Selectivitys distortion of the production function and its

Selectivity’s distortion of the production function and its influence on management advice Sheng-Ping Wang 1, 2, Mark Maunder 2, and Alexandre Aires-Da-Silva 2 1. National Taiwan Ocean University 2. Inter-American Tropical Tuna Commission

Introduction � For many situations, catch and data are only available for assessment especially for nontarget species, small scale fisheries. . . � The Schaefer surplus production model is commonly used in fisheries stock assessment. It has a symmetrical relationship between equilibrium yield and biomass where maximum sustainable yield occurs when the population is at 50% of the unexploited level 2

Introduction � The Schaefer model has been criticized because contemporary stock assessment models, which explicitly model the individual population processes, suggest that MSY is obtained at biomass levels substantially less than 50% of the unexploited level for many species. 3

Introduction � Pella and Tomlinson (1969) developed a more general surplus production model with an additional (shape) parameter that allows MSY to occur at any biomass level. MSY B 0 4

Introduction � Surplus production models represent population dynamics as a function of a single aggregated measure of biomass. e. g. carrying capacity (K or B 0), productivity rate (r), and shape parameter (m) 5

Introduction � It is well known that the production function of a stock is highly dependent on biological processes ◦ e. g. growth, natural mortality, and recruitment and density dependence (e. g. the stock-recruitment relationship). � This has led to questioning of the use of traditional surplus production models for the assessment of fish stocks. ◦ Estimation of the production function from catch and an index of relative abundance (or catch and effort data is problematic 6

Introduction � The production function is also dependent on the size (or age) of fish caught by the fishery. ◦ In general, fisheries that catch small fish produce a lower MSY compared to fisheries that catch large fish. � Fisheries that catch small fish also generally produce a lower BMSY/B 0. � Therefore, the age/size of the fish caught in a fishery needs to be taken into consideration when estimating the impact of a fishery on the stock 7

Introduction � Age-structured model can incorporate biological processes and selectivity for considerations. 8

Introduction � Typically, the selectivity increases smoothly with the size or age of the individual and either asymptotes or perhaps reducing at larger sizes. � The shape of the selectivity at size/age should be explicitly taken into consideration when evaluating equilibrium yield or the shape of the production function. 9

Objectives of this study � First we use simulation analysis to illustrate the impact of selectivity and biological parameters on the production function based on equilibrium age-structured model. 10

Objectives of this study � Then we evaluate how changes in selectivity over time influence parameter estimates and management advice from production models. ◦ The simulation analysis is roughly based on the bigeye tuna stock in the eastern Pacific Ocean. ◦ The fishery has changed from mainly a longline fishery, which captures large bigeye, to a mix of longline and purse seine, which captures small bigeye. 11

Equilibrium analysis � Sensitivity of MSY and related management quantities to biological parameters and selectivity is analyzed based on an agestructured model developed to model the population dynamics under equilibrium conditions. ◦ ◦ ◦ Beverton and Holt stock-recruitment relationship Separate fishing mortality von Bertalanffy growth function knife-edged maturity Constant natural mortality 12

Equilibrium analysis � The analysis is repeated for a variety of values for the steepness of the stock-recruitment relationship (h), the von Bertalanffy growth rate parameter (K), natural mortality (M), and the parameters of selectivity. ◦ h = 0. 5, 0. 75, and 1 ◦ K = 0. 1, 0. 2, and 0. 3 ◦ M = 0. 1, 0. 2, and 0. 3 13

Knife-edged selectivity assumption � Estimates of shape parameter (BMSY/B 0) ◦ Age at first capture is fixed at 4 yrs. M = 0. 1 K = 0. 2 K = 0. 3 M = 0. 2 K = 0. 1 K = 0. 2 K = 0. 3 M = 0. 3 K = 0. 1 K = 0. 2 K = 0. 3 h = 0. 5 h = 0. 75 h = 1. 0 0. 39 0. 38 0. 36 0. 33 0. 31 0. 29 0. 27 0. 23 0. 19 0. 38 0. 37 0. 32 0. 30 0. 29 0. 26 0. 22 0. 17 0. 38 0. 37 0. 36 0. 31 0. 30 0. 28 0. 23 0. 16 0. 15

Knife-edged selectivity assumption � Estimates MSY/BMSY) M = 0. 1 K = 0. 2 K = 0. 3 M = 0. 2 K = 0. 1 K = 0. 2 K = 0. 3 M = 0. 3 K = 0. 1 K = 0. 2 K = 0. 3 of productivity parameter (r or h = 0. 5 h = 0. 75 h = 1. 0 0. 04 0. 06 0. 07 0. 10 0. 13 0. 10 0. 17 0. 25 0. 07 0. 10 0. 12 0. 18 0. 24 0. 18 0. 32 0. 52 0. 10 0. 14 0. 18 0. 27 0. 36 0. 31 0. 63 1. 00

Equilibrium analysis � Two types of curves are used to exam the impacts of selectivity on the production function and MSY based quantities. Knife-edged selectivity � Double dome-shaped selectivity (only change the shape of curve on the right hand side) � 16

Knife-edged selectivity assumption

Knife-edged selectivity assumption

Double dome-shaped selectivity assumption

Double dome-shaped selectivity assumption

Application on bigeye tuna stock in the EPO � The dynamic age-structured model is used to simulate a age-specific biomass, fishing mortality, catch series and index of relative abundance for BET in the EPO. ◦ Beverton and Holt stock-recruitment relationship ◦ ◦ �recruitment is modeled using multiplicative lognormal process variation Gear-specific separate fishing mortality von Bertalanffy growth function Knife-edged maturity Constant natural mortality

Application on bigeye tuna stock in the EPO � The BET stock in the EPO has two main fisheries, purse seine setting on floating objects and longline. � Thus the dynamic age-structured model is developed for incorporating gear-specific selectivities. ◦ Gear-specific fishing mortality is the product of gear-specific effort, catchability and selectivity.

