BYOM function simplefun.m (the model as explicit equations)

Syntax: Xout = simplefun(t,X0,par,c)

This function calculates the output of the model system. It is linked to the script files byom_bioconc_extra.m and byom_bioconc_start.m. As input, it gets:

Time t is handed over as a vector, and scenario name c as single number, by call_deri.m (you do not have to use them in this function). Output Xout (as matrix) provides the output for each state at each t.

Contents

Start

function Xout = simplefun(t,X0,par,c)
global glo   % allow for global parameters in structure glo (handy for switches)

Unpack initial states

The state variables enter this function in the vector _X_0.

Cw0 = X0(1); % state 1 is the external concentration at t=0
Ci0 = X0(2); % state 2 is the internal concentration at t=0

Unpack parameters

The parameters enter this function in the structure par. The names in the structure are the same as those defined in the byom script file. The 1 between parentheses is needed as each parameter has 5 associated values.

kd   = par.kd(1);   % degradation rate constant, d-1
ke   = par.ke(1);   % elimination rate constant, d-1
Piw  = par.Piw(1);  % bioconcentration factor, L/kg
Ct   = par.Ct(1);   % threshold external concentration that stops degradation, mg/L

Calculate the model output

This is the actual model, specified as explicit function(s):

$$ C_w=C_{w0} \exp(-k_d t) $$

$$ C_i= a \exp(-k_d t) + (C_{i0}-a) \exp(-k_e t) $$

$$ \textrm{with:} \quad a= \frac{C_{w0} k_e P_{iw}}{k_e-k_d} $$

Note: this function is not completely identical to the solution of the system of ODE's. Here, I ignore the stop in degradation, which would require quite some fiddling around. Watch out that this solution is only valid when ke is NOT equal to kd!

Cw = Cw0 * exp(-kd*t); % concentration in water

a  = Cw0 * ke * Piw/(ke-kd); % this factor occurs twice in the solution
Ci = a*exp(-kd*t) + (Ci0 - a) * exp(-ke*t); % internal concentration

Xout = [Cw Ci]; % combine them into a matrix