Tune unstructured resources with carrying capacities (algae and detritus) to steady state
Source:R/reef-components.R
tuneUR_cc.RdFor models that use unstructured resources with carrying capacities, this function sets the algae biomass and the production rate of detritus so that each is at steady state.
Details
With a carrying capacity, the time evolution of the algae biomass \(B_A(t)\) is described by
$$ \frac{dB_A}{dt} = P_A\left( 1 - \frac{B_A}{K_A} \right) - c_A \, B_A $$
where \(K_A\) is the system's carrying capacity for algae in grams/ year,
\(c_A\) is the mass-specific rate of consumption calculated with
algae_consumption() and \(P_A\) is the rate at which algae
grows, calculated with getAlgaeProduction().
Unlike detritus, algal production on a reef is primary production and is
not driven by consumer demand: it is a property of the algae itself (see
getAlgaeProduction()), left unchanged by this function. Instead, this
tuning function solves the above steady-state condition for the algae
biomass, setting it to
\(B_A = (K_A \cdot P_A)/(P_A + K_A \cdot c_A)\) grams per meter squared,
so that production equals consumption for the given, fixed production
rate. This means that, all else equal, a decrease in grazing pressure
(lower \(c_A\)) increases the tuned algae biomass rather than reducing
\(P_A\) to compensate.
The time evolution of the detritus biomass \(B_D(t)\) is described by
$$ \frac{dB_D}{dt} = P_D\left( 1 - \frac{B_D}{K_D} \right) - c_D \, B_D $$
where \(K_D\) is the system's carrying capacity for detritus in grams/ year,
\(c_D\) is the mass-specific rate of consumption calculated with
detritus_consumption() and \(P_D\) is the rate at which detritus
is produced calculated with getDetritusProduction(). Total detritus
production is given with
$$p_D = p_{D.f} + p_{D.d} + p_{D.ext}$$
In this tuning function, the external production of detritus is set to \((c_D \cdot B_D)/(1-\frac{B_D}{K_D}) - P_{D.f} - P_{D.d}\) grams per meter squared per year so that production equals consumption at steady state. Unlike algae, detritus production genuinely is driven by the rest of the system (fish egestion, senescence) plus an external flux, so tuning the (external) production rate to match a chosen biomass remains appropriate.