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

\[ \begin{aligned} \dot{y_{0}} &= a y_{0} - \frac{\alpha_{1} y_{0} y_{1}}{k_{1} y_{0} + 1} \\ \dot{y_{1}} &= \frac{\alpha_{1} y_{0} y_{1}}{k_{1} y_{0} + 1} - \frac{\alpha_{2} y_{1} y_{2}}{k_{2} y_{1} + 1} - b y_{1} \\ \dot{y_{2}} &= \frac{\alpha_{2} y_{1} y_{2}}{k_{2} y_{1} + 1} - c \left(y_{2} - zs\right) \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.06474,\; +0.004584,\; -2.876$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.024$

Divergence ∇·f

$\nabla\!\cdot f = a + \frac{\alpha_{1} k_{1} y_{0} y_{1}}{k_{1}^{2} y_{0}^{2} + 2 k_{1} y_{0} + 1} + \frac{\alpha_{1} y_{0}}{k_{1} y_{0} + 1} - \frac{\alpha_{1} y_{1}}{k_{1} y_{0} + 1} + \frac{\alpha_{2} k_{2} y_{1} y_{2}}{k_{2}^{2} y_{1}^{2} + 2 k_{2} y_{1} + 1} + \frac{\alpha_{2} y_{1}}{k_{2} y_{1} + 1} - \frac{\alpha_{2} y_{2}}{k_{2} y_{1} + 1} - b - c$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Blasius, Huppert & Stone (1999), Nature 399, 354-359

BibTeX

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