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Spatiotemporal dynamics of reaction-diffusion equations modeling predator-prey interactions using DUNE-PDELab

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Abstract.

Reaction-diffusion systems (RDS) help us to understand the distribution of the concentration of substances in space or time under the influence of two phenomena: local chemical reactions where the substances are transmuted into one another, and diffusion which causes the substances to spread out over a surface in space. Analytical solutions for these behavioral models are still lacking, therefore it is essential to use some numerical methods to find approximate solution to these systems as the numerical results will allow us the use of parameters fitting. This study focuses on the numerical approximation based on finite-element scheme together with the powerful DUNE-PDELab package to investigate the spatio-temporal dynamics of the structure of predator-prey diffusive model. The scheme is introduced for approximation of predator-prey with Holling type-II functional response, while the growth is logistic, subject to some appropriate initial and boundary conditions. The simulations were obtained in the two-dimensional case, which demonstrate that the initial and boundary conditions play an important role in identifying the spatio-temporal dynamics of predator-prey interactions.

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References

  1. M. Chaplain, M. Ganesh, I. Graham, J. Math. Biol. 42, 387 (2001)

    MathSciNet  Google Scholar 

  2. J. Murray, Phil. Trans. R. Soc. London B 295, 473 (1981)

    ADS  Google Scholar 

  3. E. Bassiri, R. Bryant, M. George, K. Yerion, Nonlinear Anal. Real. World Appl. 10, 1730 (2009)

    MathSciNet  Google Scholar 

  4. A. Holden, M. Markus, H. Othmer (Editors), Nonlinear Wave Processes in Excitable Media (Plenum, New York, 1991)

    Google Scholar 

  5. P. Bittihn, A. Squires, G. Luther, E. Bodenschatz, V. Krinsky, U. Parlitz, S. Luther, Phil. Trans. R. Soc. A 368, 2221 (2010)

    ADS  Google Scholar 

  6. W. Gurney, A. Veitch, I. Cruickshank, G. McGeachin, Ecology 79, 2516 (1998)

    Google Scholar 

  7. A. New Jersey Medvinsky, S. Petrovskii, I. Tikhonova, H. Malchow, B.-L. Li, SIAM Rev. 44, 311 (2002)

    ADS  MathSciNet  Google Scholar 

  8. R. Cantrell, C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, in Wiley Series in Mathematical and Computational Biology (Wiley, Chichester, 2003)

  9. H. Malchow, S. Petrovskii, E. Venturino, Spatiotemporal Patterns in Ecology and Epidemiology: Theory, Models, and Simulation, in Mathematical and Computational Biology, Vol. 17 (Chapman and Hall/CRC, Boca Raton, 2008)

  10. R. May, Stability and Complexity in Model Ecosystems (Princeton University Press, 1974)

  11. J. Sherratt, X. Lambin, T. Sherratt, Am. Nat. 162, 503 (2003)

    Google Scholar 

  12. J. Sherratt, B. Eagan, M. Lewis, Phil. Trans. R. Soc. London B 352, 21 (1997)

    ADS  Google Scholar 

  13. J. Sherratt, X. Lambin, C. Thomas, T. Sherratt, Proc. R. Soc. London B 269, 327 (2002)

    Google Scholar 

  14. J. Sherratt, M. Smith, J. R. Soc. Interface 5, 483 (2008)

    Google Scholar 

  15. M. Smith, J. Sherratt, N. Armstrong, Proc. R. Soc. A 464, 365 (2008)

    ADS  Google Scholar 

  16. A. Yun, J. Shin, Y. Li, J. Kim, Int. J. Bifurc. Chaos 25, 1550117 (2011)

    Google Scholar 

  17. Y. Ding, M. Kawahara, Comput. Methods Appl. Mech. Eng. 156, 375 (1998)

    ADS  Google Scholar 

  18. H. Malchow, F. Hilker, S. Petrovskii, K. Brauer, Ecol Complex 1, 211 (2004)

    Google Scholar 

  19. F. Hilker, H. Malchow, M. Langlais, S. Petrovskii, Ecol. Complex 3, 200 (2006)

    Google Scholar 

  20. H. Malchow, B. Radtke, M. Kallache, A. Medvinsky, D. Tikhonov, S. Petrovskii, Nonlinear Anal. Real World Appl. 1, 53 (2000)

