Lotka-Volterra 2d

Lotka-Volterra model in 2D

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

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.

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.

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.

Creates a DAE object for the Lotka-Volterra 2D model.

Creates a Hamiltonian DAE object for the Lotka-Volterra 2D model.

Creates a Hamiltonian ODE object for the Lotka-Volterra 2D model.

Creates an implicit DAE object for the Lotka-Volterra 2D model.

Creates an implicit DAE object for the Lotka-Volterra 2D model.

Creates an implicit ODE object for the Lotka-Volterra 2D model.

Creates an implicit ODE object for the Lotka-Volterra 2D model for use with DG integrators.

Creates a variational DAE object for the Lotka-Volterra 2D model.

Creates a variational DAE object for the Lotka-Volterra 2D model for use with SLRK integrators.

Creates a variational ODE object for the Lotka-Volterra 2D model.

Creates an ODE object for the Lotka-Volterra 2D model.

Creates a partitioned DAE object for the Lotka-Volterra 2D model.

Creates a partitioned ODE object for the Lotka-Volterra 2D model.

poincare_invariant_1st(N; DT = Float64, plan = PoincareInvariants.DEFAULT_FIRST_PLAN)

Set up the first Poincaré invariant

\[I_1 (t) = \oint_{\gamma_t} \vartheta ,\]

the integral of this module's one-form $\vartheta$ over a loop $\gamma_t$, sampled at N points. The Lagrangian is degenerate, so the momentum is not an independent coordinate but determined by $p = \vartheta(q)$, and the loop lives in the two-dimensional configuration space alone.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. Sample a loop with f_loop, advect the sample points with PIEnsembleProblem and evaluate with compute!:

pinv = poincare_invariant_1st(200)
prob = iodeproblem(; timespan = (0.0, 1E2), timestep = 1E-1)
sol  = integrate(PIEnsembleProblem(prob, pinv, f_loop), VPRKGauss(2))
I₁   = compute!(pinv, sol, parameters(prob))

Note that I₁ is preserved by a variational integrator only to the order of the discretisation, not exactly: the numerical solution satisfies $p = \vartheta(q)$ only up to the truncation error.

The four gauges of the Lotka-Volterra 2d model differ by gauge transformations, which change $\vartheta$ by an exact form. Since the integral of an exact form over a closed loop vanishes, all four give the same $I_1$ for the same loop.

See also poincare_invariant_2nd.

poincare_invariant_2nd(N; DT = Float64, plan = PoincareInvariants.DEFAULT_SECOND_PLAN)

Set up the second Poincaré invariant

\[I_2 (t) = \int_{\sigma_t} \omega ,\]

the integral of this module's two-form $\omega$ over a surface $\sigma_t$, sampled at N points. As for the first invariant the surface lives in the two-dimensional configuration space alone. The default plan samples at Padua points and rounds N up to the next Padua number, so the invariant may use slightly more points than requested; getpointnum reports how many.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. It is used exactly like the first invariant, with f_surface in place of f_loop. If the surface is the region a loop bounds, then $I_2$ over the surface and $I_1$ over the loop agree by Stokes' theorem — f_surface is not that region for f_loop, but lies inside it.

Unlike $\vartheta$, $\omega = -d\vartheta$ is gauge invariant, so all four gauges of the Lotka-Volterra 2d model share the same two-form.

See also poincare_invariant_1st.

poincare_invariant_1st(N; DT = Float64, plan = PoincareInvariants.DEFAULT_FIRST_PLAN)

Set up the first Poincaré invariant

\[I_1 (t) = \oint_{\gamma_t} \vartheta ,\]

the integral of this module's one-form $\vartheta$ over a loop $\gamma_t$, sampled at N points. The Lagrangian is degenerate, so the momentum is not an independent coordinate but determined by $p = \vartheta(q)$, and the loop lives in the two-dimensional configuration space alone.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. Sample a loop with f_loop, advect the sample points with PIEnsembleProblem and evaluate with compute!:

pinv = poincare_invariant_1st(200)
prob = iodeproblem(; timespan = (0.0, 1E2), timestep = 1E-1)
sol  = integrate(PIEnsembleProblem(prob, pinv, f_loop), VPRKGauss(2))
I₁   = compute!(pinv, sol, parameters(prob))

Note that I₁ is preserved by a variational integrator only to the order of the discretisation, not exactly: the numerical solution satisfies $p = \vartheta(q)$ only up to the truncation error.

The four gauges of the Lotka-Volterra 2d model differ by gauge transformations, which change $\vartheta$ by an exact form. Since the integral of an exact form over a closed loop vanishes, all four give the same $I_1$ for the same loop.

See also poincare_invariant_2nd.

poincare_invariant_2nd(N; DT = Float64, plan = PoincareInvariants.DEFAULT_SECOND_PLAN)

Set up the second Poincaré invariant

\[I_2 (t) = \int_{\sigma_t} \omega ,\]

the integral of this module's two-form $\omega$ over a surface $\sigma_t$, sampled at N points. As for the first invariant the surface lives in the two-dimensional configuration space alone. The default plan samples at Padua points and rounds N up to the next Padua number, so the invariant may use slightly more points than requested; getpointnum reports how many.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. It is used exactly like the first invariant, with f_surface in place of f_loop. If the surface is the region a loop bounds, then $I_2$ over the surface and $I_1$ over the loop agree by Stokes' theorem — f_surface is not that region for f_loop, but lies inside it.

