SaltonSea

Systems / ODEs / Climate & geophysics

An eco-epidemiological fish–bird disease model whose period-doubling route to chaos was proposed to explain observed mass die-offs at the Salton Sea.

continuous · ODE3 dimensionschaotic

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= - lam x y + r x \left(1 - \frac{x + y}{k}\right) \\ \dot{y} &= lam x y - \frac{m y z}{a + y} - \mu y \\ \dot{z} &= - d z + \frac{th y z}{a + y} \end{aligned} \]

Parameters

Symbol	Default	Role
a	15	predator functional-response half-saturation
d	8.3	pelican predator death rate
k	400	fish carrying capacity
lam	0.06	disease transmission rate
m	15.5	maximum predation rate on infected fish
mu	3.4	disease-induced death rate of infected fish
r	22	intrinsic fish growth rate
th	10	predator conversion efficiency

State variables: x, y, z

Properties

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

$\nabla\!\cdot f = - d + lam x - lam y + \frac{m y z}{a^{2} + 2 a y + y^{2}} - \frac{m z}{a + y} - \mu + r + \frac{th y}{a + y} - \frac{2 r x}{k} - \frac{r y}{k}$
state-dependent

Equilibria

5 equilibria
0 stable · 5 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.SaltonSea()
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

Upadhyay, Bairagi, Kundu & Chattopadhyay (2008), Appl. Math. Comput. 196, 392-401
doi:10.1016/j.amc.2007.06.007

BibTeX

@misc{saltonsea,
  title = {SaltonSea system},
  note = {Upadhyay, Bairagi, Kundu & Chattopadhyay (2008), Appl. Math. Comput. 196, 392-401},
  doi = {10.1016/j.amc.2007.06.007}
}