Journal of Statistical Physics

, Volume 177, Issue 4, pp 569–587 | Cite as

On the Consistency of the Reaction-Telegraph Process Within Finite Domains

  • Paulo F. C. Tilles
  • Sergei V. PetrovskiiEmail author


Reaction-telegraph equation (RTE) is a mathematical model that has often been used to describe natural phenomena, with specific applications ranging from physics to social sciences. In particular, in the context of ecology, it is believed to be a more realistic model to describe animal movement than the more traditional approach based on the reaction-diffusion equations. Indeed, the reaction-telegraph equation arises from more realistic microscopic assumptions about individual animal movement (the correlated random walk) and hence could be expected to be more relevant than the diffusion-type models that assume the simple, unbiased Brownian motion. However, the RTE has one significant drawback as its solutions are not positively defined. It is not clear at which stage of the RTE derivation the realism of the microscopic description is lost and/or whether the RTE can somehow be ‘improved’ to guarantee the solutions positivity. Here we show that the origin of the problem is twofold. Firstly, the RTE is not fully equivalent to the Cattaneo system from which it is obtained; the equivalence can only be achieved in a certain parameter range and only for the initial conditions containing a finite number of Fourier modes. Secondly, the Dirichlet type boundary conditions routinely used for reaction-diffusion equations appear to be meaningless if used for the RTE resulting in solutions with unrealistic properties. We conclude that, for the positivity to be regained, one has to use the Cattaneo system with boundary conditions of Robin type or Neumann type, and we show how relevant classes of solutions can be obtained.


Reaction-telegraph equation Reaction-Cattaneo system Nonnegativity Robin boundary conditions 



This work was supported by The Royal Society (UK) through the Grant No. NF161377 (to P.F.C.T and S.V.P.). The publication has been prepared with the support of the “RUDN University Program 5-100” (to S.V.P.).


