MATLAB code
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%define the function file
function [vdot] = growth(t,v)
% a general growth model where the concentration of nutrient does not have any influence on the bacterial growth
k=1;
%The Hill function
nutrient = 21; %concentration of nutrient
n = 0.8; % the Hill coefficient, describes how cooperative the two variable are
Ka = 10.5; % the nutrient concentration occupying half of the cell
theta = (nutrient^n) / (((Ka)^n) + (nutrient^n));
%the Hill Function, models the cooperativitiy between the bacteria and nutrients
vdot = k*(theta)*v;
(In a separate M-file)
%calling the function
[t, v] = ode45('growth', [0, 5], 1);
k=1;
v_true = exp(k*t); %the analytical solution
nutrient = 21; %concentration of nutrient
n = 0.8; % the Hill coefficient, describes how cooperative the two variable are
Ka = 10.5; % the nutrient concentration occupying half of the cell
%the Hill Function, models the cooperativitiy between the bacteria and nutrient
theta = (nutrient^n) / (((Ka)^n) + (nutrient^n));
v1_true = exp(theta*t);
nutrient2 = 40;
%the Hill Function, models the cooperativitiy between the bacteriam and nutrient
theta2 = (nutrient2^n) / (((Ka)^n) + (nutrient2^n));
v2_true = exp(theta2*t);
nutrient3 = 80;
%the Hill Function, models the cooperativitiy between the bacteria and nutrient
theta3 = (nutrient3^n) / (((Ka)^n) + (nutrient3^n));
v3_true = exp(theta3*t);
nutrient4 = 100;
%the Hill Function, models the cooperativitiy between the bacteria and nutrient
theta4 = (nutrient4^n) / (((Ka)^n) + (nutrient4^n));
v4_true = exp(theta4*t);
%plot
subplot(2,2,1);
plot(t, v, 'o', t, v_true), xlabel('Time(s)');
%NB: solution of t and v in dots, solution of t and v_true is shown in line
ylabel('growth');
title('ODE model showing the growth rate where the growth rate is constant ');
subplot(2,2,2);
plot(t,v,'o', t, v1_true), xlabel('Time(s)');
%NB: solution of t and v in dots, solution of t and v_true (analytical solutionn) is shown in line
ylabel('growth');
title('ODE model showing the relationship between the growth rate and the internal concentration[21] with a Hill Function');
subplot (2,2,3);
plot(t,v,'o', t, v2_true), xlabel('Time(s)');
%NB: solution of t and v in dots, solution of t and v_true is shown in line
ylabel('growth');
title('ODE model showing the relationship between the growth rate and the internal concentration[40] with a Hill Function');
subplot (2,2,4);
plot(t,v,'o', t, v4_true), xlabel('Time(s)');
%NB: solution of t and v in dots, solution of t and v_true is shown in line
ylabel('growth');
title('ODE model showing the relationship between the growth rate and the internal concentration[100] with a Hill Function');
