Chaos in Tumor Growth Model with Time-Delayed Immune Response

This Demonstration considers a tumor growth model in which tumor cells play the role of prey, and the immune system is represented by resting and predator cells.
The model consists of two differential equations coupled to a delay differential equation:
with initial history functions . Here , , and represent tumor cells, predator cells, and resting cells, respectively; is time; and is a time delay measured from the time resting cells recognize tumor cells to the time predator cells destroy the prey. The parameters in the model are defined in the reference and are set here to be . When the delay time is small, all three cell populations approach equilibrium values, and the tumor can be considered nonmalignant. For larger delay times, all cell populations coexist in a limit cycle or periodic solution; in this case, the tumor can be termed mildly malignant.
The existence of periodic solutions implies that the tumor levels oscillate around a fixed point even in the absence of treatment; this has been observed clinically and is known as Jeff's phenomenon [2]. With larger , the hunting cells are lethargic in their response and the cell populations grow in an irregular fashion, leading to chaotic attractors; in this case the tumor is said to be malignant, and aggressive treatment is required.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


[1] M. Saleem and T. Agrawal, "Chaos in a Tumor Growth Model with Delayed Responses of the Immune System," Journal of Applied Mathematics, #891095, 2012 pp. 1–16. doi:10.1155/2012/891095.
[2] R. H. Thomlinson "Measurement and Management of Carcinoma of the Breast," Clinical Radiology, 33(5), 1982 pp. 481–493.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+