mhjensen
236 lines
11 KiB
HTML
236 lines
11 KiB
HTML
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'sections': [('What is Machine Learning?', 2, None, '___sec0'),
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('Types of Machine Learning', 2, None, '___sec1'),
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('Different algorithms', 2, None, '___sec2'),
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('Representing data, more examples', 2, None, '___sec8'),
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('Case study from Hudson bay', 2, None, '___sec10'),
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('Hudson bay data', 2, None, '___sec11'),
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('Hares and lynx in Hudson bay from 1900 to 1920',
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('Why now create a computer model for the hare and lynx '
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('The traditional (top-down) approach', 2, None, '___sec15'),
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('Basic dynamics of the population of hares',
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<a class="navbar-brand" href="How2ReadData-bs.html">Data Analysis and Machine Learning: Introduction and Representing data</a>
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<a href="#" class="dropdown-toggle" data-toggle="dropdown">Contents <b class="caret"></b></a>
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<ul class="dropdown-menu">
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs001.html#___sec0" style="font-size: 80%;">What is Machine Learning?</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs002.html#___sec1" style="font-size: 80%;">Types of Machine Learning</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs003.html#___sec2" style="font-size: 80%;">Different algorithms</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs004.html#___sec3" style="font-size: 80%;">Software and needed installations</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs005.html#___sec4" style="font-size: 80%;">Python installers</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs006.html#___sec5" style="font-size: 80%;">Installing R, C++, cython or Julia</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs007.html#___sec6" style="font-size: 80%;">Installing R, C++, cython or Julia</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs008.html#___sec7" style="font-size: 80%;">Introduction to Jupyter notebook and available tools</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs009.html#___sec8" style="font-size: 80%;">Representing data, more examples</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs010.html#___sec9" style="font-size: 80%;">Predator-Prey model from ecology</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs011.html#___sec10" style="font-size: 80%;">Case study from Hudson bay</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs012.html#___sec11" style="font-size: 80%;">Hudson bay data</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs013.html#___sec12" style="font-size: 80%;">Plotting the data</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs014.html#___sec13" style="font-size: 80%;">Hares and lynx in Hudson bay from 1900 to 1920</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs015.html#___sec14" style="font-size: 80%;">Why now create a computer model for the hare and lynx populations?</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs016.html#___sec15" style="font-size: 80%;">The traditional (top-down) approach</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs017.html#___sec16" style="font-size: 80%;">Basic mathematics notation</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs018.html#___sec17" style="font-size: 80%;">Basic dynamics of the population of hares</a></li>
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<!-- navigation toc: --> <li><a href="#___sec18" style="font-size: 80%;">Basic dynamics of the population of lynx</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs020.html#___sec19" style="font-size: 80%;">Evolution equations</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs021.html#___sec20" style="font-size: 80%;">Adapt the model to the Hudson Bay case</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs022.html#___sec21" style="font-size: 80%;">The program</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs023.html#___sec22" style="font-size: 80%;">The plot</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs024.html#___sec23" style="font-size: 80%;">Linear regression in Python</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs025.html#___sec24" style="font-size: 80%;">Linear Least squares in R</a></li>
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<!-- navigation toc: --> <li><a href="._How2ReadData-bs026.html#___sec25" style="font-size: 80%;">Non-Linear Least squares in R</a></li>
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<!-- !split -->
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<h2 id="___sec18" class="anchor">Basic dynamics of the population of lynx </h2>
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<p>
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<div class="panel panel-default">
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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We assume that the primary growth for the lynx population depends on sufficient food for raising lynx kittens, which implies an adequate source of nutrients from predation on hares. Thus, the growth of the lynx population does not only depend of how many lynx there are, but on how many hares they can eat.
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In a time interval \( \Delta t HL \) hares and lynx can meet, and in a
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fraction \( b\Delta t HL \) the lynx eats the hare. All of this does not
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contribute to the growth of lynx, again just a fraction of
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\( b\Delta t HL \) that we write as
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\( d\Delta t HL \). In addition, lynx die just as in the population
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dynamics with one isolated animal population, leading to a loss
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\( -c\Delta t L \).
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<p> <!-- subsequent paragraphs come in larger fonts, so start with a paragraph -->
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The accounting of lynx then looks like
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$$ \Delta L = d\Delta t H^nL^n - c\Delta t L^n$$
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