.. _sphx_glr_api_gallery_ode_dynamics_population:

Population models
-----------------

Verhulst's logistic growth and the Lotka-Volterra predator-prey cycles.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Integrating \dot N = rN(1 - N/K) numerically reproduces Verhulst&#x27;s closed-form sigmoid exactly: growth looks exponential at first, is fastest at N = K/2, and saturates at the carrying capacity K from any positive starting population.">

.. only:: html

  .. image:: /api/gallery/ode_dynamics/population/images/thumb/sphx_glr_plot_01_verhulst_logistic_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/population/plot_01_verhulst_logistic`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Verhulst's logistic growth</div>
    </div>


.. raw:: html

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

.. only:: html

  .. image:: /api/gallery/ode_dynamics/population/images/thumb/sphx_glr_plot_02_lotka_volterra_thumb.png
    :alt:

  :doc:`/api/gallery/ode_dynamics/population/plot_02_lotka_volterra`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Lotka-Volterra predator-prey cycles</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/ode_dynamics/population/plot_01_verhulst_logistic
   /api/gallery/ode_dynamics/population/plot_02_lotka_volterra