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) stands for 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 linearised system. and linsy which is a Scilab macro [ydot]=linsy(t,x) which computes the dynamics of the linearised 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 due to insecticide which destroys both prey 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