MaynardSmith¶

Systems / Maps / Population maps

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

discrete · map2 dimensionscycles

Definition¶

\[ \begin{aligned} x' &= y \\ y' &= a y + b - x^{2} \end{aligned} \]

Parameters¶

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

State variables: x, y

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

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

Reference¶

Maynard Smith (1968), Mathematical Ideas in Biology (Cambridge University Press)
doi:10.1017/cbo9780511565144

BibTeX

@misc{maynardsmith,
  title = {MaynardSmith system},
  note = {Maynard Smith (1968), Mathematical Ideas in Biology (Cambridge University Press)},
  doi = {10.1017/cbo9780511565144}
}