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    <h1>Logistic kinetics</h1>
    
    <div class="hidden name"><code>logistic.solution.Rd</code></div>
    </div>

    <div class="ref-description">
    
    <p>Function describing exponential decline from a defined starting value, with 
  an increasing rate constant, supposedly caused by microbial growth</p>
    
    </div>

    <pre class="usage"><span class='fu'>logistic.solution</span>(<span class='no'>t</span>, <span class='no'>parent.0</span>, <span class='no'>kmax</span>, <span class='no'>k0</span>, <span class='no'>r</span>)</pre>
    
    <h2 class="hasAnchor" id="arguments"><a class="anchor" href="#arguments"></a>Arguments</h2>
    <table class="ref-arguments">
    <colgroup><col class="name" /><col class="desc" /></colgroup>
    <tr>
      <th>t</th>
      <td><p>Time.</p></td>
    </tr>
    <tr>
      <th>parent.0</th>
      <td><p>Starting value for the response variable at time zero.</p></td>
    </tr>
    <tr>
      <th>kmax</th>
      <td><p>Maximum rate constant.</p></td>
    </tr>
    <tr>
      <th>k0</th>
      <td><p>Minumum rate constant effective at time zero.</p></td>
    </tr>
    <tr>
      <th>r</th>
      <td><p>Growth rate of the increase in the rate constant.</p></td>
    </tr>
    </table>
    
    <h2 class="hasAnchor" id="note"><a class="anchor" href="#note"></a>Note</h2>

    <p>The solution of the logistic model reduces to the
  <code><a href='SFO.solution.html'>SFO.solution</a></code> if <code>k0</code> is equal to
  <code>kmax</code>.</p>
    
    <h2 class="hasAnchor" id="value"><a class="anchor" href="#value"></a>Value</h2>

    <p>The value of the response variable at time <code>t</code>.</p>
    
    <h2 class="hasAnchor" id="references"><a class="anchor" href="#references"></a>References</h2>

    <p>FOCUS (2014) &#8220;Generic guidance for Estimating Persistence and
  Degradation Kinetics from Environmental Fate Studies on Pesticides in EU
  Registration&#8221; Report of the FOCUS Work Group on Degradation Kinetics,
  Version 1.1, 18 December 2014
  <a href='http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics'>http://esdac.jrc.ec.europa.eu/projects/degradation-kinetics</a></p>
    

