Ricker¶

Systems / Maps / Population maps

The Ricker recruitment map — over-compensatory density dependence driving fishery population cycles and chaos.

discrete · map1 dimensionchaoticcycles

Ricker return map · bifurcation diagram — return map · bifurcation diagram

Definition¶

@staticmethod
def _step(X, a):
    x = X
    return x * np.exp(a - x)

Parameters¶

Symbol	Default	Role
`a`	`3.3`	intrinsic growth rate (the bifurcation parameter)

State variables: x

Properties¶

Lyapunov spectrum

TODO — lyapunov_spectrum failed: AttributeError

Kaplan–Yorke dimension

TODO — requires a numeric Lyapunov spectrum

Divergence ∇·f

n/a — discrete map — flow divergence undefined (per-step contraction is |det J|)

Equilibria

2 fixed points
0 stable · 2 unstable

Use it in TSDynamics¶

import tsdynamics as ts

system = ts.systems.Ricker()
traj = system.run(steps=10_000)
ts.plot(traj)

spectrum = ts.analysis.lyapunov_spectrum(system)
ts.analysis.kaplan_yorke_dimension(spectrum)

Reference¶

Ricker (1954), J. Fish. Res. Board Can. 11, 559-623
doi:10.1139/f54-039

BibTeX

@misc{ricker,
  title = {Ricker system},
  note = {Ricker (1954), J. Fish. Res. Board Can. 11, 559-623},
  doi = {10.1139/f54-039}
}