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:
  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  R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, USA, 2011. Google Scholar  J. C. Butcher, Implicit Runge-Kutta processes, Mathematics of Computation, 18 (1964), 50-64.  doi: 10.1090/S0025-5718-1964-0159424-9. Google Scholar  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. Google Scholar  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  R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. Google Scholar  J. J. Elser, Y. Kuang and J. D. Nagy, Biological stoichiometry: An ecological perspective on tumor dynamics, BioScience, 53 (2003), 1112-1120.   Google Scholar  G. M. Lieberman, Second Order Parabolic Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. Google Scholar  R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994. Google Scholar  J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar  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  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  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  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  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  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  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  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  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  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.   Google Scholar  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  K. Y. Volokh, Stresses in growing soft tissues, Acta Biomaterialia, 2 (2006), 493-504.  doi: 10.1016/j.actbio.2006.04.002. Google Scholar  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. Google Scholar  J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar  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:
  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  R. L. Burden and J. D. Faires, Numerical Analysis, Brooks/Cole, USA, 2011. Google Scholar  J. C. Butcher, Implicit Runge-Kutta processes, Mathematics of Computation, 18 (1964), 50-64.  doi: 10.1090/S0025-5718-1964-0159424-9. Google Scholar  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. Google Scholar  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  R. D. Driver, Ordinary and Delay Differential Equations, Springer-Verlag, New York, 1977. Google Scholar  J. J. Elser, Y. Kuang and J. D. Nagy, Biological stoichiometry: An ecological perspective on tumor dynamics, BioScience, 53 (2003), 1112-1120.   Google Scholar  G. M. Lieberman, Second Order Parabolic Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. doi: 10.1142/3302. Google Scholar  R. E. Mickens, Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994. Google Scholar  J. D. Murray, Mathematical Biology, Springer-Verlag, Berlin, 1989. doi: 10.1007/978-3-662-08539-4. Google Scholar  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  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  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  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  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  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  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  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  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  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.   Google Scholar  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  K. Y. Volokh, Stresses in growing soft tissues, Acta Biomaterialia, 2 (2006), 493-504.  doi: 10.1016/j.actbio.2006.04.002. Google Scholar  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. Google Scholar  J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1. Google Scholar  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 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
 $m=20.00$ $n=10.00$ $k_h=10.00$ $k_t=3.00$ $f=0.6667$ $P=150$ $m_1=20.00$ $\beta_1=1.00$ $c=0.005$ $dz=0.20$ $g=100.00$ $\alpha=0.05$
 $m=20.00$ $n=10.00$ $k_h=10.00$ $k_t=3.00$ $f=0.6667$ $P=150$ $m_1=20.00$ $\beta_1=1.00$ $c=0.005$ $dz=0.20$ $g=100.00$ $\alpha=0.05$
