Operating Model

MixME uses standard age-structured equations to model the dynamics of fleets and stocks. These equations describe the mortality and harvesting processes for each age-class in continuous-time within a single time-step - usually taken to be a single year - and yield the catches (in terms of landings and discard numbers) for each fleet in that time-step and the survivors at the beginning of the next time-step. Recruitment occurs at be beginning of each time-step according to a parametric stock-recruitment relationship.

The important implication is that changes in parameter values, such as natural mortality or catchability, can only occur between time-steps.

Population dynamics

For a stock $s = 1,\dots,n^s$ of age group $a = 1,\dots,A$, where $A$ is a plus-group of the oldest age-groups, the number of individuals $N$ in each age group at the beginning of year $y$ is:

$$ \begin{align} N_{s,1,y} &= R_{s,y} \\ N_{s,a,y} &= N_{s,a-1,y-1}\exp^{-(M_{s,a-1,y-1}+F_{s,a-1,y-1})}\text{, where } 1 < a < A \\ N_{s,A,y} &= N_{s,A-1,y-1}\exp^{-(M_{s,A-1,y-1}+F_{s,A-1,y-1})} + N_{s,A,y-1}\exp^{-(M_{s,A,y-1}+F_{s,A,y-1})} \end{align} $$

where $R$ is the number of recruits predicted from a stock-recruitment relationship, $F$ is the instantaneous rate of fishing mortality and $M$ is the instantaneous rate of natural mortality.

Harvesting

The overall rate of fishing mortality for each age group of a given population is the sum of the fleet-specific partial fishing mortalities,

$$F_{s,a,y} = \sum_f{F_{f,s,a,y}}$$

where fleet $f = 1,\dots,n^f$. The fleet-specific partial selection pattern for each stock is:

$$Sel_{f,s,a,y} = \frac{F_{f,s,a,y}}{\sum_a{F_{f,s,a,y}}}$$

This allows for easy calculation of overall species selection patterns. The fleet-specific partial fishing mortality is:

$$F_{f,s,a,y} = Sel_{f,s,a,y} \cdot q_{f,s,y} \cdot E_{f,y}$$

where $q$ is the fleet- and stock-specific catchability, and $E$ is the fleet effort. The catch $C$ for each fleet is given by the Baranov catch equation:

$$C_{f,s,a,y} = \frac{F_{f,s,a,y}}{F_{s,a,y} + M_{s,a,y}} \cdot N_{s,a,y}\left(1-\exp^{-(F_{s,a,y}+M_{s,a,y})}\right)$$

Fishing fleet dynamics

Fleet-specific effort is constrained by the availability of quota.

Discarding processes

encompasses both the discarding of species of little or no commercial value and the selective discarding of fish from commercially valuable stocks. Such selective discarding has been the subject of extensive empirical(?) modelling work and can be broadly divided into two types of behaviours.

Firstly, high-grading is discarding of low-value catch in favour of more valuable fish. These may be individuals that fall below a size or weight management threshold or that have limited market value. In many cases, high-grading is carried out to enable landings of higher commercial value.

Secondly, over-quota discarding is the discarding of marketable catch in response to restrictive quota, and occurs to enable the vessel to keep fishing, potentially targeting more valuable fish and maximise the overall value of the landed catch.

(\cite{Batsleer2015}).

Within fisheries models, the two discarding processes are typically summarised into a single landings fraction, or discards fraction, the proportion of catch that is landed. Objects in the FLR ecosystem track the numbers of landed and discarded fish. Landings and discards have distinct weights at age, with discards having a lower weight than landings to reflect the smaller-sized fish, and the overall catch weight for a given age class is the weighted mean of the landings and discards weights. These numbers are all summaries or modelled from empirical data, worked up in assessment working groups.

However, when over-quota discarding occurs, these are fish of marketable size that would be landed if quota were available. It may be expected that the individual sizes of over-quota discards will be closer to those of landed fish than high-grade discards.

In MixME, the two discarding processes are applied separately, but discards numbers and weights from each process are then combined. \texttt{discards.n}. However, the total weight of overquota discards for each stock is tracked and recorded over the course of the simulation.

