Depletion Estimates of

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Depletion Estimates of Population Size and
Turnover

• Goals
• Describe and explain
• Constructing the estimators
• underlying assumptions
• use of auxiliary data
• tailoring models to specific cases
• Estimators for
• closed populations
• Leslie
• Delury
• open populations
• general model
• age-structure

2
Depletion Experiments

• They are not
• an excuse for overfishing
• should be performed with minimal risk
• sub-population
• with care
• controlled in a classical design

• They are
• a deliberate, controlled harvest of fish
• designed to estimate original (and current)
  abundance

3

• assume CPUE ? abundance
• How does CPUE vary with repeated sampling ?
• What is the cumulative catch when CPUE 0 ?

4
Leslie Estimator

• Ideally, ASSUME, over study period
• closed population
• no mortality, birth
• no migration

Population model Nt number at time t Kt-1
cumulative catch to time t-1 Observation
Model yt current abundance index q Nt
(observation model)
Nt N1 - Kt-1
yt q(N1 - Kt-1)
5
Graphically yt q(N1 - Kt-1)

• when K t-1 N1 then yt 0
• y and K data can come from different sources
• survey and commercial

6
DeLury Estimator

• Ideally, ASSUME, over study period
• closed population
• no mortality, birth
• no migration
• (as with Leslie) AND
• catch rate (rate of increase in K) is directly
  proportional to nominal effort (ft) by q
• if Et is the sum of all effort up to time t,
  then popn. model

dKt / dt q ft
Nt N1 e -qEt
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Substitute exponential populationmodel into the
proportional observation model
yt q Nt q (N1 e -qEt)
log(yt) log( q (N1 e -qEt))

• model in linear form
• independent E rather than K
• need slope and intercept for N1
• yt
• must be CPUE
• measured over a brief period
• so that C/f qN

8
Bias in Leslie and DeLury

• Estimators approximately unbiased if
• all fish equally vulnerable (one q)
• E or K measured exactly

• Errors in E or K tend to
• underestimate q
• overestimate N1
• less problem with Leslie
• different data for y K
• Variable q
• more vulnerable fish die first
• distribution, size, stupidity
• refugia (q0 for some fish)
• bias q upward, N1 downward
• if y and K or E from same data
• non-independence
• incorrect confidence intervals
• too small

9
Two Closed Populations
Territoriality? Change in Feeding? Beware of too
brief a sampling period.
10
Depletion Estimates inOpen Populations
General form of experiment so far
Need migration, mortality, etc. in population
model
11
Open Population Models
Similar to biomass dynamics, butmost parameters
estimated outsideof depletion experiment.
Nt Gt-1Nt-1 - Ct-1 Rt

• G growth in numbers
• survival
• em/immigration
• somatic growth if N is biomass
• any change in number proportional to N - C
• C catch
• R recruitment

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Nt Gt-1Nt-1 - Ct-1 Rt

• Assume over depletion experiment
• R single value (or single mean)
• Gt is known independently
• survival, migration, growth,etc.

N2 G1N1 - C1 R
N3 G2G1N1 - C1 R - C2 R
G2G1N1 - G2G1C1 G2C2 G2R R
which leads to the recursive form
Nt Kt-1N1 - Kt-1 KtR
Ks are recursive
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Nt Kt-1N1 - Kt-1 KtR
K1 G1 Kt Kt-1Gt, K1 G1C1 Kt
Kt-1Ct Gt K1 0 Kt Kt-1 Gt-1
1
K contribution of initial population K cumulati
ve catch (weighted) K recruitment
(weighted) G, the growth-survival term, is
thesource of the weights Note, all Ks can be
calculated fromdata at hand only R, N1 are
unknownhere
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Population Observation Models
Nt Kt-1N1 - Kt-1 KtR
yt q Nt
yt q(Kt-1N1 - Kt-1 KtR) qN1Kt-1 -
qKt-1 qRKt
Thus, we can estimate q, N1, Rfrom a multiple
regression withan intercept of 0.
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Auxiliary Data and Assumptions in the Open
Depletion Estimate

• relative recruitment time series
• Rt rtR in equations
• lack of independent G (s - survival)
• do a series of fits
• assume alternative s
• perhaps find best estimate?
• at least sensitivity
• assume parameter relationships to reduce
  dimension of fit
• initial equilibrium R 1 - G1 N1
• dont estimate R

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Example from the Tuna Fishery

• FADs - fish attracting devices
• (fish like stuff)
• Experiment
• sample around FADs on 7 successive days

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• Estimate N1, s, q, R
• ignore growth
• assume initial equilibrium R 1-sN1
• fitted range of s values
• s 0.95 (per diem)
• implied about 3 week recovery time around
  FAD
• Note all numerical change attributed to
  migrations and catch
• same model used to estimate movements and
  mortality
• time scales differ

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Bias in Open Stock Depletion Estimates

• estimating more parameters
• results more sensitive to errors
• Monte Carlo studies indicate
• errors in q or C
• underestimate q
• overestimate N1
• variable q
• similar to closed case
• overestimate q
• underestimate N1
• stock - recruitment relationship
• may generate R too high if R assumed constant