function xdot = bio(t,x)
%
% b.w. bequette
% (c) 18 July 96
%
% dynamic equations for bioreactor, integrated using ode45,
% using the following command
%
% [t,x] = ode45('bio',t0,tf,x0)
%
% where t0 is the initial time (usually 0), tf is
% the final time and x0 is the initial condition vector
% x0(1) = biomass initial condition
% x0(2) = substrate initial condition
%
% biomass ("bugs") consumes the substrate
%
% state variables
%
% x(1) = biomass
% x(2) = substrate
%
% 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
%
% the function vector consists of 2 equations
%
  f = zeros(2,1);
%
% parameter values
%
   D = 0.3;
   mumax = 0.53;
   Y = 0.4;
   km = 0.12;
   sf = 4.0;
   k1 = 0.4545;
%
%  Substrate Inhibition expression for specific growth rate
%
   mu = mumax*x(2)/(km+x(2)+k1*x(2)*x(2));
%
%  dynamic equations
%
   dx1dt = [(mu - D)*x(1);
   dx2dt = (sf - x(2))*D - mu*x(1)/Y;
   xdot = [dx1dt; dx2dt];