BYOM function derivatives.m (the model in ODEs)

Syntax: dX = derivatives(t,X,par,c)

This function calculates the derivatives for 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 and scenario name c are handed over as single numbers by call_deri.m (you do not have to use them in this function). Output dX (as vector) provides the differentials for each state at t.

Contents

Start

function dX = derivatives(t,X,par,c)
global glo   % allow for global parameters in structure glo (handy for switches)

Unpack states

The state variables enter this function in the vector X. Here, we give them a more handy name.

Cw = X(1); % state 1 is the external concentration at previous time point
Ci = X(2); % state 2 is the internal concentration at previous time point

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

% % Optionally, include the threshold concentration as a global (see script)
% Ct   = glo.Ct;      % threshold external concentration that stops degradation, mg/L

Calculate the derivatives

This is the actual model, specified as a system of two ODEs:

$$ \frac{d}{dt}C_w=-k_d C_w \quad \textrm{as long as } C_w>C_t $$

$$ \frac{d}{dt}C_i=k_e(P_{iw}C_w-C_i) $$

if Cw > Ct % if we are above the critical internal concentration ...
    dCw = -kd * Cw; % let the external concentration degrade
else           % otherwise ...
    dCw = 0; % make the change in external concentration zero
end

dCi = ke * (Piw*Cw-Ci); % first order bioconcentration

dX = [dCw;dCi]; % collect both derivatives in one vector dX