<p>This work investigates the dynamical properties of a maturity-structured cell population model governed by partial differential equations involving conformable derivatives with respect to both temporal and spatial variables. We first rigorously formulate the conformable model and provide a comprehensive biological interpretation of its parameters, underscoring the pertinence of conformable calculus in accurately capturing anomalous transport phenomena inherent in cellular maturation dynamics. The validity and robustness of the conformable framework are substantiated through detailed numerical simulations, which demonstrate its effectiveness in modeling subdiffusive behaviors characteristic of heterogeneous biological environments. Our theoretical analysis initially concentrates on the homogeneous case, wherein we derive sufficient conditions ensuring the existence of chaos and hypercyclicity within the conformable weighted space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {C}_{0,\mu _s}([0,1];\, \mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mrow> <mn>0</mn> <mo>,</mo> <msub> <mi>μ</mi> <mi>s</mi> </msub> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <mspace width="0.166667em" /> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Although the study primarily addresses the homogeneous setting, these results constitute a foundational step toward the treatment of the more complex non-homogeneous case. By establishing appropriate conjugacy relations, we extend the chaotic and hypercyclic dynamics to the space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {C}_0([0,1];\, \mathbb {C})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">C</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> <mo>;</mo> <mspace width="0.166667em" /> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Finally, we elucidate the biological implications of the conditions underpinning chaotic behavior, thereby forging a substantive connection between the mathematical framework and the underlying cellular processes.</p>

Dynamical analysis of a maturity-structured cell population model with conformable time–space derivatives: chaos and hypercyclicity

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发表日期：

2026年7月2日

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Manal Menchih,
Ahmed Kajouni,
Khalid Hilal

摘要

This work investigates the dynamical properties of a maturity-structured cell population model governed by partial differential equations involving conformable derivatives with respect to both temporal and spatial variables. We first rigorously formulate the conformable model and provide a comprehensive biological interpretation of its parameters, underscoring the pertinence of conformable calculus in accurately capturing anomalous transport phenomena inherent in cellular maturation dynamics. The validity and robustness of the conformable framework are substantiated through detailed numerical simulations, which demonstrate its effectiveness in modeling subdiffusive behaviors characteristic of heterogeneous biological environments. Our theoretical analysis initially concentrates on the homogeneous case, wherein we derive sufficient conditions ensuring the existence of chaos and hypercyclicity within the conformable weighted space $\mathcal {C}_{0,\mu _s}([0,1];\, \mathbb {C})$ $C_{0, μ_{s}} ([0, 1]; C)$ . Although the study primarily addresses the homogeneous setting, these results constitute a foundational step toward the treatment of the more complex non-homogeneous case. By establishing appropriate conjugacy relations, we extend the chaotic and hypercyclic dynamics to the space $\mathcal {C}_0([0,1];\, \mathbb {C})$ $C_{0} ([0, 1]; C)$ . Finally, we elucidate the biological implications of the conditions underpinning chaotic behavior, thereby forging a substantive connection between the mathematical framework and the underlying cellular processes.