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In this talk I want to highlight how some simple mathematical fisheries equations can provide an aid to clear thinking when it comes to:

• fisheries science
• fisheries management and the illusive MSY
• fisheries data

The biomass of the fish at any one time is a product of:

• Existing biomass (B)
• Added biomass in form of growth of existing fish (G)
• Added biomass in form of new recruitment coming in (R)
• Loss of biomass in form of natural mortality (M)
  • Predation, starvation, desease
• Loss of biomass in form of catch (C)

So over time we have gains in form of growth and recuitment and loss because of natural causes and fishing.

Over time (here over one time step, from time t to t+1) we can expressed this in terms of a mass-balance equation:

B\(_{t+1}\) = B\(_t\) + G\(_t\) + R\(_t\) - M\(_t\) - C\(_t\)

Expressing fish population dynamics in this form raises questions:

• Which of these variables do we currently have measures of?
• Which of these variables can we possibly obtain measures of?
• Which of these variables can we likely never obtain any measurements of?
• Which of these variables can we manage?

In some cases we try to apply this simplification:

B\(_{t+1}\) = B\(_t\) + G\(_t\) + R\(_t\) - M\(_t\) - C\(_t\)

B\(_{t+1}\) = B\(_t\) + f(B\(_t\)) - C\(_t\)

and then use some index (Survey index or catch-per-unit-effort):

CPUE\(_t\) = q B\(_t\)

The f(B\(_t\)) is often refered to as production, its form being assumption driven

What data does one need to solve this riddle?

In its most simplistic term we can think of the catch over some very short time being a function of the mean biomass over that time:

Catch = Fishing pressure x Biomass, more succinctly

C = F B

Think here of the fishing pressure as the proportion of the biomass that is removed by fishing:

F = C / B

often termed harvest rate or fishing pressure.

We have here three variables. If we only have measures of one, we know from our school mathematics that one can not derive the other two terms.

Imagine we have a total catch (data) of some 200 t over a month. This catch could have arisen from different scenarios, e.g.:

Scenario 1: Fishing pressure = 0.2, Biomass = 1000 t
            C = FB
            Catch = 0.2 x 1000 t = 200 t
            
Scenario 2: Fishing pressure = 0.4, Biomass =  500 t
            C = FB
            Catch = 0.4 x  500 t = 200 t

So a certain catch can come from a high stock size using a low fishing pressure or from a low stock size using high fishing pressure.

Ergo, catch alone does not inform us about biomass nor fishing pressure.

Most national fisheries policies refer to MSY. So

C = F B

becomes specifically:

MSY = F\(_{MSY}\) B\(_{MSY}\)

• MSY: Maximum Sustainable Yield
• F\(_{MSY}\): Fishing pressure that results in MSY
• B\(_{MSY}\): Biomass that results in MSY

So in order to achieve MSY we need to know F\(_{MSY}\) and/or B\(_{MSY}\)

The fishing pressure can be split into two components:

F = qE, where:

• Effort (E): The question of how much?
• Catchability (q): The question of how?
  • We can think of this term as fishing efficiency
  • Formally this is fishing pressure exerted by one unit of effort, i.e.

q = F/E

Scenario 1: 10 small vessels go out for 1 day 
            Total effort: 10 fishing days
            Catchability: 0.01
            F = q E: 0.01 x 10 = 0.1

Scenario 2: 1 large vessels go out fishing for 1 day.
            Total effort: 1 fishing days.
            Catchability: 0.10
            F = qE = 0.10 x 1 = 0.1

Ergo we can exert the same fishing pressure (a)nd hence obtain the same catch) by having high effort and low catchability or low effort and high catchability.

More importantly, we have a societal choise to make (q vs E, i.e. how vs how much)

We have C = FB and F = qE

hence

C = qEB

and when expressed in the form of the illusive goal:

MSY = \(q_{MSY}\) E\(_{MSY}\) B\(_{MSY}\)

We can regulate directly or indirectly using any of these components

• Biomass
  • Minimum SSB, stocking, biomanipulation, enhancement, MPA’s …
• Catch
  • TAC, ITQ, …
• Fishing pressure (q or Effort)
  • Size of capture: mesh size
  • Effort control: number of boats, fishing days, …
  • Closed area (MPA’s)
  • Closed season

Going from: q -> E -> C -> B means ever more management cost
                                             data needs

We often have within the national management system:

• One arm trying to put some cap or reduce effort (number of boats, days at sea, …)
• The other arm is in development (fuel subsidy, improvement of gear, …), i.e. increasing catchability

The net effect may be that the fishing pressure may not be reduced

The MPA’s panacea:

• MPA’s are often sold as resulting in increased total catch
• It does increase local density, but the above is still being debated

The direct effect is often just a displacement of effort and increase in the cost of fishing

We know that catches as well as fish stock are composed of different sizes of fish.

• The size composition of the catch is a reflection of:
  • The size composition in the stock.
  • The area fished (nursery, feeding vs spawning areas).
  • The characteristics of the gear used.
  • The fishing pressure exerted

Higher effort                          Lower effort

The steepness of the size spectrum is an indirect measure of the fishing pressure, i.e.

\(F = f(sizespectrum)\)

If we can solve this we have two estimates of the fundamental catch equation:

\(C = FB\)

and we can thus solve the riddle:

\(B = C / F\)

Adding measurements on the size composition of the catch hence reduces the assumption one needs to make in order to solve the riddle in the fundamental fisheries equation.

Solving the fundamental catch equation is not simple, achieved only by indirect means and requires data time series (fleet and gear composition, effort, catch, cpue, size composition, survey data, …)

In this context older data are of extreme value.

To facilitate solving the riddle

• Retrieve old data as far back as possible
• Store all data in a managed database
• Make all data fully available to your fisheries scientists