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A simple model of carcinogenic mutations with time delay and diffusion.

Monika Joanna Piotrowska1, Urszula Foryś, Marek Bodnar

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Area of Science:

  • Mathematical Biology
  • Cancer Modeling
  • Dynamical Systems

Background:

  • Lotka-Volterra models are extended to include cell mutations and environmental conditions.
  • Focus is on unfavorable conditions, such as those induced by chemotherapy.
  • Delays represent interactions between benign and other cells.

Purpose of the Study:

  • To analyze a system of delay differential equations (DDEs) modeling normal to malignant cell mutations.
  • To compare the dynamics of ordinary differential equations (ODEs), DDEs, and DDEs with diffusion.
  • To investigate the impact of delay and diffusion on system stability and dynamics.

Main Methods:

  • Utilized delay differential equations (DDEs) and diffusion to model cell mutations.
  • Compared dynamics of ODEs, DDEs, and DDEs with diffusion.
  • Focused on analyzing systems with a positive steady state and Hopf bifurcation.

Main Results:

  • A globally stable system without delay and diffusion becomes destabilized by increasing delay.
  • Hopf bifurcation leads to oscillatory kinetic dynamics for specific delay values.
  • The system with delay and diffusion suggests the emergence of spatially non-homogeneous periodic solutions.

Conclusions:

  • Delay is a critical factor that can destabilize stable cancer models.
  • Hopf bifurcation and oscillations indicate complex dynamics in response to treatment.
  • Spatially non-homogeneous periodic solutions may arise in diffusion-influenced cancer progression.