  Yang Kuang, John D. Nagy, James J. Elser. Biological stoichiometry of tumor dynamics: Mathematical models and analysis. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 221-240. doi: 10.3934/dcdsb.2004.4.221  Joseph A. Connolly, Neville J. Ford. Comparison of numerical methods for fractional differential equations. Communications on Pure & Applied Analysis, 2006, 5 (2) : 289-307. doi: 10.3934/cpaa.2006.5.289  Nicola Guglielmi, Christian Lubich. Numerical periodic orbits of neutral delay differential equations. Discrete & Continuous Dynamical Systems, 2005, 13 (4) : 1057-1067. doi: 10.3934/dcds.2005.13.1057  Alfonso Ruiz-Herrera. Chaos in delay differential equations with applications in population dynamics. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1633-1644. doi: 10.3934/dcds.2013.33.1633  Iasson Karafyllis, Lars Grüne. Feedback stabilization methods for the numerical solution of ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 283-317. doi: 10.3934/dcdsb.2011.16.283  Xiaoying Han, Jinglai Li, Dongbin Xiu. Error analysis for numerical formulation of particle filter. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1337-1354. doi: 10.3934/dcdsb.2015.20.1337  Tomás Caraballo, P.E. Kloeden, Pedro Marín-Rubio. Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay. Discrete & Continuous Dynamical Systems, 2007, 19 (1) : 177-196. doi: 10.3934/dcds.2007.19.177  Eugen Stumpf. Local stability analysis of differential equations with state-dependent delay. Discrete & Continuous Dynamical Systems, 2016, 36 (6) : 3445-3461. doi: 10.3934/dcds.2016.36.3445  Songbai Guo, Jing-An Cui, Wanbiao Ma. An analysis approach to permanence of a delay differential equations model of microorganism flocculation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021208  Roman Srzednicki. A theorem on chaotic dynamics and its application to differential delay equations. Conference Publications, 2001, 2001 (Special) : 362-365. doi: 10.3934/proc.2001.2001.362  Gang Huang, Yasuhiro Takeuchi, Rinko Miyazaki. Stability conditions for a class of delay differential equations in single species population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2451-2464. doi: 10.3934/dcdsb.2012.17.2451  Qiumei Huang, Xiuxiu Xu, Hermann Brunner. Continuous Galerkin methods on quasi-geometric meshes for delay differential equations of pantograph type. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5423-5443. doi: 10.3934/dcds.2016039  Evelyn Sander, E. Barreto, S.J. Schiff, P. So. Dynamics of noninvertibility in delay equations. Conference Publications, 2005, 2005 (Special) : 768-777. doi: 10.3934/proc.2005.2005.768  Ilknur Koca. Numerical analysis of coupled fractional differential equations with Atangana-Baleanu fractional derivative. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 475-486. doi: 10.3934/dcdss.2019031  Sun Yi, Patrick W. Nelson, A. Galip Ulsoy. Delay differential equations via the matrix lambert w function and bifurcation analysis: application to machine tool chatter. Mathematical Biosciences & Engineering, 2007, 4 (2) : 355-368. doi: 10.3934/mbe.2007.4.355  Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521  Abdelhai Elazzouzi, Aziz Ouhinou. Optimal regularity and stability analysis in the $\alpha-$Norm for a class of partial functional differential equations with infinite delay. Discrete & Continuous Dynamical Systems, 2011, 30 (1) : 115-135. doi: 10.3934/dcds.2011.30.115  Pankaj Kumar, Shiv Raj. Modelling and analysis of prey-predator model involving predation of mature prey using delay differential equations. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021035  Iacopo P. Longo, Sylvia Novo, Rafael Obaya. Topologies of continuity for Carathéodory delay differential equations with applications in non-autonomous dynamics. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5491-5520. doi: 10.3934/dcds.2019224  Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś, Ewa Nizińska. Model of tumour angiogenesis -- analysis of stability with respect to delays. Mathematical Biosciences & Engineering, 2013, 10 (1) : 19-35. doi: 10.3934/mbe.2013.10.19

2020 Impact Factor: 2.425
{"ft_lang_label":"__label__en","ft_lang_prob":0.5999972,"math_prob":0.6292785,"size":15421,"snap":"2021-43-2021-49","text_gpt3_token_len":4831,"char_repetition_ratio":0.17960693,"word_repetition_ratio":0.6245815,"special_character_ratio":0.35471112,"punctuation_ratio":0.26781788,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97091824,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-28T00:23:12Z\",\"WARC-Record-ID\":\"<urn:uuid:955786e1-e9b4-4c4a-914c-f9a299e1505b>\",\"Content-Length\":\"122463\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6d85eb9f-ed5c-4197-8adf-a80523e41b72>\",\"WARC-Concurrent-To\":\"<urn:uuid:357480c7-1e8b-4861-9f26-bac6972451a8>\",\"WARC-IP-Address\":\"107.161.80.18\",\"WARC-Target-URI\":\"https://www.aimsciences.org/article/doi/10.3934/dcdss.2019038\",\"WARC-Payload-Digest\":\"sha1:7Z3AFZKQIZ7MBGDNT4JVA5MLP7C6XVVT\",\"WARC-Block-Digest\":\"sha1:X33CYLXIOCIJUUUSX43BMTGB4SALK4WA\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358323.91_warc_CC-MAIN-20211127223710-20211128013710-00247.warc.gz\"}"}