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Lotka-Volterra 2d

Lotka–Volterra models are used in mathematical biology for modelling population dynamics of animal species, as well as many other fields where predator-prey and similar models appear. The dynamics of the growth of two interacting species can be modelled by the following noncanonical Hamiltonian system

\[\dot{q} = \begin{pmatrix} \hphantom{-} 0 & + q_1 q_2 \\ - q_1 q_2 & \hphantom{+} 0 \\ \end{pmatrix} \nabla H (q) , \quad H (q) = a_1 \, q_1 + a_2 \, q_2 + b_1 \, \log q_1 + b_2 \, \log q_2 .\]

Sub-models

The Euler-Lagrange equations of the Lotka-Volterra model can be obtained from different Lagrangians, which are connected by gauge transformations. Although they all lead to the same equations of motion, they lead to different variational integrators. Therefore different models based on different Lagrangians are implemented.

GeometricProblems.LotkaVolterra2d — Module

Lotka-Volterra model in 2D

\[\begin{aligned} L (q, \dot{q}) &= \bigg( q_2 + \frac{\log q_2}{q_1} \bigg) \, \dot{q_1} + q_1 \, \dot{q_2} - H(q) , \\ H(q) &= a_1 \, q_1 + a_2 \, q_2 + b_1 \, \log q_1 + b_2 \, \log q_2 \end{aligned}\]

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GeometricProblems.LotkaVolterra2dSymmetric — Module

Lotka-Volterra model in 2D with symmetric Lagrangian

\[\begin{aligned} L (q, \dot{q}) &= \frac{1}{2} \frac{\log q_2}{q_1} \, \dot{q_1} - \frac{1}{2} \frac{\log q_1}{q_2} \, \dot{q_2} - H(q) , \\ H(q) &= a_1 \, q_1 + a_2 \, q_2 + b_1 \, \log q_1 + b_2 \, \log q_2 \end{aligned}\]

This Lagrangian is a slight generalization of Equation (5) in José Fernández-Núñez, Lagrangian Structure of the Two-Dimensional Lotka-Volterra System, International Journal of Theoretical Physics, Vol. 37, No. 9, pp. 2457-2462, 1998.

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GeometricProblems.LotkaVolterra2dSingular — Module

Lotka-Volterra model in 2D with "singular" Lagrangian

\[\begin{aligned} L (q, \dot{q}) &= \frac{\log q_2}{q_1} \, \dot{q_1} - H(q) , \\ H(q) &= a_1 \, q_1 + a_2 \, q_2 + b_1 \, \log q_1 + b_2 \, \log q_2 \end{aligned}\]

This Lagrangian is equivalent to the Lagrangian of the symmetric Lotka-Volterra model. It differs only by a gauge transformation with the term $- 1/2 \, d(\log(q_1) \log(q_2))/dt$. It leads to the same Euler-Lagrange equations but to a different variational integrator.

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GeometricProblems.LotkaVolterra2dGauge — Module

Lotka-Volterra model in 2D with symmetric Lagrangian with gauge term

\[\begin{aligned} L (q, \dot{q}) &= \bigg( q_2 + \frac{1}{2} \frac{\log q_2}{q_1} \bigg) \, \dot{q_1} + \bigg( q_1 - \frac{1}{2} \frac{\log q_1}{q_2} \bigg) \, \dot{q_2} - H(q) , \\ H(q) &= a_1 \, q_1 + a_2 \, q_2 + b_1 \, \log q_1 + b_2 \, \log q_2 \end{aligned}\]

This Lagrangian is equivalent to the Lagrangian of the symmetric Lotka-Volterra model. It differs only by a gauge transformation with the term $d(q_1 q_2)/dt$. It leads to the same Euler-Lagrange equations but to a different variational integrator.

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User Functions

Plotting Functions

GeometricProblems.LotkaVolterra2dPlots.plot_lotka_volterra_2d! — Method

Plots the solution of a 2D Lotka-Volterra model together with the energy error.

Arguments:

  • sol <: GeometricSolution
  • equ <: GeometricProblem

Keyword aguments:

  • nplot=1: plot every nplotth time step
  • xlims=:auto: xlims for solution plot
  • ylims=:auto: ylims for solution plot
  • latex=true: use LaTeX guides
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GeometricProblems.LotkaVolterra2dPlots.plot_lotka_volterra_2d! — Method

Plots the solution of a 2D Lotka-Volterra model together with the energy error.

Arguments:

  • sol <: GeometricSolution
  • equ <: GeometricProblem

Keyword aguments:

  • nplot=1: plot every nplotth time step
  • xlims=:auto: xlims for solution plot
  • ylims=:auto: ylims for solution plot
  • latex=true: use LaTeX guides
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GeometricProblems.LotkaVolterra2dPlots.plot_lotka_volterra_2d — Method

Plots the solution of a 2D Lotka-Volterra model together with the energy error.

Arguments:

  • sol <: GeometricSolution
  • equ <: GeometricProblem

Keyword aguments:

  • nplot=1: plot every nplotth time step
  • xlims=:auto: xlims for solution plot
  • ylims=:auto: ylims for solution plot
  • latex=true: use LaTeX guides
source