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Investigation into the Critical Domain Problem for the Reaction-Telegraph Equation Using Advanced Numerical Algorithms

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Abstract

Reaction-telegraph equation (RTE)—a nonlinear partial differential equation of mixed parabolic-hyperbolic type—is believed to be a better mathematical framework to describe population dynamics than the more traditional reaction–diffusion equations. Being motivated by ecological problems such as habitat fragmentation and alien species introduction (biological invasions), here we consider the RTE on a bounded domain with the goal to reveal the dependence of the critical domain size (that separates extinction from persistence) on biologically meaningful parameters of the equation. Since an analytical study does not seem to be possible, we investigate into this critical domain problem by means of computer simulations using an advanced numerical algorithm. We show that the population dynamics described by the RTE is significantly different from those of the corresponding reaction–diffusion equation. The properties of the critical domain are revealed accordingly.

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References

  1. Sornette, D.: Critical Phenomena in Natural Sciences, 2nd edn. Springer, Berlin (2004)

    MATH  Google Scholar 

  2. Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York (2002)

    MATH  Google Scholar 

  3. Serber, R.: The Los Alamos Primer: The First Lectures on How to Build an Atomic Bomb. University of California Press, Oakland (1992)

    Google Scholar 

  4. Tikhonov, A.N., Samarskii, A.A.: Equations of Mathematical Physics. Dover, New York (1990)

    MATH  Google Scholar 

  5. Kierstead, H., Slobodkin, L.B.: The size of water masses containing plankton blooms. J. Mar. Res. 12, 141–147 (1953)

    Google Scholar 

  6. Petrovskii, S., Shigesada, N.: Some exact solutions of a generalized Fisher equation related to the problem of biological invasion. Math. Biosci. 172, 73–94 (2001)

    Article  MathSciNet  Google Scholar 

  7. Fahrig, L.: Effects of habitat fragmentation on biodiversity. Annu. Rev. Ecol. Evol. Syst. 34, 487–515 (2003)

    Article  Google Scholar 

  8. Lamont, B.B., Klinkhamer, P.G., Witkowski, E.: Population fragmentation may reduce fertility to zero in Banksia goodiia: demonstration of the Allee effect. Oecologia 94(3), 446–450 (1993)

    Article  Google Scholar 

  9. Pimentel, D. (ed.): Biological Invasions: Economic and Environmental Costs of Alien Plant, Animal, and Microbe Species. CRC Press, Boca Raton (2002)

    Google Scholar 

  10. Lewis, M.A., Kareiva, P.: Allee dynamics and the spread of invading organisms. Theor. Popul. Biol. 43, 141–158 (1993)

    Article  Google Scholar 

  11. Shigesada, N., Kawasaki, K.: Biological Invasions: Theory and Practice. Oxford University Press, Oxford (1997)

    Google Scholar 

  12. Mangel, M.: The Theoretical Biologists Toolbox: Quantitative Methods for Ecology and Evolutionary Biology. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  13. May, R.M.: Stability and Complexity in Model Ecosystems, vol. 6. Princeton University Press, Princeton (1973)

    Google Scholar 

  14. Maynard Smith, J.: Models in Ecology. Cambridge University Press, Cambridge (1974)

    MATH  Google Scholar 

  15. Lewis, M.A., Petrovskii, S.V., Potts, J.: The Mathematics Behind Biological Invasions. Interdisciplinary Applied Mathematics, vol. 44. Springer, Berlin (2016)

    Book  Google Scholar 

  16. Petrovskii, S.V., Li, B.-L.: Exactly Solvable Models of Biological Invasion, p. 217p. Chapman & Hall/CRC Press, NP (2006)

    MATH  Google Scholar 

  17. Malchow, H., Petrovskii, S.V., Venturino, E.: Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation. CRC Press, Boca Raton (2008)

    MATH  Google Scholar 

  18. Eli, H.E.: Are diffusion models too simple? A comparison with telegraph models of invasion. Am. Nat. 142(5), 779–795 (1993)

