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.
Definition¶
\[
\begin{aligned}
\dot{x} &= - a_{12} x y - a_{13} x z + x \left(1 - x\right) \\
\dot{y} &= - a_{21} x y + r_{2} y \left(1 - y\right) \\
\dot{z} &= - a_{31} x z - d_{3} z + \frac{r_{3} x z}{k_{3} + x}
\end{aligned}
\]
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 |
State variables: x, y, z
Properties¶
Lyapunov spectrum
TODO — lyapunov_spectrum failed: AttributeError
Kaplan–Yorke dimension
TODO — requires a numeric Lyapunov spectrum
Divergence ∇·f
$\nabla\!\cdot f = - a_{12} y - a_{13} z - a_{21} x - a_{31} x - d_{3} - 2 r_{2} y + r_{2} + \frac{r_{3} x}{k_{3} + x} - 2 x + 1$
state-dependent
state-dependent
Equilibria
8 equilibria
0 stable · 8 unstable
0 stable · 8 unstable
Use it in TSDynamics¶
import tsdynamics as ts
system = ts.systems.ItikBanksTumor()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)
spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)
Reference¶
Itik & Banks (2010), Int. J. Bifurcation Chaos 20, 71-79
doi:10.1142/s0218127410025417