8.7.3.17. Grazing
Grazing loss of plankton \(j\):
\[G_j = \sum_{z\in\op{pred}} G_{j,z}\]
where
\[G_{j,z} = g^{\max}_z
\frac{({p}_{j,z} {c}_j)^s}{A_z}
\frac{p_z^h}{p_z^h + {k^{{{\text{graz}}}}_z}^h}
(1 - {\mathrm{e}}^{-i_{{{\text{graz}}}} p_z})^{e_{\op{inhib}}}
f^{{{\text{graz}}}}_z(T)^{e^{\op{graz}}_j}
{c}_z\]
with
\[A_z = \biggl[ \sum_j ({p}_{j,z} {c}_j)^s \biggr]_{\ge c^{\min}_{\op{graz}}}\]
\[p_z = \biggl[ \sum_j {p}_{j,z} {c}_j - c^{\min}_{\op{graz}} \biggr]_{\ge 0}\]
\(s\) is 1 for non-switching and 2 for switching grazers
(#define DARWIN_GRAZING_SWITCH). The exponent \(h\) defaults to
1.
Note: For non-switching grazers (\(s=1\)), Ben has an additional
factor
\[\frac{S^{[j]}_z}{S^{{{\text{phy}}}}_z + S^{{\text{zoo}}}_z}\]
in \(G_{j,z}\) where
\[ \begin{align}\begin{aligned}S^{{{\text{phy}}}}_z &= \sum_{j\in{{\text{phy}}}} {p}_{j,z} {c}_j\\S^{{\text{zoo}}}_z &= \sum_{j\in{{\text{zoo}}}} {p}_{j,z} {c}_j\end{aligned}\end{align} \]
and \(S^{[j]}_z\) is the sum for the class plankton \(j\)
belongs to. This is not implemented yet!
Gains from grazing:
\[ \begin{align}\begin{aligned}g^{{\mathrm{C}}}_z &= \sum_j G_{j,z} a_{j,z} {{\text{reg}}}^{Q{\mathrm{C}}}_z\\g^{{\mathrm{P}}}_z &= \sum_j G_{j,z} a_{j,z} {{\text{reg}}}^{Q{\mathrm{P}}}_z Q^{\mathrm{P}}_j
\qquad\text{if }\op{DARWIN\_ALLOW\_PQUOTA}\\&\dots\\g^{\op{DOC}} &= \sum_{j,z} G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{C}}}_z) (1 - f^{\text{exp graz}}_{j,z})\\g^{\op{DOP}} &= \sum_{j,z} \begin{cases}
G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{P}}}_z) (1 - f^{\text{exp graz}}_{j,z}) Q^{\mathrm{P}}_j
&\text{if }\op{DARWIN\_ALLOW\_PQUOTA}\\ G_{j,z} (R^{{\mathrm{P}}:{\mathrm{C}}}_j - a_{j,z} R^{{\mathrm{P}}:{\mathrm{C}}}_z) (1 - f^{\text{exp graz}}_{j,z})
&\text{else}
\end{cases}\\&\dots\\g^{\op{POC}} &= \sum_{j,z} G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{C}}}_z) f^{\text{exp graz}}_{j,z}\\g^{\op{POP}} &= \sum_{j,z} \begin{cases}
G_{j,z} (1 - a_{j,z} {{\text{reg}}}^{Q{\mathrm{P}}}_z) f^{\text{exp graz}}_{j,z} Q^{\mathrm{P}}_j
&\text{if }\op{DARWIN\_ALLOW\_PQUOTA}\\ G_{j,z} (R^{{\mathrm{P}}:{\mathrm{C}}}_j - a_{j,z} R^{{\mathrm{P}}:{\mathrm{C}}}_z) f^{\text{exp graz}}_{j,z}
&\text{else}
\end{cases}\\&\dots\\g^{\op{POSi}} &= \sum_{j,z} \begin{cases}
G_{j,z} Q^{\op{Si}}_j &\text{if }\op{DARWIN\_ALLOW\_SIQUOTA}\\ G_{j,z} R^{{\op{Si}}:{\mathrm{C}}}_j &\text{else}
\end{cases}\\g^{\op{PIC}} &= \sum_{j} G_{j} R^{{\text{PIC:POC}}}_j\end{aligned}\end{align} \]
where
\[ \begin{align}\begin{aligned}{{\text{reg}}}^{Q{\mathrm{P}}}_z &= \left( \left[ \frac{Q^{{\mathrm{P}}\max}_z - Q^{{\mathrm{P}}}_z}
{Q^{{\mathrm{P}}\max}_z - Q^{{\mathrm{P}}\min}_z}
\right]_0^1 \right)^{h_{\op{G}}}\\&\dots\\{{\text{reg}}}^{Q{\mathrm{C}}}_z &= \left( \min\left\{
\frac{Q^{{\mathrm{P}}}_z - Q^{{\mathrm{P}}\min}_z}{Q^{{\mathrm{P}}\max}_z - Q^{{\mathrm{P}}\min}_z},
\frac{Q^{{\mathrm{N}}}_z - Q^{{\mathrm{N}}\min}_z}{Q^{{\mathrm{N}}\max}_z - Q^{{\mathrm{N}}\min}_z},
\frac{Q^{\op{Fe}}_z - Q^{\op{Fe}\min}_z}{Q^{\op{Fe}\max}_z - Q^{\op{Fe}\min}_z}
\right\}_0^1 \right)^{h_{\op{G}}}\\& \qquad\text{(only quota elements)}\end{aligned}\end{align} \]
and \(h_{\op{G}}\) is the Hill number for grazing (hillnumGraz,
default 1).
