Various models have been built to explain the growth of algal cultures. Basically, most of the models can be split into two groups. Firstly, Monod’s model, that describes the algal growth based on the amount of limiting substrate or nutrient available in the media. Secondly, Droop’s model that includes both internal and external nutrient content to determine the growth of algae.
This post aims at explanation and application of Monod’s model of growth. Algal growth curve generally consists of a lag phase, an exponential phase, a stationary phase and death phase
Monod model considers a logarthmic equation for culture density with a specific growth rate constant(μ) and half-saturation constant of substrate(Ks). It also includes a constant for yield of algae culture per unit nutrient consumed(Yxs). Inverse of this quantity provides the ratio of nutrient consumed per cell or culture density. Once the equations and constants are known, simulations can be done in order to validate or extrapolate the experimental values.
MatLab has inbuilt numerical integration functions namely ODE45, ODE23, ODE15S etc. For our case, ODE45 will mostly do the work. To simulate Monod based algal model carry out the following steps.
1. Create a m-file with the name monod.m and copy the following code :
function dydt = monod(t,x)
dydt = zeros(2,1);
μ = % Provide specific growth rate value in 1/day
Ks = % Provide half-saturation constant value in mg/L
Yxs = % Provide algae yield per mg of nutrient
dydt(1) = μ * (x(2)/(x(2) + Ks)) * x(1);
dydt(2) = – dydt(1)/ Yxs ;
2. Create another m-file and save it as monodop.m . Copy and paste the following code.
ts = [ 0 7 ]; % Time span in days.
x01 = ; % Enter initial concentration of algal culture in mg/L
x02 = ; % Enter initial concentration of substrate in mg/L.
x0 = [ x01 x02];
[ t x ] = ode45(‘monod’,ts,x0);
ylabel(‘Algal concentration, mg/L’);
title(‘Algal concentration with time);
ylabel(‘Substrate concentration, mg/L’);
title(‘Substrate concentration with time);
3. Provide all the required constants value in monod.m and initial conditions in monodop.m file.
4. Run the ‘monod.m’ file first followed by ‘monodop.m’ file.
The above code displays algae growth curve for given values of constant. To compare the curve with experimental values use hold on command, that helps in plotting two sets of data on same plot.
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