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<!DOCTYPE MAN SYSTEM "../../manrev.dtd">
<MAN>
  <LANGUAGE>eng</LANGUAGE>
  <TITLE>systems</TITLE>
  <TYPE>Scilab Function</TYPE>
  <DATE>April 1993</DATE>
  <SHORT_DESCRIPTION name="systems"> a collection of dynamical system</SHORT_DESCRIPTION>
  <CALLING_SEQUENCE>
    <CALLING_SEQUENCE_ITEM>[]=systems()  </CALLING_SEQUENCE_ITEM>
  </CALLING_SEQUENCE>
  <DESCRIPTION>
    <P>
    A call to this function will load into Scilab a set of macros which
    describes dynamical systems. Their parameters can be initiated by 
    calling the routine tdinit().
  </P>
  </DESCRIPTION>
  <SECTION label="Bioreact">
    <VERBATIM>
<![CDATA[
[ydot]=biorecat(t,x)
   ]]>
    </VERBATIM>
    <P>
    a bioreactor model, 
  </P>
    <ITEMIZE>
      <ITEM label="x(1)">
        <SP>is the biomass concentration</SP>
      </ITEM>
      <ITEM label="x(2)">
        <SP>is the sugar concentration</SP>
      </ITEM>
    </ITEMIZE>
    <VERBATIM>
<![CDATA[
                    xdot(1)=mu_td(x(2))*x(1)- debit*x(1);
                    xdot(2)=-k*mu_td(x(2))*x(1)-debit*x(2)+debit*x2in;
   ]]>
    </VERBATIM>
    <P>
    where mu_td is given by 
  </P>
    <VERBATIM>
<![CDATA[
                    mu_td(x)=x/(1+x);
   ]]>
    </VERBATIM>
  </SECTION>
  <SECTION label="Compet">
    <VERBATIM>
<![CDATA[
[xdot]=compet(t,x [,u])
   ]]>
    </VERBATIM>
    <P>
    a competition model. <VERB>x(1),x(2)</VERB> stands for two populations which  grows on a same resource. <VERB>1/u</VERB> is the level of that resource ( 1 is the default value).
  </P>
    <VERBATIM>
<![CDATA[
xdot=0*ones(2,1);
xdot(1) = ppr*x(1)*(1-x(1)/ppk) - u*ppa*x(1)*x(2) ,
xdot(2) = pps*x(2)*(1-x(2)/ppl) - u*ppb*x(1)*x(2) ,
   ]]>
    </VERBATIM>
    <P>
     &quot;The macro <VERB>[xe]=equilcom(ue)</VERB>&quot; computes an equilibrium point of the competition model and a fixed  level of the resource ue ( default value is 1)
  </P>
    <P>
     &quot;The macro <VERB>[f,g,h,linsy]=lincomp([ue])</VERB>&quot; gives the linearisation of the competition model ( with output y=x) around the equilibrium point xe=equilcom(ue). This macro returns [f,g,h] the three matrices of the linearised system. and linsy which is a Scilab macro [ydot]=linsy(t,x) which computes the  dynamics of the linearised system
  </P>
  </SECTION>
  <SECTION label="Cycllim">
    <VERBATIM>
<![CDATA[
[xdot]=cycllim(t,x)
   ]]>
    </VERBATIM>
    <P>
    a model with a limit cycle 
  </P>
    <VERBATIM>
<![CDATA[
  xdot=a*x+qeps(1-||x||**2)x
   ]]>
    </VERBATIM>
  </SECTION>
  <SECTION label="Linear">
    <VERBATIM>
<![CDATA[
[xdot]=linear(t,x)
   ]]>
    </VERBATIM>
    <P>
    a linear system 
  </P>
  </SECTION>
  <SECTION label="Blinper">
    <VERBATIM>
<![CDATA[
[xdot]=linper(t,x)
   ]]>
    </VERBATIM>
    <P>
    a linear system with quadratic perturbations.
  </P>
  </SECTION>
  <SECTION label="Pop">
    <VERBATIM>
<![CDATA[
[xdot]=pop(t,x)
   ]]>
    </VERBATIM>
    <P>
    a fish population model
  </P>
    <VERBATIM>
<![CDATA[
xdot= 10*x*(1-x/K)- peche(t)*x
   ]]>
    </VERBATIM>
  </SECTION>
  <SECTION label="Proie">
    <P>
    a Predator prey model with external insecticide.
  </P>
    <VERBATIM>
<![CDATA[
[xdot]=p_p(t,x,[u]
   ]]>
    </VERBATIM>
    <ITEMIZE>
      <ITEM label="x(1)">
        <SP>The prey ( that we want to kill )</SP>
      </ITEM>
      <ITEM label="x(2)">
        <SP>the predator ( that we want to preserve )</SP>
      </ITEM>
      <ITEM label="u">
        <SP>mortality rate due to insecticide which  destroys both prey and predator ( default value u=0)</SP>
      </ITEM>
    </ITEMIZE>
    <VERBATIM>
<![CDATA[
xdot(1) = ppr*x(1)*(1-x(1)/ppk) - ppa*x(1)*x(2) - u*x(1);
xdot(2) = -ppm*x(2)             + ppb*x(1)*x(2) - u*x(2);
   ]]>
    </VERBATIM>
    <P>
     The macro <VERB>[xe]=equilpp([ue])</VERB> computes the equilibrium point of the <VERB>p_p</VERB> system for the value <VERB>ue</VERB> of the command. The default value for <VERB>ue</VERB> is 0.</P>
    <VERBATIM>
<![CDATA[
                    xe(1) =  (ppm+ue)/ppb;
                    xe(2) =  (ppr*(1-xe(1)/ppk)-ue)/ppa;
   ]]>
    </VERBATIM>
  </SECTION>
  <SECTION label="Lincom">
    <VERBATIM>
<![CDATA[
[xdot]=lincom(t,x,k)
   ]]>
    </VERBATIM>
    <P>
    linear system with a feedback</P>
    <VERBATIM>
<![CDATA[
 xdot= a*x +b*(-k*x)
   ]]>
    </VERBATIM>
  </SECTION>
  <SEE_ALSO>
    <SEE_ALSO_ITEM>
      <LINK>tdinit</LINK>
    </SEE_ALSO_ITEM>
  </SEE_ALSO>
</MAN>
