TurchinHanski

Systems / ODEs / Chemical & biological systems

An empirically grounded vole–weasel predator–prey model under seasonal forcing that reproduces the latitudinal gradient from stable cycles to chaos.

continuous · ODE3 dimensionschaotic cycles

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{y_{0}} &= - \frac{a y_{0} y_{1}}{d + y_{0}} - \frac{g y_{0}^{2}}{h^{2} + y_{0}^{2}} - r y_{0}^{2} + r y_{0} \left(- e \sin{\left(y_{2} \right)} + 1\right) \\ \dot{y_{1}} &= s y_{1} \left(- e \sin{\left(y_{2} \right)} + 1\right) - \frac{s y_{1}^{2}}{y_{0}} \\ \dot{y_{2}} &= 2 \pi \end{aligned} \]

Parameters

Symbol	Default	Role
a	8	specialist predator attack rate
d	0.04	half-saturation of the specialist response
e	0.5	amplitude of the seasonal modulation
g	0.1	maximum generalist predation rate
h	0.8	half-saturation of generalist predation
r	8.12	intrinsic prey growth rate
s	1.25	specialist predator growth/self-limitation rate

Properties

Lyapunov spectrum

$+0.1135,\; -2.484,\; 0$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 2.046$

Divergence ∇·f

$\nabla\!\cdot f = \frac{a y_{0} y_{1}}{d^{2} + 2 d y_{0} + y_{0}^{2}} - \frac{a y_{1}}{d + y_{0}} - e r \sin{\left(y_{2} \right)} - e s \sin{\left(y_{2} \right)} + \frac{2 g y_{0}^{3}}{h^{4} + 2 h^{2} y_{0}^{2} + y_{0}^{4}} - \frac{2 g y_{0}}{h^{2} + y_{0}^{2}} - 2 r y_{0} + r + s - \frac{2 s y_{1}}{y_{0}}$
state-dependent

Equilibria

none found (no equilibria)

Define it in TSDynamics

import tsdynamics as ts

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

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

Reference

Turchin & Hanski (1997), Am. Nat. 149, 842-874

BibTeX

@misc{turchinhanski,
  title = {TurchinHanski system},
  note = {Turchin & Hanski (1997), Am. Nat. 149, 842-874}
}