    Article  Google Scholar 

  19. Kac, M.: A stochastic model related to the telegrapher equation. Rocky Mt. J. Math. 4, 497–509 (1974)

    Article  MathSciNet  Google Scholar 

  20. Kareiva, P.M., Shigesada, N.: Analyzing insect movement as a correlated random walk. Oecologia 56, 234–238 (1983)

    Article  Google Scholar 

  21. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Time fractional third-order evolution equation: symmetry analysis, explicit solutions, and conservation laws. J. Comput. Nonlinear Dyn. 13, 021011 (2017)

    Article  Google Scholar 

  22. Harris, P.A., Garra, R.: Nonlinear heat conduction equations with memory: physical meaning and analytical results. J. Math. Phys. 58, 063501 (2017)

    Article  MathSciNet  Google Scholar 

  23. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Lie symmetry analysis and conservation laws for the time fractional simplified modified Kawahara equation. Open Phys. 16, 302–310 (2018)

    Article  Google Scholar 

  24. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Optimal system, nonlinear self-adjointness and conservation laws for generalized shallow water wave equation. Open Phys. 16, 364–370 (2018)

    Article  Google Scholar 

  25. Baleanu, D., Inc, M., Yusuf, A., Aliyu, A.I.: Space-time fractional Rosenou–Haynam equation: Lie symmetry analysis, explicit solutions and conservation laws. Adv. Differ. Equ. 2018, 46 (2018)

    Article  MathSciNet  Google Scholar 

  26. Di Crescenzo, A., Martinucci, B., Zacks, S.: Telegraph process with elastic boundary at the origin. Methodol. Comput. Appl. Probab. 20, 333–352 (2018)

    Article  MathSciNet  Google Scholar 

  27. Giusti, A.: Dispersion relations for the time-fractional Cattaneo–Maxwell heat equation. J. Math. Phys. 59, 013506 (2018)

    Article  MathSciNet  Google Scholar 

  28. Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Lie symmetry analysis, explicit solutions and conservation laws for the space–time fractional nonlinear evolution equations. Phys. A 496, 371–383 (2018)

    Article  MathSciNet  Google Scholar 

  29. Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Investigation of the logarithmic-KdV equation involving Mittag–Leffler type kernel with Atangana–Baleanu derivative. Phys. A 506, 520–531 (2018)

    Article  MathSciNet  Google Scholar 

  30. Inc, M., Yusuf, A., Aliyu, A.I., Selahattin, G., Baleanu, D.: Optical solitary wave solutions for the conformable perturbed nonlinear Schrödinger equation with power law nonlinearity. J. Adv. Phys. 7, 49–57 (2018)

    Article  Google Scholar 

  31. Sobolev, S.L.: On hyperbolic heat-mass transfer equation. Int. J. Heat Mass Transf. 122, 629–630 (2018)

    Article  Google Scholar 

  32. Tchier, F., Inc, M., Yusuf, A., Aliyu, A.I., Baleanu, D.: Time fractional third-order variant Boussinesq system: symmetry analysis, explicit solutions, conservation laws and numerical approximations. Eur. Phys. J. Plus 133, 240 (2018)

    Article  Google Scholar 

  33. 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)

    MathSciNet  Google Scholar 

  34. Hajipour, M., Jajarmi, A., Baleanu, D., Sun, H.-G.: On an accurate discretization of a variable-order fractional reaction–diffusion equation. Comm. Nonlinear Sci. Numer. Simul. 69, 119–133 (2019)

    Article  MathSciNet  Google Scholar 

  35. Gelfand, I.M.: Some questions of analysis and differential equations. Uspehi Mat. Nauk 3(87), 3–19 (1959)

    MathSciNet  Google Scholar 

  36. Méndez V., Fedotov S., Horsthemke W.: Reactions and transport: diffusion, inertia, and subdiffusion. In: Reaction-Transport Systems. Springer Series in Synergetics. Springer, Berlin (2010)