  1. 1.
    Hastings, A.: An ecological theory journal at last. Theor. Ecol. 1, 1–4 (2008)Google Scholar
  2. 2.
    Hastings, A.: Population Biology: Concepts and Models. Springer-Verlag, New York (1997)zbMATHGoogle Scholar
  3. 3.
    Kot, M.: Elements of Mathematical Ecology. Cambridge University Press, Cambridge (2001)zbMATHGoogle Scholar
  4. 4.
    Mangel, M.: The Theoretical Biologist’s Toolbox: Quantitative Methods for Ecology and Evolutionary Biology. Cambridge University Press, Cambridge (2006)Google Scholar
  5. 5.
    Smith, J.M.: Models in Ecology. Cambridge University Press, Cambridge (1974)zbMATHGoogle Scholar
  6. 6.
    Pyke, G.H.: Understanding movements of organisms: it’s time to abandon the Levy foraging hypothesis. Methods in Ecology and Evolution 6, 1–16 (2015)Google Scholar
  7. 7.
    Kareiva, P.M.: Local movement in herbivorous insecta: applying a passive diffusion model to mark-recapture field experiments. Oecologia 57, 322–327 (1983)ADSGoogle Scholar
  8. 8.
    Bearup, D., Benefer, C.M., Petrovskii, S.V., Blackshaw, R.: Revisiting Brownian motion as a description of animal movement: a comparison to experimental movement data. Methods Ecol. Evol. 7, 1525–1537 (2016)Google Scholar
  9. 9.
    Hastings, A., Cuddington, K., Davies, K.F., Dugaw, C.J., Elmendorf, S., Freestone, A., Harrison, S., Holland, M., Lambrinos, J., Malvadkar, U., Melbourne, B.A., Moore, K., Taylor, C., Thomson, D.: The spatial spread of invasions: new developments in theory and evidence. Ecol. Lett. 8, 91–101 (2005)Google Scholar
  10. 10.
    Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford University Press, Oxford (1997)Google Scholar
  11. 11.
    Skellam, J.G.: Random dispersal in theoretical populations. Biometrika 38, 196–218 (1951)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hastings, A., Harisson, S., McCann, K.: Unexpected spatial patterns in an insect outbreak match a predator diffusion model. Proc. R. Soc. Lond. B 264, 1837–1840 (1997)ADSGoogle Scholar
  13. 13.
    Malchow, H., Petrovskii, S.V., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, Simulations. CRC Press, Boca Raton (2008)zbMATHGoogle Scholar
  14. 14.
    Meron, E.: Nonlinear Physics of Ecosystems. CRC Press, Boca Raton (2015)zbMATHGoogle Scholar
  15. 15.
    Kareiva, P.M., Shigesada, N.: Analyzing insect movement as a correlated random walk. Oecologia 56, 234–238 (1983)ADSGoogle Scholar
  16. 16.
    Othmer, H.G., Dunbar, S.R., Alt, W.: Models of dispersal in biological systems. J. Math. Biol. 26, 263–298 (1988)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Goldstein, S.: On diffusion by discontinuous movements, and on the telegraph equation. Q. J. Mech. Appl. Math. 4, 129–156 (1951)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Kac, M.: A stochastic model related to the telegraph’s equation. Rocky Mt. J. Math. 4, 497–509 (1956)Google Scholar
  19. 19.
    Joseph, D.D., Preziosi, L.: Heat waves. Rev. Mod. Phys. 61, 41–73 (1989)ADSMathSciNetzbMATHGoogle Scholar
  20. 20.
    Mendez, V., Fedotov, S., Horsthemke, W.: Reaction-Transport Systems: Mesoscopic Foundations, Fronts, and Spatial Instabilities. Springer, Berlin (2010)Google Scholar
  21. 21.
    Weiss, G.H.: Some applications of persistent random walks and the telegrapher’s equation. Physica A 311, 381–410 (2002)ADSMathSciNetzbMATHGoogle Scholar
  22. 22.
    Masoliver, J., Lindenberg, K.: Continuous time persistent random walk: a review and some generalizations. Eur. Phys. J. B 90, 107 (2017)ADSMathSciNetGoogle Scholar
  23. 23.
    Angelani, L.: Run-and-tumble particles, telegrapher’s equation and absorption problems with partially reflecting boundaries. J. Phys. A 48, 495003 (2015)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Malakar, K., Jemseena, V., Kundu, A., Kumar, K.V., Sabhapandit, S., Majumdar, S.N., Redner, S., Dhar, A.: Steady state, relaxation and first-passage properties of a run-and-tumble particle in one-dimension. J. Statis. Mech. 2018, 043215 (2018)MathSciNetGoogle Scholar
  25. 25.
    Evans, M.R., Majumdar, S.N.: Run and tumble particle under resetting: a renewal approach. J. Phys. A 51, 475003 (2018)ADSMathSciNetzbMATHGoogle Scholar
  26. 26.
    Dhar, A., Kundu, A., Majumdar, S.N., Sabhapandit, S., Schehr, G.: Run-and-tumble particle in one-dimensional confining potential: steady state, relaxation and first passage properties. Phys. Rev. E 99, 032132 (2019)ADSGoogle Scholar
  27. 27.
