HastingsPowell

Systems / ODEs / Chemical & biological systems

A three-trophic food chain whose tea-cup attractor is a landmark of ecological chaos.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - \frac{a_{1} y_{0} y_{1}}{b_{1} y_{0} + 1} + y_{0} \left(1 - y_{0}\right) \\ \dot{y_{1}} &= \frac{a_{1} y_{0} y_{1}}{b_{1} y_{0} + 1} - \frac{a_{2} y_{1} y_{2}}{b_{2} y_{1} + 1} - d_{1} y_{1} \\ \dot{y_{2}} &= \frac{a_{2} y_{1} y_{2}}{b_{2} y_{1} + 1} - d_{2} y_{2} \end{aligned} \]

Parameters

Symbol	Default	Role
a1	5	maximum attack rate of the consumer on the resource
a2	0.1	maximum attack rate of the predator on the consumer
b1	3	half-saturation of consumer feeding
b2	2	half-saturation of predator feeding
d1	0.4	death rate of the consumer
d2	0.01	death rate of the top predator

Properties

Lyapunov spectrum

$-0.0014,\; +0.0001783,\; -0.151$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.127$

Divergence ∇·f

$\nabla\!\cdot f = \frac{a_{1} b_{1} y_{0} y_{1}}{b_{1}^{2} y_{0}^{2} + 2 b_{1} y_{0} + 1} + \frac{a_{1} y_{0}}{b_{1} y_{0} + 1} - \frac{a_{1} y_{1}}{b_{1} y_{0} + 1} + \frac{a_{2} b_{2} y_{1} y_{2}}{b_{2}^{2} y_{1}^{2} + 2 b_{2} y_{1} + 1} + \frac{a_{2} y_{1}}{b_{2} y_{1} + 1} - \frac{a_{2} y_{2}}{b_{2} y_{1} + 1} - d_{1} - d_{2} - 2 y_{0} + 1$
state-dependent

Equilibria

6 equilibria
0 stable · 6 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Hastings & Powell (1991), Ecology 72, 896-903

BibTeX

@misc{hastingspowell,
  title = {HastingsPowell system},
  note = {Hastings & Powell (1991), Ecology 72, 896-903}
}