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  <title><![CDATA[AV's Ph.D Journal]]></title>
  <link href="http://alevax.github.io/atom.xml" rel="self"/>
  <link href="http://alevax.github.io/"/>
  <updated>2014-09-22T19:25:10+02:00</updated>
  <id>http://alevax.github.io/</id>
  <author>
    <name><![CDATA[Ing. Alessandro Vasciaveo]]></name>
    
  </author>
  <generator uri="http://octopress.org/">Octopress</generator>

  
  <entry>
    <title type="html"><![CDATA[Logistic Map]]></title>
    <link href="http://alevax.github.io/blog/2014/04/22/logistic-map/"/>
    <updated>2014-04-22T18:01:37+02:00</updated>
    <id>http://alevax.github.io/blog/2014/04/22/logistic-map</id>
    <content type="html"><![CDATA[<script type="math/tex; mode=display"> x(k+1) = \lambda x(k)[1-x(k)] </script>

<ul>
  <li><script type="math/tex"> k </script>, population growth cycle</li>
  <li><script type="math/tex"> \lambda </script>, linked to the rate of population growth</li>
  <li><script type="math/tex">% &lt;![CDATA[
 0 <= x(k) <= 1  %]]&gt;</script>, state variable proportional to the number of individuals in the population</li>
</ul>

<p>In order for <script type="math/tex"> x </script> to remain bounded, it should be <script type="math/tex">% &lt;![CDATA[
 0<=\lambda<=4  %]]&gt;</script>.</p>

<p>The logistic law describes well the dynamics of evolving populations where there is a growth per cycle proportional to <script type="math/tex"> \lambda x(k) </script> with a corrective negative term (feedback) <script type="math/tex"> -\lambda x^2(k) </script> which is proportional to the square of the size already attained by the population.</p>
]]></content>
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