Calculates the detritus biomass at the next time step based on the current detritus biomass.
Arguments
- params
A MizerParams object
- n
A matrix of current species abundances (species x size)
- n_other
Other dynamic components.
- rates
A list of rates as returned by
getRates()- dt
Time step size
- ...
Unused
Details
The time evolution of the detritus biomass \(B\) is described by
$$dB_D/dt = P_D - c_D \cdot B_D $$
where \(c_D\) is the mass-specific rate of consumption, calculated
with detritus_consumption()and \(P_D\) is the rate at which the
rest of the system produces detritus biomass, calculate with
getDetritusProduction().
The dynamical equation is solved analytically to
$$B_D(t+dt) = B(t)\exp(-c_D \cdot dt) + \frac{P_D}{c_D} (1-\exp(-c_D \cdot dt)).$$
This avoids the stability problems that would arise if we used the Euler method to solve the equation numerically.
B_D(t+dt) above is a convex combination of B_D(t) and P_D/c_D, and
is therefore guaranteed non-negative only when P_D >= 0. P_D (from
getDetritusProduction()) is feces + decomp + external: feces and
decomp scale with current fish abundance and mortality, but external
is a fixed constant, tuned once at fixed abundances by
tuneUR()/tuneUR_cc() and never updated again. In a live, non-frozen
simulation (e.g. a multi-year mizer::project() call with fishing),
feces + decomp can shrink enough that the fixed external term drives
total production negative, breaking the non-negativity guarantee above
and eventually producing negative/NaN detritus biomass. Production is
therefore floored at zero (production <- max(production, 0)) before it
enters the analytic update above. This has no effect at any steady state
tuned by tuneUR()/tuneUR_cc() (there, production equals consumption,
which is non-negative by construction) – it only ever engages once a
live simulation has moved away from that steady state, which is exactly
the situation where an unbounded negative production would otherwise be
unphysical: external represents net exchange with the surrounding
system (see the "flux of external detritus is negative" warning in
tuneUR()/tuneUR_cc(), documented there as detritus flowing off the
reef), and the true floor on a net production/export rate is zero
(nothing can be produced from, or exported past, an empty pool) rather
than an unboundedly negative constant.