Operating Model equations

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.

Fishing fleet dynamics

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 numbers $C$ for each fleet is given by the Baranov catch equation:

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

Effort dynamics

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

Discarding processes

Discarding processes, the disposal of a portion of catch at sea, can be broadly divided into several categories:

1. non-target: the discarding of species of little or no commercial value
2. regulatory: the discarding of undersized fish from managed stocks to comply with landing regulations
3. high-grading: the discarding of lower-value (but still marketable) fish in favour of more valuable fish to achieve landings of higher commercial value
4. over-quota: discarding of marketable catch in response to restrictive quota. This allows the vessel to keep fishing, potentially to target more valuable fish and maximise the overall value of the landed catch

Within fisheries models, discarding processes are typically described by the proportion of catch that is landed, also known as the landings fraction. FLR catch objects 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 typically summarised or modelled from empirical data, and 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, non-target and and regulatory discarding are described by the landings fraction, whereas over-quota discarding is modelled dynamically. High-grading is not explicitly modelled in MixME.

Landings fraction

In the absence of over-quota discarding, landings $L_{f,s,a,y}$ and discards $D_{f,s,a,y}$ numbers at age are given by:

$$ \begin{align} L_{f,s,a,y} &= C_{f,s,a,y} \cdot \delta_{f,s,a,y} \\ D_{f,s,a,y} &= C_{f,s,a,y} (1 - \delta_{f,s,a,y}) \end{align} $$

where $\delta$ is the landings fraction. The overall masses of landings $L^w$, discards $D^w$ and catch $C^w$ are:

$$ \begin{align} L_{f,s,a,y}^w &= L_{f,s,a,y} \cdot w_{f,s,a,y}^d \\ D_{f,s,a,y}^w &= D_{f,s,a,y} \cdot w_{f,s,a,y}^l \\ C_{f,s,a,y}^w &= L_{f,s,a,y}^w + D_{f,s,a,y}^w \end{align} $$

where $w^l$ and $w^d$ are the stock- and fleet-specific individual mean weight of landing and discards, respectively.

Over-quota discarding

Over-quota discarding only occurs if quota is exceeded. The calculations 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. If advice is catch-based, then both marketable and non-marketable fish that exceed quota are considered 'over-quota'.

Landings-based advice

If advice is landings-based, the mass of over-quota catch $C_{f,s,y}^{w,o}$ of each stock $s$ caught by each fleet $f$ is given by:

$$C_{f,s,y}^{w,o} = \sum_a(L^w_{f,s,a,y})- Q_{f,s,y}$$

where $L_{f,s,a,y}^w$ is the mass of marketable catch by stock and fleet, and $Q_{f,s}$ is the stock-specific quota for each fleet. The age-structured mass of over-quota catch $O_{f,s,a,y}$ for each stock and fleet may then be estimated given the proportional distribution of marketable biomass across ages:

$$ C_{f,s,a,y}^{w,o} = \left(\sum_a(L^w_{f,s,a,y})- Q_{f,s,y}\right)\frac{L^w_{f,s,a,y}}{\sum_a(L^w_{f,s,a,y})} $$

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 comprise smaller individuals compared to older age classes. The over-quota numbers at age $C_{f,s,a,y}^o$ for each stock and fleet are therefore given by:

$$C_{f,s,a,y}^o = \frac{C_{f,s,a,y}^{w,o}}{w^l_{f,s,a,y}}$$

The updated landings and discards numbers at age given over-quota discarding are:

$$ \begin{align} L_{f,s,a,y}^* = L_{f,s,a,y} - C_{f,s,a,y}^o \\ D_{f,s,a,y}^* = D_{f,s,a,y} + C_{f,s,a,y}^o \end{align} $$

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 updated individual mean weight at age of discards $w_{f,s,a,y}^{d*}$ is therefore a weighted average of marketable $w^l_{f,s,a,y}$ and non-marketable $w^d_{f,s,a,y}$ individual mean weight at age:

$$ w_{f,s,a,y}^{d*} = \left( w^d_{f,s,a,y} \cdot \left(1 - \frac{C^o_{f,s,a,y}}{D_{f,s,a,y} + C^o_{f,s,a,y}} \right) \right) + \left( w^l_{f,s,a,y} \cdot \left(\frac{C^o_{f,s,a,y}}{D_{f,s,a,y} + C^o_{f,s,a,y}} \right) \right) $$

Catch-based advice

If advice is catch-based, the mass of over-quota catch is:

$$C_{f,s,y}^{w,o} = \sum_a\left(L^w_{f,s,a,y} + D^w_{f,s,a,y}\right)- Q_{f,s,y}$$

where $L^w_{f,s,a,y}$ and $D^w_{f,s,a,y}$ are the masses of marketable and non-marketable catch by stock and fleet. Hence, $C_{f,s,y}^{w,o}$ is composed of both marketable $L_{f,s,a,y}^{w,o}$ and non-marketable fish $D_{f,s,a,y}^{w,o}$ that must be considered separately because the two fractions may have a very different individual mean weight at age structure.

$$ \begin{align} L_{f,s,a,y}^{w,o} &= C_{f,s,y}^{w,o} \left(\frac{L_{f,s,a,y}^w}{\sum_a{L_{f,s,a,y}^w}}\right)\left(\frac{L_{f,s,a,y}^w}{L_{f,s,a,y}^w + D_{f,s,a,y}^w}\right) \\ D_{f,s,a,y}^{w,o} &= C_{f,s,y}^{w,o} \left(\frac{D_{f,s,a,y}^w}{\sum_a{D_{f,s,a,y}^w}}\right)\left(1 - \frac{L_{f,s,a,y}^w}{L_{f,s,a,y}^w + D_{f,s,a,y}^w}\right) \end{align} $$

This yields the numbers of over-quota marketable $L_{f,s,a,y}^o$ and non-marketable $D_{f,s,a,y}^o$ discards:

$$ \begin{align} L^o_{f,s,a,y} &= \frac{L_{f,s,a,y}^{w,o}}{w^l_{f,s,a,y}} \\ D^o_{f,s,a,y} &= \frac{D_{f,s,a,y}^{w,o}}{w^d_{f,s,a,y}} \end{align} $$

The updated landings and discards numbers at age given over-quota discarding are:

$$ \begin{align} L_{f,s,a,y}^* = L_{f,s,a,y} - L^o_{f,s,a,y} \\ D_{f,s,a,y}^* = D_{f,s,a,y} + L^o_{f,s,a,y} \end{align} $$

Finally, the overall discards weight at age is updated:

$$ w_{f,s,a,y}^{d*} = \left( w^d_{f,s,a,y} \cdot \left(1 - \frac{L^o_{f,s,a,y}}{L^o_{f,s,a,y} + D_{f,s,a,y}} \right) \right) + \left( w^l_{f,s,a,y} \cdot \left(\frac{L^o_{f,s,a,y}}{L^o_{f,s,a,y} + D_{f,s,a,y}} \right) \right) $$