    <h2 class="hasAnchor" id="examples"><a class="anchor" href="#examples"></a>Examples</h2>
    <pre class="examples"><div class='input'>  <span class='co'># Reproduce the plot on page 57 of FOCUS (2014)</span>
  <span class='fu'><a href='https://www.rdocumentation.org/packages/graphics/topics/plot'>plot</a></span>(<span class='kw'>function</span>(<span class='no'>x</span>) <span class='fu'>logistic.solution</span>(<span class='no'>x</span>, <span class='fl'>100</span>, <span class='fl'>0.08</span>, <span class='fl'>0.0001</span>, <span class='fl'>0.2</span>),
       <span class='kw'>from</span> <span class='kw'>=</span> <span class='fl'>0</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>ylim</span> <span class='kw'>=</span> <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='fl'>0</span>, <span class='fl'>100</span>),
       <span class='kw'>xlab</span> <span class='kw'>=</span> <span class='st'>"Time"</span>, <span class='kw'>ylab</span> <span class='kw'>=</span> <span class='st'>"Residue"</span>)</div><div class='input'>  <span class='fu'><a href='https://www.rdocumentation.org/packages/graphics/topics/plot'>plot</a></span>(<span class='kw'>function</span>(<span class='no'>x</span>) <span class='fu'>logistic.solution</span>(<span class='no'>x</span>, <span class='fl'>100</span>, <span class='fl'>0.08</span>, <span class='fl'>0.0001</span>, <span class='fl'>0.4</span>),
       <span class='kw'>from</span> <span class='kw'>=</span> <span class='fl'>0</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>add</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>lty</span> <span class='kw'>=</span> <span class='fl'>2</span>, <span class='kw'>col</span> <span class='kw'>=</span> <span class='fl'>2</span>)</div><div class='input'>  <span class='fu'><a href='https://www.rdocumentation.org/packages/graphics/topics/plot'>plot</a></span>(<span class='kw'>function</span>(<span class='no'>x</span>) <span class='fu'>logistic.solution</span>(<span class='no'>x</span>, <span class='fl'>100</span>, <span class='fl'>0.08</span>, <span class='fl'>0.0001</span>, <span class='fl'>0.8</span>),
       <span class='kw'>from</span> <span class='kw'>=</span> <span class='fl'>0</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>add</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>lty</span> <span class='kw'>=</span> <span class='fl'>3</span>, <span class='kw'>col</span> <span class='kw'>=</span> <span class='fl'>3</span>)</div><div class='input'>  <span class='fu'><a href='https://www.rdocumentation.org/packages/graphics/topics/plot'>plot</a></span>(<span class='kw'>function</span>(<span class='no'>x</span>) <span class='fu'>logistic.solution</span>(<span class='no'>x</span>, <span class='fl'>100</span>, <span class='fl'>0.08</span>, <span class='fl'>0.001</span>, <span class='fl'>0.2</span>),
       <span class='kw'>from</span> <span class='kw'>=</span> <span class='fl'>0</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>add</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>lty</span> <span class='kw'>=</span> <span class='fl'>4</span>, <span class='kw'>col</span> <span class='kw'>=</span> <span class='fl'>4</span>)</div><div class='input'>  <span class='fu'><a href='https://www.rdocumentation.org/packages/graphics/topics/plot'>plot</a></span>(<span class='kw'>function</span>(<span class='no'>x</span>) <span class='fu'>logistic.solution</span>(<span class='no'>x</span>, <span class='fl'>100</span>, <span class='fl'>0.08</span>, <span class='fl'>0.08</span>, <span class='fl'>0.2</span>),
       <span class='kw'>from</span> <span class='kw'>=</span> <span class='fl'>0</span>, <span class='kw'>to</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='kw'>add</span> <span class='kw'>=</span> <span class='fl'>TRUE</span>, <span class='kw'>lty</span> <span class='kw'>=</span> <span class='fl'>5</span>, <span class='kw'>col</span> <span class='kw'>=</span> <span class='fl'>5</span>)</div><div class='input'>  <span class='fu'><a href='https://www.rdocumentation.org/packages/graphics/topics/legend'>legend</a></span>(<span class='st'>"topright"</span>, <span class='kw'>inset</span> <span class='kw'>=</span> <span class='fl'>0.05</span>,
         <span class='kw'>legend</span> <span class='kw'>=</span> <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/paste'>paste0</a></span>(<span class='st'>"k0 = "</span>, <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='fl'>0.0001</span>, <span class='fl'>0.0001</span>, <span class='fl'>0.0001</span>, <span class='fl'>0.001</span>, <span class='fl'>0.08</span>),
                         <span class='st'>", r = "</span>, <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='fl'>0.2</span>, <span class='fl'>0.4</span>, <span class='fl'>0.8</span>, <span class='fl'>0.2</span>, <span class='fl'>0.2</span>)),
         <span class='kw'>lty</span> <span class='kw'>=</span> <span class='fl'>1</span>:<span class='fl'>5</span>, <span class='kw'>col</span> <span class='kw'>=</span> <span class='fl'>1</span>:<span class='fl'>5</span>)</div><div class='img'><img src='logistic.solution-1.png' alt='' width='700' height='433' /></div><div class='input'>
  <span class='co'># Fit with synthetic data</span>
  <span class='no'>logistic</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinmod.html'>mkinmod</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fu'><a href='mkinsub.html'>mkinsub</a></span>(<span class='st'>"logistic"</span>))

  <span class='no'>sampling_times</span> <span class='kw'>=</span> <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='fl'>0</span>, <span class='fl'>1</span>, <span class='fl'>3</span>, <span class='fl'>7</span>, <span class='fl'>14</span>, <span class='fl'>28</span>, <span class='fl'>60</span>, <span class='fl'>90</span>, <span class='fl'>120</span>)
  <span class='no'>parms_logistic</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='kw'>kmax</span> <span class='kw'>=</span> <span class='fl'>0.08</span>, <span class='kw'>k0</span> <span class='kw'>=</span> <span class='fl'>0.0001</span>, <span class='kw'>r</span> <span class='kw'>=</span> <span class='fl'>0.2</span>)
  <span class='no'>parms_logistic_optim</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='kw'>parent_0</span> <span class='kw'>=</span> <span class='fl'>100</span>, <span class='no'>parms_logistic</span>)
  <span class='no'>d_logistic</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinpredict.html'>mkinpredict</a></span>(<span class='no'>logistic</span>,
    <span class='no'>parms_logistic</span>, <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/c'>c</a></span>(<span class='kw'>parent</span> <span class='kw'>=</span> <span class='fl'>100</span>),
    <span class='no'>sampling_times</span>)
  <span class='no'>d_2_1</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='add_err.html'>add_err</a></span>(<span class='no'>d_logistic</span>,
    <span class='kw'>sdfunc</span> <span class='kw'>=</span> <span class='kw'>function</span>(<span class='no'>x</span>) <span class='fu'><a href='sigma_twocomp.html'>sigma_twocomp</a></span>(<span class='no'>x</span>, <span class='fl'>0.5</span>, <span class='fl'>0.07</span>),
    <span class='kw'>n</span> <span class='kw'>=</span> <span class='fl'>1</span>, <span class='kw'>reps</span> <span class='kw'>=</span> <span class='fl'>2</span>, <span class='kw'>digits</span> <span class='kw'>=</span> <span class='fl'>5</span>, <span class='kw'>LOD</span> <span class='kw'>=</span> <span class='fl'>0.1</span>, <span class='kw'>seed</span> <span class='kw'>=</span> <span class='fl'>123456</span>)<span class='kw'>[[</span><span class='fl'>1</span>]]

