Lotka-Volterra Systems - QuiescentModelUsingStart

model QuiescentModelUsingStart "Find steady state solutions to LotkaVolterra equations"
  parameter Real alpha=0.1 "Reproduction rate of prey";
  parameter Real beta=0.02 "Mortality rate of predator per prey";
  parameter Real gamma=0.4 "Mortality rate of predator";
  parameter Real delta=0.02 "Reproduction rate of predator per prey";
  Real x(start=10) "Prey population";
  Real y(start=10) "Predator population";
initial equation
  der(x) = 0;
  der(y) = 0;
equation
  der(x) = x*(alpha-beta*y);
  der(y) = y*(delta*x-gamma);
end QuiescentModelUsingStart;

Because of various limitation of SysPhS-1.1 some workarounds were required to get this one to work.

The exported Modelica code is:

model QuiescentModelUsingStart
  parameter Real alpha(start=0.1,fixed=true);
  parameter Real beta(start=0.02,fixed=true);
  parameter Real gamma(start=0.4,fixed=true);
  parameter Real delta(start=0.02,fixed=true);
  Real x(start=20.0,fixed=true);
  Real y(start=5.0,fixed=true);
  Real dx(start=0.0,fixed=true);
  Real dy(start=0.0,fixed=true);
equation
  der(x)=x*(alpha-beta*y);
  der(y)=y*(delta*x-gamma);
  dx=der(x);
  dy=der(y);
end QuiescentModelUsingStart;

Because of the lack of explicit support for 'initial equation', additional variables dx and dy with equations dx = der(x) and dy = der(y) with start values 0 were introduced in the SysPhS version. But they are not in fact compatible with the non-fixed 'start' values in the Modelica University version:

  Real x(start=10) "Prey population";
  Real y(start=10) "Predator population";
initial equation
  der(x) = 0;
  der(y) = 0;
equation
  der(x) = x*(alpha-beta*y);
  der(y) = y*(delta*x-gamma);

That doesn't matter for the Modelica University version, but it does matter if you try to mimic it with SysPhS, because currently (in SysPhS-1.1) default values are always assumed to be 'fixed', not mere "suggestions" for iteration variables:

Wolfram SystemModeler correctly complains if you use x(start=10,fixed=true) and y(start=10,fixed=true).

To get around this, in the SysPhS version a HACK was used, the "correct" steady values x=20 and y=5 were chosen as forced default values. Then it validates in Wolfram SystemModeler and runs (and you can see it indeed flat-lines at those values).