Blasius

Systems / ODEs / Physical systems

A three-level food chain with Holling type-II responses whose chaotic "teacup" attractor is a landmark of ecological chaos.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= a x - \frac{\alpha_{1} x y}{k_{1} x + 1} \\ \dot{y} &= \frac{\alpha_{1} x y}{k_{1} x + 1} - \frac{\alpha_{2} y z}{k_{2} y + 1} - b y \\ \dot{z} &= \frac{\alpha_{2} y z}{k_{2} y + 1} - c \left(z - zs\right) \end{aligned} \]

Parameters

Symbol	Default	Role
a	1	resource growth / interaction rate
alpha1	0.2	half-saturation constant of the lower response
alpha2	1	half-saturation constant of the upper response
b	1	predator-prey interaction rate
c	10	top-predator interaction rate
k1	0.05	predator mortality rate
k2	0	top-predator mortality offset
zs	0.006	top-predator baseline (immigration) level

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = a + \frac{\alpha_{1} k_{1} x y}{k_{1}^{2} x^{2} + 2 k_{1} x + 1} + \frac{\alpha_{1} x}{k_{1} x + 1} - \frac{\alpha_{1} y}{k_{1} x + 1} + \frac{\alpha_{2} k_{2} y z}{k_{2}^{2} y^{2} + 2 k_{2} y + 1} + \frac{\alpha_{2} y}{k_{2} y + 1} - \frac{\alpha_{2} z}{k_{2} y + 1} - b - c$
state-dependent

Equilibria

4 equilibria
0 stable · 4 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.Blasius()
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

Blasius, Huppert & Stone (1999), Nature 399, 354-359
doi:10.1038/20676

BibTeX

@misc{blasius,
  title = {Blasius system},
  note = {Blasius, Huppert & Stone (1999), Nature 399, 354-359},
  doi = {10.1038/20676}
}