High-grade discarding is assumed to occur as a proportion of catches, defined by the landings selection at age. The \texttt{discards.wt} slot of the \texttt{FLCatch} object stores the individual mean weights at age of high-grade discards. Over-quota discarding only occurs if quota is exceeded. The calculation of over-quota discards will depend on the advice basis. If advice is landings-based, then only the marketable catches that exceed quota are considered 'over-quota' because unmarketable catch does not consume quota. The overall over-quota biomass $B^o_{f,s}$ of each stock $s$ caught by each fleet $f$ is given by:

\begin{equation} B^o_{f,s} = \sum_a^A\left(L_{f,s,a} \cdot w^l_{f,s,a}\right)- Q_{f,s} \end{equation}

where $L_{f,s,a}$ is numbers at age and $w^l_{f,s,a}$ is the individual mean weight of the marketable catch by stock and fleet, and $Q_{f,s}$ is the stock-specific quota for each fleet. The over-quota biomass at age $B^o_{f,s,a}$ for each stock and fleet may then be estimated given the proportional distribution of marketable biomass across ages:

\begin{equation} B^o_{f,s} = \left(\sum_a^A\left(L_{f,s,a} \cdot w^l_{f,s,a}\right)- Q_{f,s}\right)\frac{L_{f,s,a} \cdot w^l_{f,s,a}}{\sum_a^A\left(L_{f,s,a} \cdot w^l_{f,s,a}\right)} \end{equation}

Partitioning is carried out in terms of the proportional distribution of biomass rather than numbers to avoid spuriously inflating the biomass of younger age classes, where harvested numbers are typically higher but comprising smaller individuals than older ages. The over-quota numbers at age $N^o_{f,s,a}$ for each stock and fleet are therefore given by:

\begin{equation} N^o_{f,s,a} = \frac{B^o_{f,s,a}}{w^l_{f,s,a}} \end{equation}

Over-quota catch contributes to overall discards, and the individual mean weight at age of discards by stock and fleet in that year should be updated to reflect the discarding of marketable-sized fish. The overall individual mean weight at age of discards $w^d_{f,s,a}$ is therefore a weighted average of marketable $w^l_{f,s,a}$ and non-marketable $w^{d*}_{f,s,a}$ individual mean weight at age:

\begin{equation} w^d_{f,s,a} = \left( w^{d*}{f,s,a} \cdot \left(1 - \frac{N^o{f,s,a}}{D_{f,s,a} + N^o_{f,s,a}} \right) \right) + \left( w^l_{f,s,a} \cdot \left(\frac{N^o_{f,s,a}}{D_{f,s,a} + N^o_{f,s,a}} \right) \right) \end{equation}

If advice is catch-based, then both marketable and non-marketable fish that exceed quota are considered 'over-quota', and over-quota biomass is:

\begin{equation} B^o_{f,s} = \sum_a^A\left(L_{f,s,a} \cdot w^l_{f,s,a} + D_{f,s,a} \cdot w^d_{f,s,a}\right)- Q_{f,s} \end{equation}

where $D_{f,s,a}$ is numbers at age and $w^{d*}{f,s,a}$ is the individual mean weight of the non-marketable catch by stock and fleet. Hence, $B^o{f,s}$ is composed of both marketable $B^{o,l}{f,s,a}$ and non-marketable fish $B^{o,d}{f,s,a}$ that must be considered separately because the two fractions may have a very different individual mean weight at age structure.

\begin{equation} \begin{split} B^l_{f,s,a} &= L_{f,s,a} \cdot w^l_{f,s,a} \ B^d_{f,s,a} &= D_{f,s,a} \cdot w^d*{f,s,a} \
B^{o,l}{f,s,a} &= B^o_{f,s} \left(\frac{B^l_{f,s,a}}{\sum_a^A{B^l_{f,s,a}}}\right)\left(\frac{B^l_{f,s,a}}{B^l_{f,s,a} + B^d_{f,s,a}}\right) \ B^{o,d}{f,s,a} &= B^o{f,s} \left(\frac{B^d_{f,s,a}}{\sum_a^A{B^d_{f,s,a}}}\right)\left(\frac{B^l_{f,s,a}}{B^l_{f,s,a} + B^d_{f,s,a}}\right) \end{split} \end{equation}

This yields the numbers of over-quota marketable and non-marketable discards:

\begin{equation} \begin{split} N^{o,l}{f,s,a} &= \frac{B^{o,l}{f,s,a}}{w^l_{f,s,a}} \ N^{o,d}{f,s,a} &= \frac{B^{o,d}{f,s,a}}{w^{d*}_{f,s,a}} \end{split} \end{equation}

Finally the overall discards weight at age is updated:

\begin{equation} w^d_{f,s,a} = \left( w^{d*}{f,s,a} \cdot \left(1 - \frac{N^{o,d}{f,s,a} + D_{f,s,a}}{N^{o,l}{f,s,a} + N^{o,d}{f,s,a} + D_{f,s,a}} \right) \right) + \left( w^l_{f,s,a} \cdot \left(\frac{N^{o,d}{f,s,a} + D{f,s,a}}{N^{o,l}{f,s,a} + N^{o,d}{f,s,a} + D_{f,s,a}} \right) \right) \end{equation}