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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

The Spatial Pattern Dynamics of Reaction-Diffusion Instability in Tumor Cell--Immune Cell System

Journal of Applied Nonlinear Dynamics 10(2) (2021) 245--261 | DOI:10.5890/JAND.2021.06.005

Md. Nazmul Hasan$^1$ , Khan Rubayat Rahaman$^2$, Sabbir Janee$^3$

$^1$ Department of Mathematics, Jahangirnagar University, Savar, Dhaka, Bangladesh

$^2$ St. Marys University, 923 Robie Street, Halifax, NS, Canada B3H 3C3

$^3$ Department of Biotechnology and Genetic Engineering, Jahangirnagar University, Savar, Dhaka, Bangladesh

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Abstract

In this paper, spatial patterns of a diffusive Tumor cell and immune cell model with sigmoid ratio-dependent functional response are investigated. The asymptotic stability behavior of the corresponding nonspatial model around the unique positive interior equilibrium point in homogeneous steady state is obtained. We have obtained the optimal condition under which the system loses stability and a Turing pattern occurs. Numerical simulations have been carried out in order to show the significant role of reaction-diffusion coefficients and other important parameters of the system. Various contour figures of spatial patterns through Turing instability are portrayed and analyzed in order to substantiate the applicability of the present model. The paper ends with an extended discussion of biological implications of the immune system.

