systems(1)

NAME
systems - a collection of dynamical system

CALLING SEQUENCE
[]=systems()

DESCRIPTION
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().

BIOREACT
[ydot]=biorecat(t,x)
a bioreactor model,

x(1)
    is the biomass concentration

x(2)
    is the sugar concentration
                        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;
    where mu_td is given by
                        mu_td(x)=x/(1+x);

COMPET
[xdot]=compet(t,x [,u])
a competition model. x(1),x(2) represemt two populations which grows on a same resource. 1/u is the level of that resource ( 1 is the default value).
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) ,

The macro [xe]=equilcom(ue)
    computes an equilibrium point of the competition model and a fixed level of the resource ue ( default value is 1)

The macro [f,g,h,linsy]=lincomp([ue])
    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 linearized system. and linsy which is a Scilab macro [ydot]=linsy(t,x) which computes the dynamics of the linearized system

CYCLlIM
[xdot]=cycllim(t,x)
a model with a limit cycle
  xdot=a*x+qeps(1-||x||**2)x

LINEAR
[xdot]=linear(t,x)
a linear system

BLINPER
[xdot]=linper(t,x)
a linear system with quadratic perturbations.

POP
[xdot]=pop(t,x)
a fish population model
xdot= 10*x*(1-x/K)- peche(t)*x

PROIE
a Predator prey model with external insecticide.
[xdot]=p_p(t,x,[u]

x(1)
    The prey ( that we want to kill )

x(2)
    the predator ( that we want to preserve )

u   mortality rate du to insecticide which destroys both pray and predator ( default value u=0)
    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);
    The macro [xe]=equilpp([ue]) computes the equilibrium point of the p_p system for the value ue of the command. The default value for ue is 0.
                        xe(1) =  (ppm+ue)/ppb;
                        xe(2) =  (ppr*(1-xe(1)/ppk)-ue)/ppa;

LINCOM
[xdot]=lincom(t,x,k)
linear system with a feedback
	xdot= a*x +b*(-k*x)

SEE ALSO
tdinit.