  <span class='no'>m</span> <span class='kw'>&lt;-</span> <span class='fu'><a href='mkinfit.html'>mkinfit</a></span>(<span class='st'>"logistic"</span>, <span class='no'>d_2_1</span>)</div><div class='output co'>#&gt; Model cost at call  1 :  789.6044 
#&gt; Model cost at call  2 :  789.6043 
#&gt; Model cost at call  7 :  716.9934 
#&gt; Model cost at call  12 :  697.1186 
#&gt; Model cost at call  15 :  697.1185 
#&gt; Model cost at call  16 :  697.1184 
#&gt; Model cost at call  17 :  661.1574 
#&gt; Model cost at call  20 :  661.1573 
#&gt; Model cost at call  22 :  620.0542 
#&gt; Model cost at call  25 :  620.0541 
#&gt; Model cost at call  29 :  616.6874 
#&gt; Model cost at call  32 :  616.6874 
#&gt; Model cost at call  33 :  616.6874 
#&gt; Model cost at call  34 :  615.1671 
#&gt; Model cost at call  37 :  615.1671 
#&gt; Model cost at call  39 :  612.0795 
#&gt; Model cost at call  42 :  612.0795 
#&gt; Model cost at call  43 :  612.0795 
#&gt; Model cost at call  44 :  605.9119 
#&gt; Model cost at call  45 :  593.0433 
#&gt; Model cost at call  46 :  548.0815 
#&gt; Model cost at call  47 :  504.9062 
#&gt; Model cost at call  50 :  504.9061 
#&gt; Model cost at call  51 :  504.9061 
#&gt; Model cost at call  53 :  485.929 
#&gt; Model cost at call  55 :  485.929 
#&gt; Model cost at call  56 :  485.929 
#&gt; Model cost at call  58 :  485.241 
#&gt; Model cost at call  60 :  485.241 
#&gt; Model cost at call  61 :  485.2409 
#&gt; Model cost at call  62 :  485.2409 
#&gt; Model cost at call  63 :  484.0717 
#&gt; Model cost at call  69 :  483.9062 
#&gt; Model cost at call  74 :  483.5646 
#&gt; Model cost at call  79 :  483.4908 
#&gt; Model cost at call  84 :  483.4859 
#&gt; Model cost at call  85 :  483.4859 
#&gt; Model cost at call  89 :  483.4848 
#&gt; Model cost at call  90 :  483.4836 
#&gt; Model cost at call  92 :  483.4836 
#&gt; Model cost at call  94 :  483.4836 
#&gt; Model cost at call  97 :  483.4833 
#&gt; Model cost at call  100 :  483.4833 
#&gt; Model cost at call  105 :  483.4832 
#&gt; Model cost at call  108 :  483.4832 
#&gt; Model cost at call  114 :  483.4832 
#&gt; Model cost at call  128 :  483.4832 
#&gt; Optimisation by method Port successfully terminated.</div><div class='input'>  <span class='fu'><a href='plot.mkinfit.html'>plot_sep</a></span>(<span class='no'>m</span>)</div><div class='img'><img src='logistic.solution-2.png' alt='' width='700' height='433' /></div><div class='input'>  <span class='fu'><a href='https://www.rdocumentation.org/packages/base/topics/summary'>summary</a></span>(<span class='no'>m</span>)$<span class='no'>bpar</span></div><div class='output co'>#&gt;              Estimate   se_notrans    t value       Pr(&gt;t)        Lower
#&gt; parent_0 1.057896e+02 2.3743105248 44.5559374 6.656664e-16 1.006602e+02
#&gt; kmax     6.398190e-02 0.0193490291  3.3067243 2.836921e-03 3.329058e-02
#&gt; k0       1.612775e-04 0.0009640761  0.1672871 4.348592e-01 3.972250e-10
#&gt; r        2.263946e-01 0.2822811886  0.8020181 2.184792e-01 1.531165e-02
#&gt;                Upper
#&gt; parent_0 110.9190170
#&gt; kmax       0.1229682
#&gt; k0        65.4803698
#&gt; r          3.3474197</div></pre>
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    <h2>Contents</h2>
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      <li><a href="#arguments">Arguments</a></li>
      
      <li><a href="#note">Note</a></li>

      <li><a href="#value">Value</a></li>

      <li><a href="#references">References</a></li>
      
      <li><a href="#examples">Examples</a></li>
    </ul>

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