ItikBanksTumor

Systems / ODEs / Chemical & biological systems

A Lotka–Volterra-type cancer model of competing tumour, healthy, and immune cell populations whose interactions produce a strange attractor.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - a_{12} y_{0} y_{1} - a_{13} y_{0} y_{2} + y_{0} \left(1 - y_{0}\right) \\ \dot{y_{1}} &= - a_{21} y_{0} y_{1} + r_{2} y_{1} \left(1 - y_{1}\right) \\ \dot{y_{2}} &= - a_{31} y_{0} y_{2} - d_{3} y_{2} + \frac{r_{3} y_{0} y_{2}}{k_{3} + y_{0}} \end{aligned} \]

Parameters

Symbol	Default	Role
a12	1	competition of healthy cells on the tumour
a13	2.5	competition of immune cells on the tumour
a21	1.5	competition of the tumour on healthy cells
a31	0.2	inactivation rate of immune cells by the tumour
d3	0.5	death rate of effector immune cells
k3	1	half-saturation of the immune response
r2	0.6	growth rate of healthy host cells
r3	4.5	maximum immune recruitment rate

Properties

Lyapunov spectrum

$+0.001739,\; -0.4975,\; -0.6006$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.003$

Divergence ∇·f

$\nabla\!\cdot f = - a_{12} y_{1} - a_{13} y_{2} - a_{21} y_{0} - a_{31} y_{0} - d_{3} - 2 r_{2} y_{1} + r_{2} + \frac{r_{3} y_{0}}{k_{3} + y_{0}} - 2 y_{0} + 1$
state-dependent

Equilibria

7 equilibria
2 stable · 5 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.ItikBanksTumor()
traj = sys.integrate(final_time=100.0, dt=0.01)

exps = sys.lyapunov_spectrum()
ts.kaplan_yorke_dimension(exps)

Reference

Itik & Banks (2010), Int. J. Bifurcation Chaos 20, 71-79

BibTeX

@misc{itikbankstumor,
  title = {ItikBanksTumor system},
  note = {Itik & Banks (2010), Int. J. Bifurcation Chaos 20, 71-79}
}