# American Institute of Mathematical Sciences

June  2019, 12(3): 591-613. doi: 10.3934/dcdss.2019038

## Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics

 Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, 7535, South Africa

* Corresponding author: mkowolax@yahoo.com (K. M. Owolabi)

Received  May 2017 Revised  September 2017 Published  September 2018

Fund Project: The research contained in this report is supported by South African National Research Foundation.

In this paper, we extend a system of coupled first order non-linear system of delay differential equations (DDEs) arising in modeling of stoichiometry of tumour dynamics, to a system of diffusion-reaction system of partial delay differential equations (PDDEs). Since tumor cells are further modified by blood supply through the vascularization process, we determine the local uniform steady states of the homogeneous tumour growth model with respect to the vascularization process. We show that the steady states are globally stable, determine the existence of Hopf bifurcation of the homogeneous tumour growth model with respect to the vascularization process. We derive, analyse and implement a fitted operator finite difference method (FOFDM) to solve the extended model. This FOFDM is analyzed for convergence and we observe seen that it has second-order accuracy. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.

Citation: Kolade M. Owolabi, Kailash C. Patidar, Albert Shikongo. Efficient numerical method for a model arising in biological stoichiometry of tumour dynamics. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 591-613. doi: 10.3934/dcdss.2019038
##### References:
 [1] G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199.  doi: 10.1016/S0377-0427(00)00468-4. [2] R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, USA, 2011. [3] J. C. Butcher, Implicit Runge-Kutta processes, Mathematics of Computation, 18 (1964), 50-64.  doi: 10.1090/S0025-5718-1964-0159424-9. [4] M. Chen, M. Fan and Y. Kuang, Global dynamics in a stoichiometric food chain model with two limiting nutrients, Mathematical Biosciences, 289 (2017), 9-19.  doi: 10.1016/j.mbs.2017.04.004. [5] M. Cherif, Stoichiometry and population growth in osmotrophs and non-osmotrophs, John Wiley & Sons, Ltd, 2016 (2016), a0026353. doi: 10.1002/9780470015902.a0026353. [6] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. [7] J. J. Elser, Y. Kuang and J. D. Nagy, Biological stoichiometry: An ecological perspective on tumor dynamics, BioScience, 53 (2003), 1112-1120. [8] G. M. Lieberman, Second Order Parabolic Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [9] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994. [10] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [11] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [12] K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, SpringerPlus, 5 (2016), 1643.  doi: 10.1186/s40064-016-3295-x. [13] K. M. Owolabi and K. C. Patidar, Numerical simulations of multicomponent ecological models with adaptive, methods, Theoretical Biology and Medical Modelling, 13 (2016), 1.  doi: 10.1186/s12976-016-0027-4. [14] K. M. Owolabi and K. C. Patidar, Solution of pattern waves for diffusive fisher-like non-linear equations with adaptive methods, International Journal of Nonlinear Sciences and Numerical Simulation, 17 (2016), 291-304.  doi: 10.1515/ijnsns-2015-0173. [15] K. M. Owolabi, Mathematical study of multispecies dynamics modeling predator-prey spatial interactions, Journal of Numerical Mathematics, 25 (2017), 1-16.  doi: 10.1515/jnma-2015-0094. [16] K. C. Patidar, On the use of non-standard finite difference methods, Journal of Difference Equations and Applications, 11 (2005), 735-758.  doi: 10.1080/10236190500127471. [17] K. C. Patidar, Nonstandard finite difference methods: recent trends and further developments, Journal of Difference Equations and Applications, 22 (2016), 817-849.  doi: 10.1080/10236198.2016.1144748. [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [19] H. Smith, An Introduction to Delay Differential Equations with Sciences Applications, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [20] I. J. Stamper, M. R. Owen, P. K. Maini and H. M. Byrne, Oscillatory dynamics in a model of vascular tumour growth-implications for chemotherapy, Biology Direct, 5 (2010), 5-27. [21] G. S. Virk, Runge Kutta method for delay-differential systems, Control Theory and Applications, IEE Proceedings D, 132 (1985), 119-123.  doi: 10.1049/ip-d.1985.0021. [22] K. Y. Volokh, Stresses in growing soft tissues, Acta Biomaterialia, 2 (2006), 493-504.  doi: 10.1016/j.actbio.2006.04.002. [23] T. Wedeking, S. Löchte, C. P. Richter, M. Bhagawati, J. Piehler and C. You, Single cell GFP-trap reveals stoichiometry and dynamics of cytosolic protein complexes, Nano Letters, 15 (2015), 3610-3615.  doi: 10.1021/acs.nanolett.5b01153. [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [25] T. T. Yusuf and F. Benyah, Optimal strategy for controlling the spread of HIV/AIDS disease: A case study of South Africa, Journal of Biological Dynamics, 6 (2012), 475-494.  doi: 10.1080/17513758.2011.628700.

