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 & 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.  Google Scholar

[2]

R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, USA, 2011. Google Scholar

[3]

J. C. Butcher, Implicit Runge-Kutta processes, Mathematics of Computation, 18 (1964), 50-64.  doi: 10.1090/S0025-5718-1964-0159424-9.  Google Scholar

[4]

M. ChenM. 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.  Google Scholar

[5]

M. Cherif, Stoichiometry and population growth in osmotrophs and non-osmotrophs, John Wiley & Sons, Ltd, 2016 (2016), a0026353. doi: 10.1002/9780470015902.a0026353.  Google Scholar

[6]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[7]

J. J. ElserY. Kuang and J. D. Nagy, Biological stoichiometry: An ecological perspective on tumor dynamics, BioScience, 53 (2003), 1112-1120.   Google Scholar

[8]

G. M. Lieberman, Second Order Parabolic Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[9]

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.  Google Scholar

[10]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. Smith, An Introduction to Delay Differential Equations with Sciences Applications, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[20]

I. J. StamperM. R. OwenP. K. Maini and H. M. Byrne, Oscillatory dynamics in a model of vascular tumour growth-implications for chemotherapy, Biology Direct, 5 (2010), 5-27.   Google Scholar

[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.  Google Scholar

[22]

K. Y. Volokh, Stresses in growing soft tissues, Acta Biomaterialia, 2 (2006), 493-504.  doi: 10.1016/j.actbio.2006.04.002.  Google Scholar

[23]

T. WedekingS. LöchteC. P. RichterM. BhagawatiJ. 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.  Google Scholar

[24]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, USA, 2011. Google Scholar

[3]

J. C. Butcher, Implicit Runge-Kutta processes, Mathematics of Computation, 18 (1964), 50-64.  doi: 10.1090/S0025-5718-1964-0159424-9.  Google Scholar

[4]

M. ChenM. 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.  Google Scholar

[5]

M. Cherif, Stoichiometry and population growth in osmotrophs and non-osmotrophs, John Wiley & Sons, Ltd, 2016 (2016), a0026353. doi: 10.1002/9780470015902.a0026353.  Google Scholar

[6]

R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977.  Google Scholar

[7]

J. J. ElserY. Kuang and J. D. Nagy, Biological stoichiometry: An ecological perspective on tumor dynamics, BioScience, 53 (2003), 1112-1120.   Google Scholar

[8]

G. M. Lieberman, Second Order Parabolic Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302.  Google Scholar

[9]

R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.  Google Scholar

[10]

J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

H. Smith, An Introduction to Delay Differential Equations with Sciences Applications, Springer, New York, 2011. doi: 10.1007/978-1-4419-7646-8.  Google Scholar

[20]

I. J. StamperM. R. OwenP. K. Maini and H. M. Byrne, Oscillatory dynamics in a model of vascular tumour growth-implications for chemotherapy, Biology Direct, 5 (2010), 5-27.   Google Scholar

[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.  Google Scholar

[22]

K. Y. Volokh, Stresses in growing soft tissues, Acta Biomaterialia, 2 (2006), 493-504.  doi: 10.1016/j.actbio.2006.04.002.  Google Scholar

[23]

T. WedekingS. LöchteC. P. RichterM. BhagawatiJ. 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.  Google Scholar

[24]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[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.  Google Scholar

Figure 1.  Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 3,d_x = 2,b_1 = 6,d_1 = 0.5$
Figure 2.  Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 6,d_x = 1,b_1 = 6,d_1 = 1$
Figure 3.  Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 6,d_x = 0.5,b_1 = 3,d_1 = 2$
Table 1.  Parameter values[7]
$m=20.00$ $n=10.00$ $k_h=10.00$
$k_t=3.00$ $f=0.6667$ $P=150$
$m_1=20.00$ $\beta_1=1.00$ $c=0.005$
$dz=0.20$ $g=100.00$ $\alpha=0.05$
$m=20.00$ $n=10.00$ $k_h=10.00$
$k_t=3.00$ $f=0.6667$ $P=150$
$m_1=20.00$ $\beta_1=1.00$ $c=0.005$
$dz=0.20$ $g=100.00$ $\alpha=0.05$
[1]

Tetsuya Ishiwata, Young Chol Yang. Numerical and mathematical analysis of blow-up problems for a stochastic differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 909-918. doi: 10.3934/dcdss.2020391

[2]

Nahed Naceur, Nour Eddine Alaa, Moez Khenissi, Jean R. Roche. Theoretical and numerical analysis of a class of quasilinear elliptic equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 723-743. doi: 10.3934/dcdss.2020354

[3]

Tetsuya Ishiwata, Takeshi Ohtsuka. Numerical analysis of an ODE and a level set methods for evolving spirals by crystalline eikonal-curvature flow. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 893-907. doi: 10.3934/dcdss.2020390

[4]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[5]

John Mallet-Paret, Roger D. Nussbaum. Asymptotic homogenization for delay-differential equations and a question of analyticity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3789-3812. doi: 10.3934/dcds.2020044

[6]

Mugen Huang, Moxun Tang, Jianshe Yu, Bo Zheng. A stage structured model of delay differential equations for Aedes mosquito population suppression. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3467-3484. doi: 10.3934/dcds.2020042

[7]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[8]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[9]

Yukihiko Nakata. Existence of a period two solution of a delay differential equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1103-1110. doi: 10.3934/dcdss.2020392

[10]

Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168

[11]

Kerioui Nadjah, Abdelouahab Mohammed Salah. Stability and Hopf bifurcation of the coexistence equilibrium for a differential-algebraic biological economic system with predator harvesting. Electronic Research Archive, 2021, 29 (1) : 1641-1660. doi: 10.3934/era.2020084

[12]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026

[13]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[14]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[15]

Chang-Yuan Cheng, Shyan-Shiou Chen, Rui-Hua Chen. Delay-induced spiking dynamics in integrate-and-fire neurons. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020363

[16]

Guihong Fan, Gail S. K. Wolkowicz. Chaotic dynamics in a simple predator-prey model with discrete delay. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 191-216. doi: 10.3934/dcdsb.2020263

[17]

Qiang Long, Xue Wu, Changzhi Wu. Non-dominated sorting methods for multi-objective optimization: Review and numerical comparison. Journal of Industrial & Management Optimization, 2021, 17 (2) : 1001-1023. doi: 10.3934/jimo.2020009

[18]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[19]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, 2021, 14 (1) : 89-113. doi: 10.3934/krm.2020050

[20]

Wei Ouyang, Li Li. Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods. Journal of Industrial & Management Optimization, 2021, 17 (1) : 169-184. doi: 10.3934/jimo.2019105

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (173)
  • HTML views (301)
  • Cited by (1)

[Back to Top]