ATN Equations used by Network3D web service

Parameter definitions

Gleaned from Foodweb.WebDynamics.Bioenergetic.ModelImplBioenergetic and Dynamics.bioenergetic.SystemParams

Node-specific properties

See ModelImplBioenergetic.setNodeSpecificProperty()

Symbol	Definition	Default
$x_i$	metabolic rate	0.5
$r_i$	growth rate	1.0
$K_i$	carrying capacity	1.0
$a_{xi}$	fraction activity metabolic rate	1.0
$b_{xi}$	fraction basal metabolic rate	1.0

Link-specific properties

See ModelImplBioenergetic.setLinkSpecificProperty()

Symbol	Name	Default
$\alpha_{ij}$	relative half saturation density	0.0
$y_{ij}$	maximum ingestion rate	6.0
$e_{ij}$	assimilation efficiency	1.0
$B_{0ij}$	half saturation density	0.5
$q_{ij}$	functional response control	0.0

Dynamics.bioenergetic.ModelDerivative

Functional response

$$ F_{ij} = \frac {B_j^{1 + q_{ij}}} {\sum_m \alpha_{im} B_m^{1 + q_{im}} + B_{0ij}^{1 + q_{ij}}} $$

Growth function

$$ G_i(B) = 1 - \frac{B_i}{K_i} $$

Producers

$$ B_i' = \sum_j ( - x_j y_{ji} \alpha_{ji} F_{ji} B_j / e_{ij} ) + r_i B_i G_i(B) $$

Consumers

$$ B_i' = \sum_j ( x_i y_{ij} \alpha_{ij} F_{ij} B_i - x_j y_{ji} \alpha_{ji} F_{ji} B_j / e_{ij} ) - x_i B_i $$

Notes

The full producer equation, based on the code, is:

$$ B_i' = \sum_j ( x_i y_{ij} \alpha_{ij} F_{ij} B_i - x_j y_{ji} \alpha_{ji} F_{ji} B_j / e_{ij} ) + r_i B_i G_i(B) - x_i B_i $$

Because $y_{ij}$ logically has to be 0 if $i$ does not eat $j$, the first term within the summation reduces to 0. Also, the Network3D application also automatically sets $x_i$ for producers to 0. (Rich Williams confirmed that the producer equation should not include a $- x_i B_i$ term.)

The ModelDerivative code uses $e_{ij}$. I think this is a mistake, because the literature uses $e_{ji}$.

ATNEquations

My code in WoB server uses the following rewritten from ModelDerivative, based on the notes above.

Functional response

$$ F_{ij} = \frac {B_j^{1 + q_{ij}}} {\sum_{m \in prey}\alpha_{im} B_m^{1 + q_{im}} + B_{0ij}^{1 + q_{ij}}} $$

Growth function

$$ G_i(B) = 1 - \frac{B_i}{K_i} $$

Growth function with system-wide carrying capacity

as described by Boit et al. 2012 and Kuparinen et al. 2016

$$ G_i(B) = 1 - \frac{\sum_{j \in producers} c_{ij} B_j}{K_s} $$

Producers

$$ B_i' = r_i B_i G_i(B) - \sum_{j \in predators} (x_j y_{ji} \alpha_{ji} F_{ji} B_j / e_{ji}) $$

Consumers

$$ B_i' = \sum_{j \in prey} (x_i y_{ij} \alpha_{ij} F_{ij} B_i) - \sum_{j \in predators} (x_j y_{ji} \alpha_{ji} F_{ji} B_j / e_{ji}) - x_i B_i $$