# How can I add two differential equations to the system in a given time interval?

1 view (last 30 days)
Jack95 on 1 Feb 2023
Commented: Jack95 on 3 Feb 2023
Hello everyone! I am i'm modeling a process in which we initially have 3 equations: acetogen biomass production C(1), hydrogen substrate consumption C(2), acetate production C(3).
The code is as follows:
Tspan = [0:30];
C = [50 300 100];
Miumax=0.07;
Ks=10;
Yxs=0.1;
alfa=1-Yxs;
betha=0;
[t,C]=ode45(@(t,C)ParameterJack(t,C,Miumax,Ks,Yxs,alfa,betha),Tspan,C);
figure
hold on
yyaxis left
plot(t,C(:,1),'-g',t,C(:,2),'-b')
ylabel('acetogens (mgCOD/L), hidrogen (mgCOD/L)')
ylim([min(C(:,1)) max(ylim)])
ylim([min(C(:,2)) max(ylim)])
yyaxis right
plot(t,C(:,3));
ylabel('acetate (mgCOD/L)')
ylim([min(C(:,3)) max(ylim)])
xlabel('tempo (giorni)')
grid
legend('acetogeni','idrogeno','acetato', 'Location','best')
function dCndt=ParameterJack(t,C,Miumax,Ks,Yxs,alfa,betha)
mu = (Miumax*C(2))/(Ks+C(2));
alfa=1-Yxs;
Miumax=0.07;
Ks=10;
Yxs=0.175
dCndt(1,:) = mu*C(1);
dCndt(2,:) = -C(1)*(mu/Yxs);
dCndt(3,:) = (C(1)*((alfa*mu)+betha))
end
after 20 days I have that competing organisms (methanogens) C(4) are activated and produce methane (5), and the code becomes:
Tspan = [0:30];
C0 = [50 300 100 10 0];
Miumax=0.07;
Ks=10;
Yxs=0.1;
alfa=9;
betha=0;
y=0.2
km=100;
ka=3
a2=0.8
a1=0.559;
u=0.1
[t,C]=ode45(@(t,C)ParameterJack(t,C,Miumax,Ks,km,ka,a2,a1,Yxs,u,y,alfa,betha),Tspan,C0);
figure
hold on
yyaxis left
plot(t,C(:,1),'-g',t,C(:,2),'-b',t,C(:,4),'-m');
ylabel('acetogens (mgCOD/L), hidrogen (mgCOD/L),metanogens')
ylim([min(C(:,1)) max(ylim)])
ylim([min(C(:,2)) max(ylim)])
yyaxis right
plot(t,C(:,3),'r', t,C(:,5),'-o');
ylabel('acetate (mgCOD/L), methane (mgCOD/L)')
ylim([min(C(:,3)) max(ylim)])
hold off
xlabel('tempo (giorni)')
grid
legend('acetogens','Hidrogen','metanogens','acetate','methane', 'Location','best')
function dCdt=ParameterJack(t,C,Miumax,Ks,km,u,ka,y,a2,a1,Yxs,alfa,betha)
mum = (0.1)*(C(2))/(km+C(2));
mu = (Miumax*C(2))/(Ks+C(2));
alfa=9;
Miumax=0.07;
u=0.1;
ka=50;
Ks=10;
Yxs=0.175
y=0.2;
km=100;
a2=0.0121
a1=0.159
dCdt(1,:) = mu*C(1);
dCdt(2,:) = -C(1)*(mu/Yxs)-(C(4)*(mum/y));
dCdt(3,:) = (C(1)*((alfa*mu)+betha))-C(3)*a2*C(4);
dCdt(4,:) = mum*C(4)+(u)*((C(3)*(1-a2)/((C(3)*(1-a2))+ka))*C(4));
dCdt(5,:) = (C(3)*a2)+(C(2)*a1);
end
I would like to understand how I can represent this thing in the same graph, where the growth of methanogens and the production of methane starts on day 20 and not on day 0.
Thanks everyone for the help!

Alan Stevens on 1 Feb 2023
Like this? (though I'm not sure I've interpreted which constants apply during which interval correctly!)
Tspan = 0:30;
C0 = [50 300 100 10 0];
Miumax=0.07;
Ks=10;
Yxs=0.1;
alfa=9;
betha=0;
y=0.2;
km=100;
ka=3;
a2=0.8;
a1=0.559;
u=0.1;
[t,C]=ode45(@(t,C)ParameterJack(t,C,betha),Tspan,C0);
figure
hold on
yyaxis left
plot(t,C(:,1),'-g',t,C(:,2),'-b',t,C(:,4),'-m');
ylabel('acetogens (mgCOD/L), hidrogen (mgCOD/L),metanogens')
ylim([min(C(:,1)) max(ylim)])
ylim([min(C(:,2)) max(ylim)])
yyaxis right
plot(t,C(:,3),'r', t,C(:,5),'-o');
ylabel('acetate (mgCOD/L), methane (mgCOD/L)')
ylim([0 max(ylim)])
hold off
xlabel('tempo (giorni)')
grid
legend('acetogens','Hidrogen','metanogens','acetate','methane', 'Location','best')
function dCdt=ParameterJack(t,C,betha)
Miumax=0.07;
alfa=9;
u=0.1;
Ks = 10;
y=0.2;
km=100;
Yxs = 0.175;
mum = (0.1)*(C(2))/(km+C(2));
mu = (Miumax*C(2))/(Ks+C(2));
if t<20
term2 = 0;
term3 = 0;
term4 = 0;
term5 = 0;
else
ka = 50;
a2=0.0121;
a1=0.159;
term2 = (C(4)*(mum/y));
term3 = C(3)*a2*C(4);
term4 = mum*C(4)+(u)*((C(3)*(1-a2)/((C(3)*(1-a2))+ka))*C(4));
term5 = (C(3)*a2)+(C(2)*a1);
end
dCdt(1,:) = mu*C(1);
dCdt(2,:) = -C(1)*(mu/Yxs)-term2;
dCdt(3,:) = (C(1)*((alfa*mu)+betha))-term3;
dCdt(4,:) = term4;
dCdt(5,:) = term5;
end
Jack95 on 3 Feb 2023
wonderful! You've been a great help, thank you so much!

### Categories

Find more on Mathematics and Optimization in Help Center and File Exchange

R2020a

### Community Treasure Hunt

Find the treasures in MATLAB Central and discover how the community can help you!

Start Hunting!