    MathSciNet  Google Scholar 

  21. H. Malchow, S. Petrovskii, A. Medvinsky, Ecol. Model. 149, 247 (2002)

    Google Scholar 

  22. M. Smith, J. Rademacher, J. Sherratt, SIAM J. Appl. Dyn. Syst. 8, 1136 (2009)

    ADS  MathSciNet  Google Scholar 

  23. R. Cantrell, C. Cosner, W. Fagan, J. Math. Biol. 37, 491 (1998)

    MathSciNet  Google Scholar 

  24. J. Kaipio, J. Tervo, M. Vauhkonen, Math. Comput. Simul. 39, 155 (1995)

    Google Scholar 

  25. L. Shangerganesh, K. Balachandran, Acta. Appl. Math. 116, 71 (2011)

    MathSciNet  Google Scholar 

  26. A. Leung, Indiana Univ. Math. J. 31, 223 (1982)

    MathSciNet  Google Scholar 

  27. P. Ciarlet, The Finite Element Method for Elliptic Problems, in Studies in Mathematics and Its Applications, Vol. 4 (North-Holland, Amsterdam, 1979)

  28. M. Garvie, J. Burkardt, J. Morgan, Bull. Math. Biol. 77, 548 (2015)

    MathSciNet  Google Scholar 

  29. G. Strang, G. Fix, An Analysis of the Finite Element Method (Prentice-Hall, Englewood Cliffs, 1973)

  30. V. Thomee, Galerkin finite element methods for parabolic problems, 2nd ed., in Springer Series in Computational Mathematics, Vol. 25 (Springer, Berlin, 2006)

  31. S. Brenner, R. Scott, The Mathematical Theory of Finite Element Methods (Springer, 1994)

  32. E. Holmes, M. Lewis, J. Banks, R. Veit, Ecology 75, 17 (1994)

    Google Scholar 

  33. H. Meinhardt, Models of Biological Pattern Formation (Academic Press, New York, 1982)

  34. J. Murray, Mathematical Biology I: An Introduction, 3rd edition, in Interdisciplinary Applied Mathematics, Vol. 17 (Springer, New York, 2002)

  35. J. Murray, Mathematical Biology II: Spatial Models and Biomedical Applications, 3rd edition, in Interdisciplinary Applied Mathematics, Vol. 18 (Springer, New York, 2003)

  36. M. Pascual, Proc. R. Soc. London B 251, 1 (1993)

    ADS  Google Scholar 

  37. E. Peacock-Lopez, Nonlinear Dyn. Psychol. Life Sci. 15, 1 (2011)

    MathSciNet  Google Scholar 

  38. L. Segel, J. Jackson, J. Theor. Biol. 37, 545 (1972)

    Google Scholar 

  39. D.P. Kareiva, Spatial ecology: the role of space in population dynamics and inter specific interactions, in Monographs in Population Biology, Vol. 30 (Princeton University Press, Princeton, 1997)