    Chapter  Google Scholar 

  37. Murray, J.D., Sperb, R.P.: Minimum domains for spatial patterns in a class of reaction diffusion equations. J. Math. Biol. 18, 169 (1983). https://doi.org/10.1007/BF00280665

    Article  MathSciNet  MATH  Google Scholar 

  38. Strauss, W.A.: Partial Differential Equations: An Introduction, 2nd edn. Wiley, New York (2008). ISBN-13 978-0470-05456-7

    MATH  Google Scholar 

  39. Romeiro, N.M.L., Castro, R.G.S., Malta, S.M.C., Landau, L.: A linearization technique for multi-species transport problems. Transp. Porous Med. 70, 1 (2007). https://doi.org/10.1007/s11242-006-9081-4

    Article  Google Scholar 

  40. Allen, L.J.: An Introduction to Mathematical Biology. Pearson-Prentice Hall, Upper Saddle River (2007)

    Google Scholar 

  41. Thomas, J.W.: Numerical Partial Differential Equations: Finite Difference Methods. Springer, New York (1995)

    Book  Google Scholar 

  42. Scheffer, M., Straile, D., van Nes, E.H., Hosper, H.: Climatic warming causes regime shifts in lake food webs. Limnol. Oceanogr. 46, 17801783 (2001)

    Article  Google Scholar 

  43. Carpenter, S.R., et al.: Early warnings of regime shifts: a whole-ecosystem experiment. Science 332, 10791082 (2011)

    Google Scholar 

  44. Hastings, A., Abbott, K.C., Cuddington, K., Francis, T., Gellner, G., Lai, Y.C., Morozov, A., Petrovskii, S.V., Scranton, K., Zeeman, M.L.: Transient phenomena in ecology. Science 361, eaat6412 (2018)

    Article  Google Scholar 

  45. Alharbi, W., Petrovskii, S.V.: Critical domain problem for the reaction-telegraph equation model of population dynamics. Mathematics 6, 59 (2018)

    Article  Google Scholar 

  46. Hillen, T.: Existence theory for correlated random walks on bounded domains. Can. Appl. Math. Q. 18, 1–40 (2010)

    MathSciNet  MATH  Google Scholar 

  47. Hillen, T., Swan, A.: The diffusion limit of transport equations in biology. In: Preziosi, L., Chaplain, M., Pugliese, A. (eds.) Mathematical Models and Methods for Living Systems. Lecture Notes in Mathematics, vol. 2167. Springer, Cham (2016)

    MATH  Google Scholar 

  48. Tilles, P.F.C., Petrovskii, S.V.: On the consistency of the reaction-telegraph process in finite domains. In: preparation

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Acknowledgements

This work was supported by the Universidade Estadual de Londrina-BR and Brazilian Federal Agency for Support and Evaluation of Graduate Education—CAPES (Eliandro R. Cirilo/PROGRAMA DE POS-DOC—Grant No. 88881.120111/2016-01). This study was done during the research visit of E.R.C. to the University of Leicester (UK). E.R.C. therefore expresses his gratitude to the University of Leicester for the hospitality and supportive research environment. The publication has been prepared with the support of the “RUDN University Program 5-100” (to S.P.).

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Authors and Affiliations

  1. Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK

    Sergei Petrovskii

  2. Departamento de Matemática, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, PR-445, km 380, Londrina, PR, 86051-990, Brazil

    Eliandro Cirilo, Neyva Romeiro & Paulo Natti

  3. Peoples Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya St, Moscow, 117198, Russian Federation

    Sergei Petrovskii

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  1. Eliandro Cirilo
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Correspondence to Eliandro Cirilo.

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Cirilo, E., Petrovskii, S., Romeiro, N. et al. Investigation into the Critical Domain Problem for the Reaction-Telegraph Equation Using Advanced Numerical Algorithms. Int. J. Appl. Comput. Math 5, 54 (2019). https://doi.org/10.1007/s40819-019-0633-z

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