Chaotic dynamics of a cancer model with singular and non-singular kernel

Hardik Joshi; Mehmet Yavuz

doi:10.3934/dcdss.2025016

Article Contents

2025, Volume 18, Issue 5: 1416-1439. Doi: 10.3934/dcdss.2025016

This issue Previous Article Mathematical modeling of local calcium signaling in neurons using artificial neural networks Next Article Preface

Chaotic dynamics of a cancer model with singular and non-singular kernel

Hardik Joshi^1, and
Mehmet Yavuz^{2, 3, ,}

1.
Department of Mathematics, LJ Institute of Engineering and Technology, LJ University, Ahmedabad, 382210, Gujarat, India
2.
Department of Mathematics and Computer Sciences, Faculty of Science, Necmettin Erbakan University, 42090 Konya, Türkiye
3.
Department of Applied Mathematics and Informatics, Kyrgyz-Turkish Manas University, Bishkek 720038, Kyrgyzstan

^*Corresponding author: Mehmet Yavuz
^*Corresponding author: Mehmet Yavuz

Received: April 1, 2024

Revised: November 1, 2024

Early access: February 14, 2025

Published online: February 14, 2025

Abstract / Introduction Full Text(HTML) Figure(10) / Table(1) Related Papers Cited by

Abstract

In this paper, the chaos and chaotic dynamics of cancer cells are studied by using singular and non-singular kernel operators. A deterministic cancer model is considered that shows the interaction between normal cells, tumor cells, and effectors immune cells. The analysis, existence, and uniqueness of the cancer model are derived for singular and non-singular kernel operators by using fixed point and Picard–Lindelof criteria. The numerical solution of the cancer model is provided for singular and non-singular kernel operators. The time series and phase space evolution of the different cells is obtained to examine the periodic, chaotic behavior under the influence of various biological situations. The numerical result highlights the substantial role of model parameters, singular, and non-singular kernels in the transmission dynamics of cancer cells.

Keywords:

Mathematics Subject Classification: Primary: 34H10, 65P20, 92B05.

Citation:

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Figure 1. Concentration profile over time for (a) tumor cells (b) healthy cells (c) effector cells

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Figure 2. Concentration of tumor cells over time for $ (a){\rm{ }}\alpha = 0.9, {\rm{ }}(b){\rm{ }}\alpha = 0.85, {\rm{ }}(c){\rm{ }}\alpha = 0.8 $

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Figure 3. Concentration of healthy cells over time for $ (a){\rm{ }}\alpha = 0.9, {\rm{ }}(b){\rm{ }}\alpha = 0.85, {\rm{ }}(c){\rm{ }}\alpha = 0.8 $

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Figure 4. Concentration of effector cells over time for $ (a){\rm{ }}\alpha = 0.9, {\rm{ }}(b){\rm{ }}\alpha = 0.85, {\rm{ }}(c){\rm{ }}\alpha = 0.8 $

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Figure 5. Impact of the decay rate of tumor cells killed by effector immune cells at $ \alpha = 0.9 $ for (a) singular kernel (b) non-singular kernel

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Figure 6. Impact of the decay rate of healthy host cells killed by tumor cells at $ \alpha = 0.9 $ for (a) singular kernel (b) non-singular kernel

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Figure 7. Impact of the decay rate of effector immune cells killed by tumor cells at $ \alpha = 0.9 $ for (a) singular kernel (b) non-singular kernel

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Figure 8. Chaotic dynamics of tumor cells versus healthy cells for $ (a){\rm{ }}\alpha = 0.99, {\rm{ }}(b){\rm{ }}\alpha = 0.9 $

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Figure 9. Chaotic dynamics of tumor cells versus effector cells for $ (a){\rm{ }}\alpha = 0.99, {\rm{ }}(b){\rm{ }}\alpha = 0.9 $

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Figure 10. Chaotic dynamics of healthy cells versus effector cells for $ (a){\rm{ }}\alpha = 0.99, {\rm{ }}(b){\rm{ }}\alpha = 0.9 $

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Table 1. Description of the model parameters

Parameter	Description of parameter
$ {\gamma _1} $	Growth rate of tumor cells
$ {\gamma _2} $	Growth rate of healthy host cells
$ {\gamma _3} $	Maximum rate of effector immune cells
$ {\xi _1} $	Carrying capacity of tumor cells
$ {\xi _2} $	Carrying capacity of healthy host cells
$ \tau $	Threshold rate of effector immune cells
$ {\delta _1} $	Decay rate of tumor cells killed by healthy host cells
$ {\delta _2} $	Decay rate of tumor cells killed by effector immune cells
$ {\delta _3} $	Decay rate of healthy host cells kill by tumor cells
$ {\delta _4} $	Decay rate of effector immune cells kill by tumor cells
$ \mu $	Natural death rate of effector immune cells

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