function f = bio_ss(x)
%
% b.w. bequette
% (c) 16 Nov 92
% revised 18 July 96
%
% find steady-states of bioreactor, using fsolve:
%  
%  x = fsolve('bio',x0)
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% where x0 is a vector of initial guesses
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% x(1) = biomass
% x(2) = substrate
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% biomass ("bugs") consumes the substrate
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% D     = dilution rate (F/V, time^-1)
% Y     = yield biomass/substrate
% mu    = specific growth rate
% mumax = parameter (both Monod and Substrate Inhibition)
% km    = parameter (both Monod and Substrate Inhibition)
% k1    = parameter (Substrate Inhibition only, k1 = 0 for Monod)
% sf    = substrate feed concentration
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% the function vector consists of 2 equations
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  f = zeros(2,1);
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% parameter values
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   D = 0.3;
   mumax = 0.53;
   Y = 0.4;
   km = 0.12;
   sf = 4.0;
   k1 = 0.4545;
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%  Substrate Inhibition expression for specific growth rate
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   mu = mumax*x(2)/(km+x(2)+k1*x(2)*x(2));
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%  steady-state equations
%  fsolve varies x(1) and x(2) to drive f(1) and f(2) to zero
%
   f(1) = (mu - D)*x(1);
   f(2) = (sf - x(2))*D - mu*x(1)/Y;