In virtually all types of fishery simulation models, there is a line of code (occasionally more than one line) that assigns a legal catch at the year, timestep, region, fishery, stock, and cohort level. In most cases, this line of code represents what we call a harvest process. Three common types of equations are the following:

1. Simple Linear Rate (used by FRAM & PSC chinook model).

   C(c,f) = HR(c,f)*N(c)

where

   C(c,f) = catch of cohort c in fishery f
   HR(c,f) = harvest rate for cohort c in fishery f
   N(c) = abundance of cohort c at start of period.

2. Non-Linear Relationship (similar to PM Model).

   C(c,f) = (1 - exp(-q(f)*E(f)))*N(c)

where

   q(c,f) = catchability coeff for cohort c in fishery f
   E(f) = effort in fishery f during period.

3. Instantaneous Rates (used by SSM).

            F(c,f)
   C(c,f) = ------ * (1 - exp(-Z(c))) * N(c)
             Z(c)

where

   F(c,f) = inst rate of fishing mort
   Z(c) = inst rate of total mortality for cohort c, and

   Z(c) = M(s) + Sum[F(c,f)] over all f
   F(c,f) = q(f)*E(f)
   M(c) = inst rate of natural mort for cohort c.


Fishing Process. For each year, timestep, region, and fishery a fishing process defines the amount of fishing effort to be input into the harvest processes for all cohorts residing in the given time and region in order to satisfy some management objective. Note that under this formulation, a fishing process does not compute any fishing mortalities--it only determines the inputs to the harvest processes. Only harvest processes compute fishing mortalities. Note also that a fishing process applies only to a single year, timestep, region, and fishery. This is the issue I think Jim was concerned about.

In the PSC Chinook Model, non-ceilinged fisheries have a fixed harvest rate management objective. Thus, the FPs are set for each fishery at config time and are passed into each harvest process without modification. On the other hand, each simple ceilinged fishery adjusts the effort level for all harvest processes in a given year, region, and timestep by a scalar (called the RT factor) in order to make the sum of the legal catches meet the management objective.


Code Issues. Up to now we had not considered implementing instantaneous rate equations. From a code perspective, instantaneous rate equations for harvest processes are fundamentally different from the linear and non-linear equations because all the required information is not autonomous within a single fishery. Specifically, the instantaneous nat mort rate for each stock is needed, along with the instantaneous fishing mort rates for all other fisheries within the same region. Thus, to implement instantaneous rate equations, a fishery object must have access to this outside information. If two fisheries operating in the same region at the same time both have quotas, whenever effort in one fishery is adjusted to meet its quota, the Z value (total mort rate) changes for all stocks. Thus, even simple quota fisheries will have to be solved together through a common algorithm. At a more fundamental level, the natural mortality and fishing mortality processes are intertwined and must be computed simultaneously. That is, one cannot compute nat morts and then move on to computing fishing morts, unless the fishing effort levels do not require adjustment to meet some objective. For all of the above reasons, I conclude that our proposed code structure is not compatible with using instantaneous rate equations.


Code Solution. First, we must combine the Nat Mort process and the Fishing Mort process into a single Mortality process during each timestep of a model run. A model can be configured to use one of two types of Mortality process. One type can compute nat morts and fishing morts independently (as most models currently do). Or, in a second type the nat and fishing morts can be computed simultaneously using instantaneous rates. At the start of the mortality process, all instantaneous rates would have to be determined. For example, the nat mort rates could be related to the physical environment of the region and/or the average size (length) of the individual in the cohort. Likewise, the fishing mort rates could be determined by fixed effort levels, or the effort levels for some fisheries could be set dynamically via some algorithm. The key point is that somehow all the rates are established prior to any computations. Once the rates are established, the nat mort computations and fishing mort computations can be made independently. If there are any constraints within the given timestep and region (eg quotas, escapement goals, allocations), then an algorithm must be written to adjust fishing effort levels at the start of the total mortality process level.


Code/Algorithm Problems. Finding effort levels to meet some constraints using instantaneous rate equations can be tricky. The fact that the total instantaneous mort rate (Z) is the sum of several individual fishery rates can lead to an infinite number of acceptable solutions. For example, if there is an allocation goal to equalize the sum of all Treaty fisheries with all Non-Treaty fisheries within a region, there can be many solutions (unless allocations WITHIN the Treaty and Non-Treaty groups are also specified). I used an Excel spreadsheet with Solver to model four fisheries, four stocks, and two timesteps (using instantaneous rate equations), and found that one must be very specific about the constraints in order to have a unique solution. The bottom line is that it is easy to make the model framework compatible with instantaneous rate equations, provided there is never any need to adjust fishing effort levels to meet management constraints. If constraints must be met, it looks like the algorithms might be tricky.


SSM Parameter Estimation. The SSM uses instantaneous rate equations and provides estimates of catchability coefficients (q) for each fishery. However, up to this point we do not have a formal description of a SSM that includes multiple fisheries operating within the same region at the same time. The two fisheries modeled in the prototype SSM (Canadian Troll, US Troll) operate simultaneously, but in different regions. I think I know what that formulation will look like, but we need to formalize it. Once the model is formulated, we need to answer the following questions:

Q1. If the SSM is fit to data for individual stocks, should the forward simulation model use separate q's for each fishery and stock? The alternative is to fit the SSM to multiple stocks assuming a common q. Is this biologically appropriate (ie are all stocks equally vulnerable to a fishing gear)? Is this feasible? How will it be done?

Q2. Regardless of how Q1 is resolved, the q estimates for each fishery will reflect all regulations associated with each fishery during the time frame when the data were collected (eg size limits, bag limits, selective fishery rules). If we desire to simulate changes in size limits, bag limits, and selective fishery rules, how will the SSM be modified to reflect these changes? The Lawson and Sampson (1996) model might be appropriate.

Q3. In forward simulation, how will the SSM handle incidental mortalities? The q's estimated by the SSM reflect legal catches and will not provide information about incidental mortalities related to fishing. I believe all incidental morts will be absorbed by the instantaneous nat mort parameter (M) in the SSM, provided it is not assumed to be zero. I suppose we can use auxiliary data to partition M into all types of non-legal catch mortalities. How will we do this?

Q4. The q's estimated by the SSM will be instantaneous rates based on a daily timestep. If a forward simulation model is configured to operate on a weekly or monthly basis, must the model use instantaneous rate equations, or can we convert the q's (and associated effort levels) to what Ricker calls "conditional rates" (ie the fraction of a cohort that dies within a given time period) and use other equations?


Final Thought. It seems that the theoretical model we seek is some combination of the detailed migration sub-model of the SSM with the detailed harvesting sub-model proposed by Lawson and Sampson (1996). The question is: If we include the detailed harvesting sub-model into the SSM, can the SSM estimate all the parameters? Can we make some simplifying assumptions and use auxiliary data to make the problem tractable?