References

  1. [1]  Liu, H. and Wang, W. (2010), The Amplitude Equations of an Epidemic Model, Science Technology and Engineering, 10(8), 1929--1933.
  2. [2]  Dutt, A.K. (2012), Amplitude equation for a diffusion-reaction system: The reversible Selkov model, AIP advances, 2(4), 1-24.
  3. [3]  Zheng, Q.Q. and Shen, J.W. (2015), Dynamics and pattern formation in a cancer network with diffusion, Communications in Nonlinear Science and Numerical Simulation, 27(1), 93--109.
  4. [4]  Sun, G.Q., Huang, C.H., and Wu, Z.Y. (2017), Pattern dynamics of a Gierer-Meinhardt model with spatial effects, Nonlinear Dynamics, 88(2), 1385-1396.
  5. [5]  Sun, G.Q., Wang, S.L., Ren, Q., Jin, Z., and Wu, Y.P. (2015), Effects of time delay and space on herbivore dynamics: linking inducible defenses of plants to herbivore outbreak, Scientific Reports, 5, 112--146.
  6. [6]  Li, L., Jin, Z., and Li, J. (2016), Periodic solutions in a herbivore-plant system with time delay and spatial diffusion, Applied Mathematical Modelling, 40, 4765--4777.
  7. [7]  Li, L. (2015), Patch invasion in a spatial epidemic model, Applied Mathematics and Computation, 258, 342--349.
  8. [8]  Lee, I. and Cho, U.I. (2000), Pattern Formations with Turing and Hopf Oscillating Pattern, Bull. Korean Chem. Soc., 21(12), 1213--1216.
  9. [9]  Maini, P.K., Woolley, T.E., Baker, R.E., Gaffney, E.A., and Lee, S.S. (2012), Turing model for biological pattern formation and the robustness problem, Interface Focus, 2, 487--496.
  10. [10]  Vanag, V.K. and Epstein, I.R. (2009), Cross-diffusion and pattern formation in reaction-diffusion systems, Physical Chemistry Chemical Physics, 11, 897-912.
  11. [11]  Fanelli, D., Cianci, C., and Patti, F.D. (2013), Turing instabilities in reaction-diffusion systems with cross diffusion, Eur.Phys. J. B, 86(142), 1--8.
  12. [12]  Shi, J., Xie, Z., and Little, K. (2011), Cross-diffusion induced instability and stability in reaction-diffusion systems, Journal of Applied Analysis and Computation, 1(1), 95--119.
  13. [13]  Thattai, M. and Oudenaarden, V. (2001), Intrinsic noise in gene regulatory networks, Proc Natl Acad Sci USA, 98(15), 8614--8619.
  14. [14]  Othmer, H. and Scriven, L. (1971), Instability and dynamic pattern in cellular networks, J. Theor. Biol., 32(507), 507--537.
  15. [15]  Nakao, H. and Mikhailov, A. (2010), Turing patterns in network-organized activator-inhibitor systems, Nature Physics, 6, 544--550.
  16. [16]  Nakao, H. and Mikhailov, A. (2008), Turing patterns on networks, arxiv:0807.1230v1 [nlin.PS].
  17. [17]  Hata, S., Nakaoand, H., and Mikhailov, A.S. (2012), Global feedback control of Turing patterns in network-organized activator-inhibitor systems, Europhysics Letters, 98, 64004.
  18. [18]  Coron, J.M.(2007), Control and Nonlinearity, American Mathematical Society. USA.
  19. [19]  Maidi, A. and Corriou, J.P. (2015), Controllability of Nonlinear Diffusion System, Can. J. Chem. Eng, 93, 427-431.
  20. [20]  Popescu, M. and Dumitrache, A. (2011), Stabilization of feedback control and stabilizability optimal solution for nonlinear quadratic problems, Commun Nonlinear Sci Numer Simulat, 16, 2319--2327.
  21. [21]  Udwadia, F.E. and Koganti, P.B. (2015), Optimal stable control for nonlinear dynamical systems: an analytical dynamics based approach, Nonlinear Dyn, 82, 547-562.
  22. [22]  Skandari, M.H.N. (2015), On the stability of a class of nonlinear control systems, Nonlinear Dyn, 80, 1245--1256.
  23. [23]  Saldi, N., Y\"{u}ksel, S., and Linder, T. (2016), Near optimality of quantized policies in stochastic control under weak continuity conditions, Journal of Mathematical Analysis and Applications, 435, 321-337.
  24. [24]  Liu, K., Fridman, E., and Johansson, K.H. (2015), Networked Control With Stochastic Scheduling, IEEE Transactions On Automatic Control, 60(11), 3071-3076.
  25. [25]  Avron, E., Kenneth, O., Retzker, A., and Shalyt, M. (2015), Lindbladians for controlled stochastic Hamiltonians, New Journal of Physics, 17, 043009.
  26. [26]  Grasso, P., Gangolli, S., and Gaunt, I. (2002), Essentials of Pathology for Toxicologists. CRC Press.
  27. [27]  Alberts, B., Johnson, A., Lewis, J., Raff, M., Roberts, K., and Walter, P. (2002), Molecular Biology of the Cell, New York and London: Garland Science.
  28. [28]  Cota, A.M. and Midwinter, M.J. (2015), The immune system, Anaesthesia and Intensive Care Medicine, 16(7), 353--355.
  29. [29]  Wood, P.J. (2015), Immunological response to infection: inflammatory and adaptive immune responses, Anaesthesia and Intensive Care Medicine, 16(7), 349-352.
  30. [30]  Shen, J. and Jung, Y.M. (2005)m Geometric and stochastic analysis of reaction-diffusion patterns, Int. J. Pure Appl. Math., 19, 195--244.
  31. [31]  Wang, Y. and Wang, J. (2012), Influence of prey refuge on predator-prey dynamics, Nonlinear Dyn., 67(2012), 191--201.
  32. [32]  Jordehi, A.R. (2015), A chaotic artificial immune system optimisation algorithm for solving global continuous optimisation problems, Neural Comput and Applic.
  33. [33]  Tolerance, M.P. (1994, Danger and the Extended Family, Annual reviews of immunology, 12(1), 991--1045.
  34. [34]  Mccoy, D.F. (1997), Artificial Immune Systems and Aerial Image Segmentation, IEEE Int. Conf. Systems. Man and Cybernetics, Flordia. USA.
  35. [35]  Kuo, R.J., Chiang, N.J., and Chen, Z.Y. (2014), Integration of artificial immune system and K-means algorithm for customer clustering, Applied Artificial Intelligence, 28, 577--596.
  36. [36]  Iasson, K., Panagiotis, D.C., and Prodromos, D. (1999), Dynamics of a reaction-diffusion system with Brusselator kinetics under feedback control, Physical Review E, 59(1), 372-380.
  37. [37]  Soh, S., Byrska, M., Kandere-Grzybowska, K., and Grzybowski, B.A. (2010), Reaction-Diffusion Systems in Intracellular Molecular Transport and Control, Angewandte Chemie International Edition, 49, 4170--4198.
  38. [38]  Wang, J.M., Su, L.L., and Li, I,X. (2013), Control of a reaction-diffusion PDE cascaded with a heat equaiton, Americal Control Conference, Washington DC. USA.
  39. [39]  Santu, G. and Swarup, P. (2016), Pattern formation and control of spatiotemporal chaos in a reaction diffusion prey-predator system supplying additional food, Chaos: Solitons and Fractals, 85, 57-67.
  40. [40]  Vanag, V.K. and Epstein, I.R. (2008), Design and control of patterns in reaction-diffusion systems, Chaos, 18, 026107.
  41. [41]  Sun, G.Q., Jusup, M., Jin, Z., Wang, Y., and Wang, Z. (2016), Pattern transitions in spatial epidemics: Mechanisms and emergent properties, Physics of Life Reviews, 19, 43-73.
  42. [42]  Giles, H., Kevin, K.L., and Bruce, R. (2012), Control Theory and Experimental Design in Diffusion Processes; arxiv:1210.3739.
  43. [43]  Bather, J.A. (1969), Diffusion Models in Stochastic Control Theory, Journal of the Royal Statistical Society, Series A (General); 132(3), 335-352.
  44. [44]  Morton, R. (1974), On the control of diffusion processes, Journal of Optimization Theory and Applications, 14(2), 151--162.
  45. [45]  Kitano, H. (2004), Biological robustness, Nat Rev Genet, 5, 826-837.
  46. [46]  Kourtidis, A., Ngok, S.P., et al. (2015), Distinct E-cadherin-based complexes regulate cell behavior through miRNA processing or Src and p120 catenin activity, Nature Cell Biology, 17(9), 1145--1157.
  47. [47]  Chrobak, J.M. and Herrero, H. (2011), A mathematical model of induced cancer-adaptive immune system competition, Journal of Biological Systems, 19(3), 521-532.
  48. [48]  Han, Z.X. (2014), Modern control theory and the implement of matlab, Publishing house of electronics industry. China.
  49. [49]  Haque, M. and Venturino, E. (2007), An ecoepidemiological model with disease in predator: the ratio-dependent case, Mathematical Methods in the Applied Sciences, 30(14), 1791--1809.
  50. [50]  Bazykin, A.D., Khibnik, A.I., and Krauskopf, B. (1998), nonlinear Dynamics of Interacting Population, vol. II, World Scientific publishing Company Incorporated.

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