Time delays in proliferation and apoptosis for solid avascular tumour

Tom 63 / 2003

Urszula Foryś, Mikhail Kolev Banach Center Publications 63 (2003), 187-196 MSC: 35K57, 62P10, 93A30 DOI: 10.4064/bc63-0-7


The role of time delays in solid avascular tumour growth is considered. The model is formulated in terms of a reaction-diffusion equation and mass conservation law. Two main processes are taken into account—proliferation and apoptosis. We introduce time delay first in underlying apoptosis only and then in both processes. In the absence of necrosis the model reduces to one ordinary differential equation with one discrete delay which describes the changes of tumour radius. Basic properties of the model depending on the magnitude of delay are studied. Nonnegativity of solutions is investigated. Steady state and the Hopf bifurcation analysis are presented. The results are illustrated by computer simulations.


  • Urszula ForyśInstitute of Applied Mathematics and Mechanics
    Faculty of Mathematics, Informatics & Mechanics
    Warsaw University
    Banacha 2
    02-097 Warszawa,
  • Mikhail KolevInstitute of Applied Mathematics and Mechanics
    Warsaw University
    Banacha 2
    02-097 Warszawa, Poland
    South-West University
    Blagoevgrad, Bulgaria

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