HastingsPowell¶
Systems / ODEs / Chemical & biological systems
A three-trophic food chain whose tea-cup attractor is a landmark of ecological chaos.
Definition¶
\[
\begin{aligned}
\dot{x} &= - \frac{a_{1} x y}{b_{1} x + 1} + x \left(1 - x\right) \\
\dot{y} &= \frac{a_{1} x y}{b_{1} x + 1} - \frac{a_{2} y z}{b_{2} y + 1} - d_{1} y \\
\dot{z} &= \frac{a_{2} y z}{b_{2} y + 1} - d_{2} z
\end{aligned}
\]
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 |
State variables: x, y, z
Properties¶
Lyapunov spectrum
TODO — lyapunov_spectrum failed: AttributeError
Kaplan–Yorke dimension
TODO — requires a numeric Lyapunov spectrum
Divergence ∇·f
$\nabla\!\cdot f = \frac{a_{1} b_{1} x y}{b_{1}^{2} x^{2} + 2 b_{1} x + 1} + \frac{a_{1} x}{b_{1} x + 1} - \frac{a_{1} y}{b_{1} x + 1} + \frac{a_{2} b_{2} y z}{b_{2}^{2} y^{2} + 2 b_{2} y + 1} + \frac{a_{2} y}{b_{2} y + 1} - \frac{a_{2} z}{b_{2} y + 1} - d_{1} - d_{2} - 2 x + 1$
state-dependent
state-dependent
Equilibria
5 equilibria
0 stable · 5 unstable
0 stable · 5 unstable
Use it in TSDynamics¶
import tsdynamics as ts
system = ts.systems.HastingsPowell()
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¶
Hastings & Powell (1991), Ecology 72, 896-903
doi:10.2307/1940591