(Meta-) Ecosystem version of the BEFW: adding detritus, recycling, and external fluxes

[alaindanet]

0. Scope and goal

@Xxiao-Li and @alaindanet are intending to develop an (1) ecosystem and (2) meta-ecosystem version of the BEFW model.

This would take two steps:

1. Turning the nutrient based FW model to an ecosystem by adding a detritus (defined below) and recycling from detritus to nutrient.
2. Turning the ecosystem into a regional-island meta-ecosystem by adding external input of Biomass, nutrient, and detritus

 0.1. Definition of ecosystem

An ecosystem is a delimited system in space where transfer of energy and matter happen between organisms and non living matter.
To the difference with a community, an ecosystem explicitly describes the feedback between living matter and non-living matter.

Non-living matters classically features nutrients and detritus. Detritus are organic dead matter which are not directly available to primary producers and need to be mineralised, e.g. by detritivores (together with decomposers), into nutrients. The nutrients are then used by primary producers.

The Detritus are produced by living organisms by excreting or not assimilating all the food they ingest (feces), not ingesting all the food they collect ($f_ij$ e.g.), through metabolic losses (respiration, renewing their cells, etc...). Here is a good reference about detritus (Thanks @Xxiao-Li ).

I assume everybody knows about nutrients :)

So at end, an ecosystem model is a model describing the cycle of living organic matter (i.e. biomass), to dead organic matter (detritus), to dead mineral matter (nutrients), back to living organic matter (i.e. primary producers and consumers).

 0.2. Definition of a meta-ecosystem

A meta-ecosystem is system of connected ecosystems. Those ecosystems are connected by fluxes of living/non-living, organic/non-organic matter: meaning connected by fluxes of organisms (i.e. biomass), nutrients and detritus.

1. Ecosystem Models

It exists a diversity of model formulation to implement detritus and recycling. The complexity of this formulation depends on the research questions. In the following, I will write down the minimal ecosystem model starting from the nutrient based BEFW2.

1.1. Minimal Ecosystem model: detritus (Based on Moore et al. 1993 & the original formulation of De Angelis)

$$ \frac{dB_i}{dt} = r_i B_i G_i - x_i B_i - d_i B_i + B_i \sum_{j \in {\mathrm{prey.}}_i} e_{ij} F_{ij} - \sum_{j \in {\mathrm{pred.}}_i} B_j F_{ji} $$

$$ \frac{dT}{dt} = -D_tT + \sum_i x_i B_i + \sum_i d_i B_i + \sum_i B_i \sum_{j \in {\mathrm{prey.}}_i} (1 - e_{ij}) F_{ij} - \sum B_j F_{jt} $$

where $\sum B_j F_{jt}$ is detritus comsumption from detritivores, and there is no nutrient in this model.

1.2. Simple Ecosystem model: nutrients + detritus (Based on Quévreux et al., 2020)

Let's consider the classic version (i.e. considering attack rates and handling time) of the model:

$$ \frac{dB_i}{dt} = r_i B_i G_i - x_i B_i - d_i B_i + B_i \sum_{j \in {\mathrm{prey.}}_i} e_{ij} F_{ij} - \sum_{j \in {\mathrm{pred.}}_i} B_j F_{ji} $$

$$ \frac{dN_l}{dt} = D_l (S_l - N_l) - \sum_{i=1}^n c_{li} G_i (N) B_i $$

Let's now modify the nutrient equation to add the inputs from the detritus:

$$ \frac{dN_l}{dt} = t_l T + D_l (S_l - N_l) - \sum_{i=1}^n c_{li} G_i (N) B_i $$

where $t_l$ and $T$ being respectively the rate of mineralisation of detritus into nutritent $l$ and the quantity of nutrient $N_l$.
So $D_l (S_l - N_l)$ is representing the external supply of nutrient, while $t_l T$ is the internal supply from nutrients. We might need to add loss rates of nutrient here??.

An alternative formulation of Nutrient supply might be simpler like:

$$ \frac{dN_l}{dt} = S_l + tT - D_lN_l - \sum_{i=1}^n c_{li} G_i (N) B_i $$

here meaning that Nutrient are supplied at constant rate locally by $S_l$, but also by detritus $tT$ and nutrients are lost by some processess (e.g. run-off, sedimentation, etc...).

The dynamics of detritus is then simply:

$$ \frac{dT}{dt} = - tT -D_tT + \sum_i x_i B_i + \sum_i d_i B_i + \sum_i B_i \sum_{j \in {\mathrm{prey.}}_i} (1 - e_{ij}) F_{ij} $$

Detritus are lost by mineralisation into nutrients (tT), lost by run-off ($D_tT$ and gain by metabolism losses, dead biomass from natural death, and finally from non dead biomass non assimilated by consumers.

2. Implementation

TODO