MaynardSmith¶

Systems / Maps / Population maps

A 2-D ecological recruitment map with delayed density dependence.

discrete · map2 dimensionscycles

Definition¶

\[ \begin{aligned} y_{0}' &= y_{1} \\ y_{1}' &= a y_{1} + b - y_{0}^{2} \end{aligned} \]

Parameters¶

Symbol	Default	Role
`a`	`0.87`	damping / linear feedback on the current state
`b`	`0.75`	additive recruitment (forcing) term

Properties¶

Lyapunov spectrum

$+0.1352,\; -0.136$
computed at build

Kaplan–Yorke dimension

$D_{KY} = 1.994$

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.MaynardSmith()
traj = sys.iterate(steps=10_000)

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

Reference¶

Maynard Smith (1968), Mathematical Ideas in Biology (Cambridge University Press)

BibTeX

@misc{maynardsmith,
  title = {MaynardSmith system},
  note = {Maynard Smith (1968), Mathematical Ideas in Biology (Cambridge University Press)}
}