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<h1 class="entry-title"><a href="/blog/2014/04/22/logistic-map/">Logistic Map</a></h1>
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<time class='entry-date' datetime='2014-04-22T18:01:37+02:00'><span class='date'><span class='date-month'>Apr</span> <span class='date-day'>22</span><span class='date-suffix'>nd</span>, <span class='date-year'>2014</span></span> <span class='time'>6:01 pm</span></time>
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<div class="entry-content"><script type="math/tex; mode=display"> x(k+1) = \lambda x(k)[1-x(k)] </script>
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<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">% <![CDATA[
0 <= x(k) <= 1 %]]></script>, state variable proportional to the number of individuals in the population</li>
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<p>In order for <script type="math/tex"> x </script> to remain bounded, it should be <script type="math/tex">% <![CDATA[
0<=\lambda<=4 %]]></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>
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