Hopf bifurcation and stability in a tumor–immune model with discrete and distributed delays

Rukiye Kara; Rukiye Kara

doi:10.3934/mbe.2026082

Mathematical Biosciences and Engineering

2026, Volume 23, Issue 8: 2259-2286. doi: 10.3934/mbe.2026082

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Research article

Hopf bifurcation and stability in a tumor–immune model with discrete and distributed delays

Rukiye Kara ^,

Department of Mathematics, Mimar Sinan Fine Arts University, Istanbul, Turkey

Received: 25 March 2026 Revised: 09 May 2026 Accepted: 09 June 2026 Published: 21 July 2026

We studied a tumor–immune interaction model with one discrete and one distributed delay, where the latter was described by an exponential kernel that captures memory effects in the immune response. We derived the characteristic equation at the interior equilibrium and identified the critical delay and frequency at which a Hopf bifurcation occurs. The direction, stability, and period of the emerging periodic solutions were determined through center manifold and normal form analysis. Numerical simulations confirm the analytical results and show how the kernel parameter modulates the amplitude and frequency of the tumor–immune oscillations. In all tested cases, the coefficients $\mu_2>0$ and $\beta_2 < 0$ indicate a stable supercritical Hopf bifurcation, while $T_2>0$ shows that the period of the bifurcating periodic solutions increases, consistently with the simulated dynamics.
- delay differential equations,
- distributed delay,
- Hopf bifurcation,
- center manifold,
- normal form
Citation: Rukiye Kara. Hopf bifurcation and stability in a tumor–immune model with discrete and distributed delays[J]. Mathematical Biosciences and Engineering, 2026, 23(8): 2259-2286. doi: 10.3934/mbe.2026082

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Abstract

We studied a tumor–immune interaction model with one discrete and one distributed delay, where the latter was described by an exponential kernel that captures memory effects in the immune response. We derived the characteristic equation at the interior equilibrium and identified the critical delay and frequency at which a Hopf bifurcation occurs. The direction, stability, and period of the emerging periodic solutions were determined through center manifold and normal form analysis. Numerical simulations confirm the analytical results and show how the kernel parameter modulates the amplitude and frequency of the tumor–immune oscillations. In all tested cases, the coefficients $\mu_2>0$ and $\beta_2 < 0$ indicate a stable supercritical Hopf bifurcation, while $T_2>0$ shows that the period of the bifurcating periodic solutions increases, consistently with the simulated dynamics.

References

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