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_debkiss_no_egg.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

Unpack states

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

WV = X(1); % state 1 is the structural body mass
cR = X(2); % state 2 is the cumulative reproduction (not used in this function)
WB = 0; % no egg phase, so egg buffer is empty

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.

dV    = glo.dV;       % dry weight density of structure
delM  = glo.delM;     % shape correction coefficient (if needed)
% delM is used to allow Lwp as a parameter instead of WVp

sJAm  = par.sJAm(1);  % specific assimilation rate
sJM   = par.sJM(1);   % specific maintenance costs
WB0   = par.WB0(1);   % initial weight of egg
Lwp   = par.Lwp(1);   % shell length at puberty
yAV   = par.yAV(1);   % yield of assimilates on volume (starvation)
yBA   = par.yBA(1);   % yield of egg buffer on assimilates
yVA   = par.yVA(1);   % yield of structure on assimilates (growth)
kap   = par.kap(1);   % allocation fraction to soma
f     = par.f(1);     % scaled food level
fB    = par.fB(1);    % scaled food level for the embryo
Lwf   = par.Lwf(1);   % half-saturation length for initial food limitation

if glo.len ~= 0 % only if the switch is set to 1 or 2!
    % Translate shell length at puberty to body weight
    WVp = glo.dV*(Lwp * glo.delM).^3; % initial length to initial dwt
    WVf = glo.dV*(Lwf * glo.delM).^3; % initial length to initial dwt
end

Calculate the derivatives

This is the actual model, specified as a system of ODEs. Note: some unused fluxes are calculated because they may be needed for future modules (e.g., respiration and ageing).

L = (WV/dV)^(1/3); % volumetric length

% select what to do with maturity maintenance
if glo.mat == 1
    sJJ = sJM * (1-kap)/kap; % add specific maturity maintenance with the suggested value
else
    sJJ = 0; % or ignore it
end

WVb = WB0 *yVA * kap; % body mass at birth

if WB > 0 % if we have an embryo
    f = fB; % assimilation at different rate
else
    if WVf > 0
        f = f / (1+WVf/WV); % hyperbolic relationship for f with body weight
    end
end

JA = f * sJAm * L^2;          % assimilation
JM = sJM * L^3;               % somatic maintenance
JV = yVA * (kap*JA-JM);       % growth

if WV < WVp                   % below size at puberty
    JR = 0;                   % no reproduction
    JJ = sJJ * L^3;           % maturity maintenance flux
    % JH = (1-kap) * JA - JJ; % maturation flux (not used!)
else
    JJ = sJJ * (WVp/dV);      % maturity maintenance
    JR = (1-kap) * JA - JJ;   % reproduction flux
end

% starvation rules may override these fluxes
if kap * JA < JM      % allocated flux to soma cannot pay maintenance
    if JA >= JM + JJ  % but still enough total assimilates to pay both maintenances
        JV = 0;       % stop growth
        if WV >= WVp  % for adults ...
            JR = JA - JM - JJ; % repro buffer gets what's left
        else
            % JH = JA - JM - JJ; % maturation gets what's left (not used!)
        end
    elseif JA >= JM   % only enough to pay somatic maintenance
        JV = 0;       % stop growth
        JR = 0;       % stop reproduction
        % JH = 0;     % stop maturation for juveniles (not used)
        JJ = JA - JM; % maturity maintenance flux gets what's left
    else              % we need to shrink
        JR = 0;       % stop reproduction
        JJ = 0;       % stop paying maturity maintenance
        % JH = 0;     % stop maturation for juveniles (not used)
        JV = (JA - JM) / yAV; % shrink; pay somatic maintenance from structure
    end
end

% we do not work with a repro buffer here, so a negative JR does not make
% sense; therefore, minimise to zero
JR = max(0,JR);

% Calcululate the derivatives
dWV = JV;             % change in body mass
dcR = yBA * JR / WB0; % continuous reproduction flux
if WB > 0             % for embryos ...
    dWB = -JA;        % decrease in egg buffer
else
    dWB = 0;          % for juveniles/adults, no change
end
if WV < WVb/4 && dWB == 0 % dont shrink below quarter size at birth
    dWV = 0; % simply stop shrinking ...
end

dX = [dWV;dcR];       % collect both derivatives in one vector