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##### References:
 [1] G. A. Bocharov and F. A. Rihan, Numerical modelling in biosciences using delay differential equations, Journal of Computational and Applied Mathematics, 125 (2000), 183-199.  doi: 10.1016/S0377-0427(00)00468-4. [2] R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, USA, 2011. [3] J. C. Butcher, Implicit Runge-Kutta processes, Mathematics of Computation, 18 (1964), 50-64.  doi: 10.1090/S0025-5718-1964-0159424-9. [4] M. Chen, M. Fan and Y. Kuang, Global dynamics in a stoichiometric food chain model with two limiting nutrients, Mathematical Biosciences, 289 (2017), 9-19.  doi: 10.1016/j.mbs.2017.04.004. [5] M. Cherif, Stoichiometry and population growth in osmotrophs and non-osmotrophs, John Wiley & Sons, Ltd, 2016 (2016), a0026353. doi: 10.1002/9780470015902.a0026353. [6] R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. [7] J. J. Elser, Y. Kuang and J. D. Nagy, Biological stoichiometry: An ecological perspective on tumor dynamics, BioScience, 53 (2003), 1112-1120. [8] G. M. Lieberman, Second Order Parabolic Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. [9] R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994. [10] J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. [11] A. Okubo and S. A. Levin, Diffusion and Ecological Problems: Modern Perspectives, Springer, New York, 2001. doi: 10.1007/978-1-4757-4978-6. [12] K. M. Owolabi, Numerical solution of diffusive HBV model in a fractional medium, SpringerPlus, 5 (2016), 1643.  doi: 10.1186/s40064-016-3295-x. [13] K. M. Owolabi and K. C. Patidar, Numerical simulations of multicomponent ecological models with adaptive, methods, Theoretical Biology and Medical Modelling, 13 (2016), 1.  doi: 10.1186/s12976-016-0027-4. [14] K. M. Owolabi and K. C. Patidar, Solution of pattern waves for diffusive fisher-like non-linear equations with adaptive methods, International Journal of Nonlinear Sciences and Numerical Simulation, 17 (2016), 291-304.  doi: 10.1515/ijnsns-2015-0173. [15] K. M. Owolabi, Mathematical study of multispecies dynamics modeling predator-prey spatial interactions, Journal of Numerical Mathematics, 25 (2017), 1-16.  doi: 10.1515/jnma-2015-0094. [16] K. C. Patidar, On the use of non-standard finite difference methods, Journal of Difference Equations and Applications, 11 (2005), 735-758.  doi: 10.1080/10236190500127471. [17] K. C. Patidar, Nonstandard finite difference methods: recent trends and further developments, Journal of Difference Equations and Applications, 22 (2016), 817-849.  doi: 10.1080/10236198.2016.1144748. [18] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [19] H. Smith, An Introduction to Delay Differential Equations with Sciences Applications, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8. [20] I. J. Stamper, M. R. Owen, P. K. Maini and H. M. Byrne, Oscillatory dynamics in a model of vascular tumour growth-implications for chemotherapy, Biology Direct, 5 (2010), 5-27. [21] G. S. Virk, Runge Kutta method for delay-differential systems, Control Theory and Applications, IEE Proceedings D, 132 (1985), 119-123.  doi: 10.1049/ip-d.1985.0021. [22] K. Y. Volokh, Stresses in growing soft tissues, Acta Biomaterialia, 2 (2006), 493-504.  doi: 10.1016/j.actbio.2006.04.002. [23] T. Wedeking, S. Löchte, C. P. Richter, M. Bhagawati, J. Piehler and C. You, Single cell GFP-trap reveals stoichiometry and dynamics of cytosolic protein complexes, Nano Letters, 15 (2015), 3610-3615.  doi: 10.1021/acs.nanolett.5b01153. [24] J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. [25] T. T. Yusuf and F. Benyah, Optimal strategy for controlling the spread of HIV/AIDS disease: A case study of South Africa, Journal of Biological Dynamics, 6 (2012), 475-494.  doi: 10.1080/17513758.2011.628700.
Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 3,d_x = 2,b_1 = 6,d_1 = 0.5$
Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 6,d_x = 1,b_1 = 6,d_1 = 1$
Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 6,d_x = 0.5,b_1 = 3,d_1 = 2$
Parameter values[7]
 $m=20.00$ $n=10.00$ $k_h=10.00$ $k_t=3.00$ $f=0.6667$ $P=150$ $m_1=20.00$ $\beta_1=1.00$ $c=0.005$ $dz=0.20$ $g=100.00$ $\alpha=0.05$
 $m=20.00$ $n=10.00$ $k_h=10.00$ $k_t=3.00$ $f=0.6667$ $P=150$ $m_1=20.00$ $\beta_1=1.00$ $c=0.005$ $dz=0.20$ $g=100.00$ $\alpha=0.05$
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