Ricker

Systems / Maps / Population maps

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

discrete · map1 dimensionchaoticcycles

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)

Properties

Lyapunov spectrum

$+0.4286$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1$

Divergence ∇·f

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

Equilibria

2 fixed points
0 stable · 2 unstable

Define it in TSDynamics

import tsdynamics as ts

sys = ts.systems.Ricker()
traj = sys.iterate(steps=10_000)

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

Reference

Ricker (1954), J. Fish. Res. Board Can. 11, 559-623

BibTeX

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