Unlike $\vartheta$, $\omega = -d\vartheta$ is gauge invariant, so all four gauges of the Lotka-Volterra 2d model share the same two-form.

See also poincare_invariant_1st.

poincare_invariant_1st(N; DT = Float64, plan = PoincareInvariants.DEFAULT_FIRST_PLAN)

Set up the first Poincaré invariant

\[I_1 (t) = \oint_{\gamma_t} \vartheta ,\]

the integral of this module's one-form $\vartheta$ over a loop $\gamma_t$, sampled at N points. The Lagrangian is degenerate, so the momentum is not an independent coordinate but determined by $p = \vartheta(q)$, and the loop lives in the two-dimensional configuration space alone.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. Sample a loop with f_loop, advect the sample points with PIEnsembleProblem and evaluate with compute!:

pinv = poincare_invariant_1st(200)
prob = iodeproblem(; timespan = (0.0, 1E2), timestep = 1E-1)
sol  = integrate(PIEnsembleProblem(prob, pinv, f_loop), VPRKGauss(2))
I₁   = compute!(pinv, sol, parameters(prob))

Note that I₁ is preserved by a variational integrator only to the order of the discretisation, not exactly: the numerical solution satisfies $p = \vartheta(q)$ only up to the truncation error.

The four gauges of the Lotka-Volterra 2d model differ by gauge transformations, which change $\vartheta$ by an exact form. Since the integral of an exact form over a closed loop vanishes, all four give the same $I_1$ for the same loop.

See also poincare_invariant_2nd.

poincare_invariant_2nd(N; DT = Float64, plan = PoincareInvariants.DEFAULT_SECOND_PLAN)

Set up the second Poincaré invariant

\[I_2 (t) = \int_{\sigma_t} \omega ,\]

the integral of this module's two-form $\omega$ over a surface $\sigma_t$, sampled at N points. As for the first invariant the surface lives in the two-dimensional configuration space alone. The default plan samples at Padua points and rounds N up to the next Padua number, so the invariant may use slightly more points than requested; getpointnum reports how many.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. It is used exactly like the first invariant, with f_surface in place of f_loop. If the surface is the region a loop bounds, then $I_2$ over the surface and $I_1$ over the loop agree by Stokes' theorem — f_surface is not that region for f_loop, but lies inside it.

Unlike $\vartheta$, $\omega = -d\vartheta$ is gauge invariant, so all four gauges of the Lotka-Volterra 2d model share the same two-form.

See also poincare_invariant_1st.

poincare_invariant_1st(N; DT = Float64, plan = PoincareInvariants.DEFAULT_FIRST_PLAN)

Set up the first Poincaré invariant

\[I_1 (t) = \oint_{\gamma_t} \vartheta ,\]

the integral of this module's one-form $\vartheta$ over a loop $\gamma_t$, sampled at N points. The Lagrangian is degenerate, so the momentum is not an independent coordinate but determined by $p = \vartheta(q)$, and the loop lives in the two-dimensional configuration space alone.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. Sample a loop with f_loop, advect the sample points with PIEnsembleProblem and evaluate with compute!:

pinv = poincare_invariant_1st(200)
prob = iodeproblem(; timespan = (0.0, 1E2), timestep = 1E-1)
sol  = integrate(PIEnsembleProblem(prob, pinv, f_loop), VPRKGauss(2))
I₁   = compute!(pinv, sol, parameters(prob))

Note that I₁ is preserved by a variational integrator only to the order of the discretisation, not exactly: the numerical solution satisfies $p = \vartheta(q)$ only up to the truncation error.

The four gauges of the Lotka-Volterra 2d model differ by gauge transformations, which change $\vartheta$ by an exact form. Since the integral of an exact form over a closed loop vanishes, all four give the same $I_1$ for the same loop.

See also poincare_invariant_2nd.

poincare_invariant_2nd(N; DT = Float64, plan = PoincareInvariants.DEFAULT_SECOND_PLAN)

Set up the second Poincaré invariant

\[I_2 (t) = \int_{\sigma_t} \omega ,\]

the integral of this module's two-form $\omega$ over a surface $\sigma_t$, sampled at N points. As for the first invariant the surface lives in the two-dimensional configuration space alone. The default plan samples at Padua points and rounds N up to the next Padua number, so the invariant may use slightly more points than requested; getpointnum reports how many.

This is implemented in the LotkaVolterra2dPoincareInvariants extension and becomes available once PoincareInvariants is loaded. It is used exactly like the first invariant, with f_surface in place of f_loop. If the surface is the region a loop bounds, then $I_2$ over the surface and $I_1$ over the loop agree by Stokes' theorem — f_surface is not that region for f_loop, but lies inside it.

Unlike $\vartheta$, $\omega = -d\vartheta$ is gauge invariant, so all four gauges of the Lotka-Volterra 2d model share the same two-form.

See also poincare_invariant_1st.