook), Tambov: Tambov. Gos. Tekh. Univ., 2007.

    Google Scholar 

  38. Baidali, S.A., Dyadik, V.F., and Yurkov, A.S., Mathematical model of the production of uranium hexafluoride, Izv. Tomsk. Politekh. Univ., 2009, vol. 315, no. 2, p.84.

    Google Scholar 

  39. Wu, J., Theory and Applications of Partial Functional Differential Equations, New York: Springer, 1996.

    Book  Google Scholar 

  40. Zubik-Kowal, B., Delay partial differential equations, Scholarpedia, 2008, vol. 3, no. 4, p. 2851.

    Article  Google Scholar 

  41. Kurangyshev, A.V., Dedushkin, A.V., and Kaznacheev, A.V., Effect of signal delay in space communication lines on the control systems of space vehicles, Molodoi Uch., 2016, no. 3, p.135.

    Google Scholar 

  42. Dolgikh, Yu.F. and Surkov, P.G., Matematicheskie modeli dinamicheskikh sistem s zapazdyvaniem: uchebnoe posobie (Mathematical Models of Dynamic Systems with Delay: A Textbook), Yekaterinburg: Ural. Gos. Univ., 2012.

    Google Scholar 

  43. Baker, C.T.H. and Paul, C.A.H., Issues in the numerical solution of evolutionary delay differential equations, Adv. Comput. Math., 1995, vol. 3, p.171.

    Article  Google Scholar 

  44. Shampine, L.F. and Thompson, S., Numerical solutions of delay differential equations, in Delay Differential Equations: Recent Advances and New Directions, New York: Springer, 2009, p.245.

    Google Scholar 

  45. Sun, Z. and Zhang, Z., A linearized compact difference scheme for a class of nonlinear delay partial differential equations, Appl. Math. Modell., 2013, vol. 37, p.742.

    Article  Google Scholar 

  46. Polyanin, A.D. and Zhurov, A.I., Non-linear instability and exact solutions to some delay reaction-diffusion systems, Int. J. Non-Linear Mech., 2014, vol. 62, p.33.

    Article  Google Scholar 

  47. Polyanin, A.D., Sorokin, V.G., and Vyazmin, A.V., Exact solutions and qualitative features of nonlinear hyperbolic reaction–diffusion equations with delay, Theor. Found. Chem. Eng., 2015, vol. 49, no. 5, pp. 622–635. https://doi.org/10.1134/S0040579515050243

    Article  CAS  Google Scholar 

  48. Tian, H., Asymptotic stability of numerical methods for linear delay parabolic differential equations, Comput. Math. Appl., 2008, vol. 56, p. 1758.

    Article  Google Scholar 

  49. Polyanin, A.D. and Zhurov, A.I., Functional constraints method for constructing exact solutions to delay reaction-diffusion equations and more complex nonlinear equations, Commun. Nonlinear Sci. Numer. Simul., 2014, vol. 19, no. 3, p.417.

    Article  Google Scholar 

  50. Polyanin, A.D. and Nazaikinskii, V.E., Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton, Fla.: CRC, 2016, 2nd ed.

    Book  Google Scholar 

  51. Polyanin, A.D. and Sorokin, V.G., Nonlinear delay reaction-diffusion equations: Traveling-wave solutions in elementary functions, Appl. Math. Lett., 2015, vol. 46, p.38.

    Article  Google Scholar 

  52. Polyanin, A.D., Exact solutions to new classes of reactiondiffusion equations containing delay and arbitrary functions, Theor. Found. Chem. Eng., 2015, vol. 49, no. 2, pp. 169–175. https://doi.org/10.1134/S0040579515020104

    Article  CAS  Google Scholar 

  53. Sorokin, V.G., Exact solutions of some nonlinear ordinary differential-difference equations, Vestn. Nats. Issled. Yad. Univ. MIFI, 2015, vol. 4, no. 6, p.493.

    Google Scholar 

  54. Polyanin, A.D. and Zhurov, A.I., The generating equations method: Constructing exact solutions to delay reaction-diffusion systems and other non-linear coupled delay PDEs, Int. J. Non-Linear Mech., 2015, vol. 71, p. 104.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences, Moscow, 119526, Russia

    A. D. Polyanin & V. G. Sorokin

  2. National Research Nuclear University MEPhI, Moscow, 115409, Russia

    A. D. Polyanin

  3. Bauman Moscow State Technical University, Moscow, 105005, Russia

    V. G. Sorokin

  4. Moscow Polytechnical University, Moscow, 107023, Russia

    A. V. Vyazmin

  5. Scientific Research Institute of Rubber Industry, Sergiev Posad raion, Moscow oblast, 141312, Russia

    A. V. Vyazmin

Authors
  1. A. D. Polyanin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. G. Sorokin
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. V. Vyazmin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. D. Polyanin.

Additional information

Original Russian Text © A.D. Polyanin, V.G. Sorokin, A.V. Vyazmin, 2018, published in Teoreticheskie Osnovy Khimicheskoi Tekhnologii, 2018, Vol. 52, No. 3, pp. 278–293.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Polyanin, A.D., Sorokin, V.G. & Vyazmin, A.V. Reaction-Diffusion Models with Delay: Some Properties, Equations, Problems, and Solutions. Theor Found Chem Eng 52, 334–348 (2018). https://doi.org/10.1134/S0040579518030132

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040579518030132

Keywords

Search

Navigation