Figure 10.12:

The two steady-state biomass and substrate concentrations versus dilution rate; stable (solid), unstable (dashed).

Code for Figure 10.12

Text of the GNU GPL.

main.m

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% straighten out the chemostat
%
% Revised 8/16/2018

mum = 1;
D = 1.05;
Ks = 1;
Sf = 5;
y = 1;


Dopt = mum*(1 - sqrt(Ks/(Ks+Sf)))
Dc = mum*Sf/(Ks+Sf)

nDs = 25;
Dvec1 = linspace (0.001, Dc, nDs)';

for i = 1:nDs
    D = Dvec1(i);
    %compute eigenvalues of jacobian for both steady states
    %first steady state is washout
    xs11(i,:) = [0, Sf];
    Ss = D*Ks/(-D + mum);
    xs21(i,:) = [y*(Sf-Ss), Ss];
    lam1(i,:) = sort(eig(jacob(xs11(i,:), mum, D, Ks, Sf, y)));
    lam2(i,:) = sort(eig(jacob(xs21(i,:), mum, D, Ks, Sf, y)));
end

Dmax = Dc;
Dvec2 = linspace (Dc, 0.99, nDs)';

for i = 1:nDs
    D = Dvec2(i);
    %compute eigenvalues of jacobian for both steady states
    %first steady state is washout
    xs12(i,:) = [0, Sf];
    Ss = D*Ks/(-D + mum);
    xs22(i,:) = [y*(Sf-Ss), Ss];
    lam12(i,:) = sort(eig(jacob(xs12(i,:), mum, D, Ks, Sf, y)));
    lam22(i,:) = sort(eig(jacob(xs22(i,:), mum, D, Ks, Sf, y)));
end

% compute production rate
DX1 = Dvec1.*xs21(:,1);
DX2 = Dvec2.*xs22(:,1);

table1 = [Dvec1, xs11, xs21, DX1];
table2 = [Dvec2, xs12, xs22, DX2];

save "chemoss.dat"  table1 table2

if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
    subplot(2,1,1)
    plot(table1(:,1), [table1(:,2), table1(:,4), table1(:,6)], ...
         table2(:,2), [table2(:,2), table2(:,4), table2(:,6)], '-')
    axis([0,1,-2,5])

    subplot(2,1,2)
    plot(table1(:,1), [table1(:,3), table1(:,5)], ...
         table2(:,2), [table2(:,3), table2(:,5)], '-')
    axis([0,1,-0.5,10])
    % TITLE
end % PLOTTING

jacob.m

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function J = jacob(x, mum, D, Ks, Sf, y)
    X = x(1);
    S = x(2);
    mug = (mum*S)/(Ks+S);
    tmp = Ks*mum/(Ks+S)^2;
    J = [mug - D, X*tmp; -mug/y, -D-X/y*tmp];