.. 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_01_verhulst_logistic.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_01_verhulst_logistic.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_ode_dynamics_population_plot_01_verhulst_logistic.py:


Verhulst's logistic growth
================================

Integrating :math:`\dot N = rN(1 - N/K)` numerically reproduces
Verhulst's closed-form sigmoid exactly: growth looks exponential at
first, is fastest at :math:`N = K/2`, and saturates at the carrying
capacity :math:`K` from any positive starting population.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.ode_dynamics.systems.population import LogisticGrowth, logistic_growth_solution








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Numerical integration versus the closed form
---------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-38

.. code-block:: Python


    r, K = 0.8, 100.0
    t = np.linspace(0.0, 15.0, 400)
    fig, ax = plt.subplots(figsize=(7, 4.5))
    for n0 in (2.0, 20.0, 150.0):
        traj = LogisticGrowth([n0], r=r, K=K).integrate((0.0, 15.0), dt=1e-2, method="rk4")
        ax.plot(traj.t, traj.y[:, 0], lw=3, alpha=0.4, label=f"RK4, N0={n0:g}")
        ax.plot(t, logistic_growth_solution(t, n0, r, K), "k--", lw=1)
        err = np.max(np.abs(traj.y[:, 0] - logistic_growth_solution(traj.t, n0, r, K)))
        print(f"N0={n0:6.1f}: max |RK4 - exact| = {err:.2e}")
    ax.axhline(K, color="gray", lw=0.8)
    ax.set_xlabel("t")
    ax.set_ylabel("N(t)")
    ax.set_title("Logistic growth approaches the carrying capacity K (dashed: exact)")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/population/images/sphx_glr_plot_01_verhulst_logistic_001.png
   :alt: Logistic growth approaches the carrying capacity K (dashed: exact)
   :srcset: /api/gallery/ode_dynamics/population/images/sphx_glr_plot_01_verhulst_logistic_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    N0=   2.0: max |RK4 - exact| = 1.62e-09
    N0=  20.0: max |RK4 - exact| = 2.50e-10
    N0= 150.0: max |RK4 - exact| = 2.62e-09





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

   **Total running time of the script:** (0 minutes 0.294 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_population_plot_01_verhulst_logistic.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_verhulst_logistic.ipynb <plot_01_verhulst_logistic.ipynb>`

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

      :download:`Download Python source code: plot_01_verhulst_logistic.py <plot_01_verhulst_logistic.py>`

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

      :download:`Download zipped: plot_01_verhulst_logistic.zip <plot_01_verhulst_logistic.zip>`


.. only:: html

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

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