LotkaVolterra

Systems / ODEs / Population dynamics

Predator and prey circling a neutrally stable centre — a conserved first integral, so every initial condition rides its own closed orbit.

continuous · ODE2 dimensionsconservativecycles

Interactive: drag to rotate

Definition

\[ \begin{aligned} \dot{x} &= \alpha x - \beta x y \\ \dot{y} &= \delta x y - \gamma y \end{aligned} \]

Parameters

Symbol	Default	Role
alpha	1.1	intrinsic growth rate of the prey
beta	0.4	predation rate
delta	0.1	conversion of consumed prey into predators
gamma	0.4	predator death rate

State variables: x, y

Properties

Lyapunov spectrum

$0,\; 0$
analytic: closed orbits of a conservative planar flow

Kaplan–Yorke dimension

$D_{KY} = 2$

Divergence ∇·f

$\nabla\!\cdot f = \alpha - \beta y + \delta x - \gamma$
state-dependent

Equilibria

2 equilibria
0 stable · 2 unstable

Use it in TSDynamics

import tsdynamics as ts

system = ts.systems.LotkaVolterra()
traj = system.run(final_time=100.0, dt=0.01)
ts.plot(traj)

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

Reference

Lotka (1920), Proc. Natl. Acad. Sci. U.S.A. 6, 410-415
doi:10.1073/pnas.6.7.410

BibTeX

@misc{lotkavolterra,
  title = {LotkaVolterra system},
  note = {Lotka (1920), Proc. Natl. Acad. Sci. U.S.A. 6, 410-415},
  doi = {10.1073/pnas.6.7.410}
}