8.7.3.17.1. Implementation
In order to reduce the number of double (predator-prey) sums as much as
possible while still maintaining some code readability, the above sums are
computed in darwin_plankton.F via \(G_j\),
\(g^{\mathrm{C}}_z\) and the following auxiliary sums:
\[ \begin{align}\begin{aligned}G^{\exp}_j &= \sum_z G_{j,z} f^{\text{exp graz}}_{j,z}
\;,\\g^{\mathrm{C}\exp}_z &= \sum_j G_{j,z} a_{j,z}
{{\text{reg}}}^{Q{\mathrm{C}}}_z f^{\text{exp graz}}_{j,z}
\;,\end{aligned}\end{align} \]
and for quotas elements additionally \(g^{\mathrm{P}}_z\), …, and
\[ \begin{align}\begin{aligned}g^{\mathrm{P}\exp}_z &= \sum_j G_{j,z} a_{j,z}
{{\text{reg}}}^{Q{\mathrm{P}}}_z Q^{\mathrm{P}}_j
f^{\text{exp graz}}_{j,z}\\&\ldots\end{aligned}\end{align} \]
The remaining terms are then computed as
\[ \begin{align}\begin{aligned}g^{\op{POC}} &= \sum_j G^{\exp}_j - \sum_z g^{\mathrm{C}\exp}_z\\g^{\op{DOC}} &= g^{\op{OC}} - g^{\op{POC}}\end{aligned}\end{align} \]
where
\[g^{\op{OC}} = \sum_j G_j - \sum_z g^{\mathrm{C}}_z
\;.\]
For other non-quota elements:
\[ \begin{align}\begin{aligned}g^{\op{POP}} &= \sum_j G^{\exp}_j R^{\mathrm{P}:\mathrm{C}}_j
- \sum_z g^{\mathrm{C}\exp}_z R^{\mathrm{P}:\mathrm{C}}_z\\g^{\op{DOP}} &= g^{\op{OP}} - g^{\op{POP}}\end{aligned}\end{align} \]
where
\[g^{\op{OP}} = \sum_j G_j R^{\mathrm{P}:\mathrm{C}}_j
- \sum_z g^{\mathrm{C}}_z R^{\mathrm{P}:\mathrm{C}}_z
\;.\]
For quota elements:
\[ \begin{align}\begin{aligned}g^{\op{POP}} &= \sum_j G^{\exp}_j Q^{\mathrm{P}}_j
- \sum_z g^{\mathrm{P}\exp}_z\\g^{\op{DOP}} &= g^{\op{OP}} - g^{\op{POP}}\end{aligned}\end{align} \]
where
\[g^{\op{OP}} = \sum_j G_j Q^{\mathrm{P}}_j
- \sum_z g^{\mathrm{P}}_z
\;.\]
8.7.3.17.2. Runtime Parameters
Grazing parameters are given in Table 8.52.
Table 8.52 Grazing parameters
Trait |
Param |
Sym |
Default |
Units |
Description |
|---|
grazemax |
a_grazemax |
\(g^{\op{max}}_z\) |
21.9/day·V-0.16 |
s-1 |
maximum grazing rate |
kgrazesat |
a_kgrazesat |
\(k^{\op{graz}}_z\) |
1.0 |
mmol C m-3 |
grazing half-saturation concentration |
tempGraz |
grp_tempGraz |
\(e^{\op{graz}}_j\) |
1 |
|
1: grazing is temperature dependent, 0: turn dependence off |
|
inhib_graz |
\(i_{\op{graz}}\) |
1.0 |
m3 / mmol C |
inverse decay scale for grazing inhibition |
|
inhib_graz_exp |
\(e_{\op{inhib}}\) |
0.0 |
|
exponent for grazing inhibition (0 to turn off inhibition) |
|
hillnumGraz |
\(h_{\op{G}}\) |
1.0 |
|
exponent for limiting quota uptake in grazing |
|
hollexp |
\(h\) |
1.0 |
|
grazing exponential 1= “Holling 2”, 2= “Holling 3” |
|
phygrazmin |
\(c^{\min}_{\op{graz}}\) |
120E-10 |
mmol C m-3 |
minimum total prey conc for grazing to occur |
See Table 8.36 for stochiometry and quota-related parameters.
Table 8.53 Trait matrices for grazing; indices (prey, pred); unitless
Trait |
Param |
Symbol |
Default |
Description |
|---|
palat |
see below |
\(p_{j,z}\) |
0 |
palatability matrix |
asseff |
grp_ass_eff |
\(a_{j,z}\) |
0.7 |
assimilation efficiency matrix |
ExportFracPreyPred |
grp_ExportFracPreyPred |
\(f^{\op{exp}\op{graz}}_{j,z}\) |
0.5 |
fraction of unassimilated prey becoming particulate organic matter |
If DARWIN_ALLOMETRIC_PALAT is defined, palatabilities are set
allometrically,
\[p_{j,z} = \left[ \frac{1}{2\sigma_{\op{pp}}}
\exp\left\{
-\frac{(\ln(V_z/V_j/r_{\op{opt}}))^2}{2\sigma_{\op{pp}}^2}
\right\}
\right]_{\ge p_{\min}}\]
grp_pred and grp_prey should be set to select which
plankton groups can graze or be grazed.
Table 8.54 Allometric palatability trait parameters (unitless)
Param |
Symbol |
Default |
Description |
|---|
a,b_ppOpt |
\(r_{\op{opt}}\) |
1024·V0 |
optimum predator-prey ratio |
a_ppSig |
\(\sigma_{\op{pp}}\) |
1 |
width of predator-prey curve |
palat_min |
\(p_{\min}\) |
0 |
min non-zero palatability, smaller palat are set to 0 (was 1D-4 in quota) |
grp_pred |
|
0 |
1: can graze, 0: not |
grp_prey |
|
1 |
1: can be grazed, 0: not |