  40. W. Fagan, R. Cantrell, C. Cosner, Am. Nat. 153, 165 (1999)

    Google Scholar 

  41. J. Dawes, M. Souza, J. Theor. Biol. 327, 11 (2013)

    Google Scholar 

  42. G. Maciel, F. Lutscher, Am. Nat. 182, 42 (2013)

    Google Scholar 

  43. M. Garvie, J. Blowey, Eur. J. Appl. Math. 16, 621 (2005)

    Google Scholar 

  44. M. Garvie, Bull. Math. Biol. 69, 931 (2007)

    MathSciNet  Google Scholar 

  45. M. Garvie, C. Trenchea, J. Biol. Dyn. 4, 559 (2010)

    MathSciNet  Google Scholar 

  46. Marcus R. Garvie, Jeff Morgan, Vandana Sharma, Comput. Math. Appl. 74, 934 (2017)

    MathSciNet  Google Scholar 

  47. V. Jansen, Theor. Popul. Ecol. 59, 119 (2001)

    Google Scholar 

  48. C. Johnson, Numerical Solution of Partial Differential Equations by the Finite Element Method (Dover, 2009)

  49. L. Malpas, R. Kennerley, G. Hirons, R. Sheldon, M. Ausden, J. Gilbert, J. Smart, J. Nat. Conserv. 21, 37 (2013)

    Google Scholar 

  50. M. Garvie, C. Trenchea, Numer. Math. 107, 641 (2007)

    MathSciNet  Google Scholar 

  51. C. Holling, Can. Entomol. 91, 385 (1959)

    Google Scholar 

  52. J. Sherratt, Ecol. Lett. 4, 30 (2001)

    Google Scholar 

  53. J. Sherratt, SIAM J. Appl. Math. 63, 1520 (2003)

    MathSciNet  Google Scholar 

  54. J. Sherratt, IMA J. Appl. Math. 73, 759 (2008)

    ADS  MathSciNet  Google Scholar 

  55. J. Smoller, Shock Waves and Reaction-Diffusion Equations, in Grundlehren der mathematis-chen wissenschaften, Vol. 258 (Springer, New York, 1983)

  56. R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, in Applied Mathematical Sciences, Vol. 68 (Springer, New York, 1997)

  57. Geuzaine et al., Int. J. Numer. Methods Eng. 79, 1309 (2009)

    Google Scholar 

  58. W. Hackbusch, Theorie und Numerik elliptischer Differential gleichungen (Teubner, 1986)

  59. K. Eriksson, D. Estep, P. Hansbo, C. Johnson, Computational Differential Equations (1996)

  60. R. Adams, Sobolev Spaces: Pure and Applied Mathematics (Academic Press, New York, 1975)

  61. C. Grossmann, H.-G. Roos, M. Stynes, Numerical Treatment of Partial Differential Equations (Springer, Berlin, 2007)

    MATH  Google Scholar 

  62. S. Larsson, V. Thomée, Partial Differential Equations with Numerical Methods, in Texts in Applied Mathematics, Vol. 45 (Springer, Berlin, 2009)

  63. P. Bastian, Lectures Notes on Scientific Computing with Partial Differential Equations (2014) https://conan.iwr.uni-heidelberg.de/data/teaching/finiteelements_ws2017/num2.pdf

  64. A. Ern, J.-L. Guermond, Theory and Practice of Finite Element Methods (Springer, 2004)

  65. Ciarlet, The finite element method for elliptic problems (SIAM, 2002)

  66. D. Braess, Finite Elemente (Springer, 2003)

  67. H. Elman, D. Silvester, A. Wathen, Finite Elements and Fast Iterative Solver (Oxford University Press, 2005)

  68. R. Rannacher, Numerische Mathematik II Numerik partieller Differentialgleichungen (2000)

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Authors and Affiliations

  1. Department of Mathematics, COMSATS University, Park Road Chak Shahzad, Islamabad, Pakistan

    Noaman Khan

  2. Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Hasa, Saudi Arabia

    Ishtiaq Ali

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  1. Noaman Khan
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  2. Ishtiaq Ali
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Correspondence to Ishtiaq Ali.

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Khan, N., Ali, I. Spatiotemporal dynamics of reaction-diffusion equations modeling predator-prey interactions using DUNE-PDELab. Eur. Phys. J. Plus 134, 574 (2019). https://doi.org/10.1140/epjp/i2019-12920-7

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