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

Kolade M. Owolabi; Kailash C. Patidar; Albert Shikongo

doi:10.3934/dcdss.2019038

Article Contents

2019, Volume 12, Issue 3: 591-613. Doi: 10.3934/dcdss.2019038

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

Received: April 30, 2017

Revised: August 31, 2017

Published: October 2018

The research contained in this report is supported by South African National Research Foundation

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Abstract

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.

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Mathematics Subject Classification: Primary: 65L05, 65L06; Secondary: 49M25.

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Figure 1. Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 3,d_x = 2,b_1 = 6,d_1 = 0.5$

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Figure 2. Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 6,d_x = 1,b_1 = 6,d_1 = 1$

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Figure 3. Numerical solution for the dynamics of homogeneous tumour growth model, when $a = 6,d_x = 0.5,b_1 = 3,d_1 = 2$

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Table 1. Parameter values[7]

$m=20.00$ $n=10.00$ $k_h=10.00$

$k_t=3.00$ $f=0.6667$ $P=150$

$m_1=20.00$ $\beta_1=1.00$ $c=0.005$

$dz=0.20$ $g=100.00$ $\alpha=0.05$

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