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

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Definition

\[ \begin{aligned} \dot{y_{0}} &= - lam y_{0} y_{1} + r y_{0} \left(1 - \frac{y_{0} + y_{1}}{k}\right) \\ \dot{y_{1}} &= lam y_{0} y_{1} - \frac{m y_{1} y_{2}}{a + y_{1}} - \mu y_{1} \\ \dot{y_{2}} &= - d y_{2} + \frac{th y_{1} y_{2}}{a + y_{1}} \end{aligned} \]

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

Properties

Lyapunov spectrum

$+0.3964,\; -0.03064,\; -4.568$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.08$

Divergence ∇·f

$\nabla\!\cdot f = - d + lam y_{0} - lam y_{1} + \frac{m y_{1} y_{2}}{a^{2} + 2 a y_{1} + y_{1}^{2}} - \frac{m y_{2}}{a + y_{1}} - \mu + r + \frac{th y_{1}}{a + y_{1}} - \frac{2 r y_{0}}{k} - \frac{r y_{1}}{k}$
state-dependent

Equilibria

3 equilibria
0 stable · 3 unstable

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Upadhyay, Bairagi, Kundu & Chattopadhyay (2008), Appl. Math. Comput. 196, 392-401

BibTeX

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