# 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:

• t is the time point, provided by the ODE solver
• X is a vector with the previous value of the states
• par is the parameter structure
• c is the external concentration (or scenario number)

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.

## 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
Ci = X(2); % state 2 is the internal concentration


## 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:

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