.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/population/plot_02_lotka_volterra.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_ode_dynamics_population_plot_02_lotka_volterra.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_ode_dynamics_population_plot_02_lotka_volterra.py:


Lotka-Volterra predator-prey cycles
=========================================

Predator and prey populations rise and fall periodically, out of phase.
Every orbit is closed because the Lotka-Volterra invariant
:math:`V = \delta x - \gamma\ln x + \beta y - \alpha\ln y` is conserved:
orbits are its level sets around the coexistence fixed point.

.. GENERATED FROM PYTHON SOURCE LINES 12-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.population import LotkaVolterra, lotka_volterra_invariant

    params = dict(alpha=1.1, beta=0.4, delta=0.1, gamma=0.4)








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Nested closed orbits and out-of-phase oscillations
---------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-45

.. code-block:: Python


    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    for x0 in (5.0, 10.0, 20.0):
        system = LotkaVolterra([x0, 2.75], **params)
        traj = system.integrate((0.0, 40.0), dt=1e-3, method="rk4")
        axes[0].plot(traj.y[:, 0], traj.y[:, 1], lw=1)
        V = lotka_volterra_invariant(traj.y[:, 0], traj.y[:, 1], **params)
        print(f"x0={x0:5.1f}: invariant drift over t=40 is {np.ptp(V):.1e}")
    fp = system.fixed_point()
    axes[0].plot(*fp, "k+", ms=12, label="coexistence fixed point")
    axes[0].set_xlabel("prey x")
    axes[0].set_ylabel("predators y")
    axes[0].set_title("Orbits are level sets of the invariant")
    axes[0].legend()

    axes[1].plot(traj.t, traj.y[:, 0], label="prey")
    axes[1].plot(traj.t, traj.y[:, 1], label="predators")
    axes[1].set_xlabel("t")
    axes[1].set_title("Predator peaks lag prey peaks")
    axes[1].legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/population/images/sphx_glr_plot_02_lotka_volterra_001.png
   :alt: Orbits are level sets of the invariant, Predator peaks lag prey peaks
   :srcset: /api/gallery/ode_dynamics/population/images/sphx_glr_plot_02_lotka_volterra_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    x0=  5.0: invariant drift over t=40 is 2.7e-15
    x0= 10.0: invariant drift over t=40 is 1.4e-14
    x0= 20.0: invariant drift over t=40 is 3.4e-13





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.398 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_population_plot_02_lotka_volterra.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_02_lotka_volterra.ipynb <plot_02_lotka_volterra.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_02_lotka_volterra.py <plot_02_lotka_volterra.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_02_lotka_volterra.zip <plot_02_lotka_volterra.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_