    Le doussal, P., Majumdar, S.N., Schehr, G.: Non-crossing run and tumble particles on a line. Phys. Rev. E 100, 012113 (2019)ADSGoogle Scholar
  28. 28.
    Berg, H.C.: E. coli in Motion. Springer, Berlin (2014)Google Scholar
  29. 29.
    Hadeler, K.P.: Reaction transport systems in biological modelling. In Mathematics Inspired by Biology, pp. 95–150. Springer, Berlin (1999)zbMATHGoogle Scholar
  30. 30.
    Holmes, E.E.: Are diffusion models too simple? A comparison with telegraph models of invasion. Am. Nat. 142, 779–795 (1993)Google Scholar
  31. 31.
    Dunbar, S.R., Othmer, H.G.: On a nonlinear hyperbolic equation describing transmission lines, cell movement, and branching random walks. In Nonlinear Oscillations in Biology and Chemistry, pp. 274–289. Springer, Berlin (1986)zbMATHGoogle Scholar
  32. 32.
    Dunbar, S.R.: A branching random evolution and a nonlinear hyperbolic equation. SIAM J. Appl. Math. 48, 1510–1526 (1988)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Mainardi, F.: Signal velocity for transient waves in linear dissipative media. Wave Motion 5, 33–41 (1983)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Sobolev, S.L.: Transport processes and travelling waves in systems with local nonequilibrium. Sov. Phys. Usp. 34(3), 217–229 (1991)ADSGoogle Scholar
  35. 35.
    Lakestani, M., Saray, B.N.: Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appl. 60, 1964–1972 (2010)MathSciNetzbMATHGoogle Scholar
  36. 36.
    Gilding, B.H., Kersner, R.: Wavefront solutions of a nonlnear telegraph equation. J. Differ. Equ. 254, 599–636 (2013)ADSzbMATHGoogle Scholar
  37. 37.
    Buono, P.-L., Eftimie, R.: Symmetries and pattern formation in hyperbolic versus parabolic models of self-organised aggregation. J. Math. Biol. 71, 847–881 (2015)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Artale Harris, P., Garra, R.: Nonlinear heat conduction equations with memory: physical meaning and analytical results. J. Math. Phys. 58, 063501 (2017)ADSMathSciNetzbMATHGoogle Scholar
  39. 39.
    Di Crescenzo, A., Martinucci, B., Zacks, S.: Telegraph process with elastic boundary at the origin. Methodol. Comput. Appl. Prob. 20, 333–352 (2018)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Giusti, A.: Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation. J. Math. Phys. 59, 013506 (2018)ADSMathSciNetzbMATHGoogle Scholar
  41. 41.
    Sobolev, S.L.: On hyperbolic heat-mass transfer equation. Int. J. Heat & Mass Transfer 122, 629–630 (2018)Google Scholar
  42. 42.
    Hajipour, M., Jajarmi, A., Malek, A., Baleanu, D.: Positivity-preserving sixth-order implicit finite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation. Appl. Math. Comput. 325, 146–158 (2018)MathSciNetGoogle Scholar
  43. 43.
    Niwa, H.: Migration dynamics of fish schools in heterothermal environments. J. Theor. Biol. 193, 215–231 (1998)Google Scholar
  44. 44.
    Murray, A.G., O’Callaghan, M., Jones, B.: Simple models of massive epidemics of herpesvirus in Australian (and New Zealand) pilchards. Environ. Int. 27, 243–248 (2001)Google Scholar
  45. 45.
    Ortega-Cejas, V., Fort, J., Mendez, V.: The role of the delay time in the modeling of biological range expansions. Ecology 85, 258–264 (2004)Google Scholar
  46. 46.
    Hillen, T.: Existence theory for correlated random walks on bounded domains. Can. Appl. Math. Q. 18, 1–40 (2010)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Hillen, T., Swan, A.: The diffusion limit of transport equations in biology. In Mathematical models and methods for living systems, pp. 73–129. Springer, Berlin (2014)zbMATHGoogle Scholar
  48. 48.
    Garabedian, P.R.: Partial Differential Equations. AMS Chelsea Publishing, Providence (1998)zbMATHGoogle Scholar
  49. 49.
    Alharbi, W.G., Petrovskii, S.V.: Critical domain problem for the reaction-telegraph equation model of population dynamics. Mathematics 6, 59 (2018)zbMATHGoogle Scholar
  50. 50.
    Cirilo, E., Petrovskii, S.V., Romeiro, N., Natti, P.: Investigation into the critical domain problem for the reaction-telegraph equation using advanced numerical algorithms. Int. J. Appl. Comput. Math. 5, 54 (2019)MathSciNetzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of LeicesterLeicesterUK
  2. 2.Departamento de MatemáticaUniversidade Federal de Santa MariaSanta MariaBrazil
  3. 3.Peoples Friendship University of Russia (RUDN University)MoscowRussian Federation

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