Application on bigeye tuna stock in the EPO � Selectivity ◦ Selectivity of longline (SLL) is assumed to be logistic curve ◦ Selectivity of purse-seine (SPS) is assumed to be descending right hand limb.

Application on bigeye tuna stock in the EPO � Selectivity ◦ The age-specific fishing mortality in 2010 is used to calculate the longline and purse-seine combined selectivity (SLL+PS).

Application on bigeye tuna stock in the EPO � The gear-specific catch is calculated without error � Gear-specific catch rate (index of relative abundance) is calculated incorporating a multiplicative lognormal observation error.

Application on bigeye tuna stock in the EPO � Pre-specific biological and fishery parameters Category of parameters von Bertalanffy growth function L∞ K t 0 Length-Weigth relationship a b Age at maturity am Virgin recruitment R 0 Steepness for spawning biomass recruitment relationship h Natural mortality M Value 1 0. 2 0 1 3 4 100 0. 75 0. 4

Application on bigeye tuna stock in the EPO � Pre-specific biological and fishery parameters Category of parameters Catchability q for longline q for purse seine Standard deviation of random residuals for Recruitment for CPUE Value 0. 0175 0. 35 0. 6 0. 2

Application on bigeye tuna stock in the EPO �

Application on bigeye tuna stock in the EPO � Gilbert’s version of the Pella-Tomlinson model is fit to the simulated data with the shape (m) and productivity (r) parameters either fixed based on the pre-specific values from the age-structured model or estimated.

Application on bigeye tuna stock in the EPO values of the shape (m) and productivity (r) parameters are obtained by equilibrium age-structured model with various selectivity assumptions. � Pre-specific ◦ Selectivity assumed to be SLL ◦ Selectivity assumed to be SPS ◦ Selectivity assumed to be SLL+PS

Application on bigeye tuna stock in the EPO � The shape and productivity parameters are based on the l vulnerable biomass l spawning biomass � 500 Longline selectivity Purse-seine selectivity Gear-combined selectivity simulation runs were carried out for each scenario.

Results by fitting to total catch and the LL catch rate Shape parameter Productivity 32

Results by fitting to total catch and the LL catch rate 33

Comparison between vulnerable and spawning biomass � Shape and productivity parameters estimated based on prespecific values obtained from various selectivity assumptions and measurement of biomass 34

Comparison between vulnerable and spawning biomass � Current biomass ratio (Bcur/BMSY) estimated based on prespecific values obtained from various selectivity assumptions and measurement of biomass 35

Comparison between vulnerable and spawning biomass 36

Using time-varied parameters � Residual sum of squares for estimation models with time-varied r and m. 37

Discussion � The results of this study indicate that the selectivity and biological processes can substantially impact the production function. � Vulnerable biomass and spawning biomass are calculated based on different equations basis. However, production model only estimates biomass based on vulnerable pattern and thus we cannot know which measurement is appropriate to be used for comparison. 38

Discussion � Estimating shape parameter of Pella. Tomlinson production model would be problematic. ◦ The estimations are biased and imprecise. ◦ Lead to the problematic estimates. . 39

Discussion � Since historical catch and catch rate were mainly contributed by LL, time-varied parameters of production calculated based on gear-combined selectivity cannot significantly improve fits of production model. ◦ Assuming the parameters of production based on LL selectivity would improve the fits of model. 40

Conclusions � Production function is substantially influenced by biological process and selectivity assumptions. � Schaefer model might not be appropriate for most scenarios. 41

Conclusions � Although Pella-Tomlinson model is much flexible, estimating shape parameters leads to problematic estimations for all selectivity assumptions. � The estimations of production model are distinct from the those of age-structured model (“true values”) since population dynamics is actually related to age-specific selectivity. 42

Thank you for your listening 43

Euqations. Equilibrium analysis

Initial conditions � • Where wa, ma and ra are the weight, maturity, and sex-ratio (proportion of females) of fish at age a. • wa is calculated based on von Bertalanffy growth function and length-weight relationship. • Maturity is assumed to be knife-edged with age-at-maturity (am). 45

Population dynamics � Where F is the fishing mortality for full -recruitment, and sa are the selectivity of fish at age a. 46

Recruitment ◦ The Beverton and Holt stock-recruitment relationship which is re-parameterized in terms of the "steepness" of the stock-recruitment relationship. X is the spawning stock biomass per recruit:

Selectivity ◦ Knife-edged selectivity ◦ Double dome-shaped selectivity

Yield � The parameters of production function and MSY-related quantities can be obtained by maximizing the yield equation.

Equations. Application on bigeye tuna stock in the EPO

Dynamic age-structured model � where Ft, g is the fishing mortality for fully-selected fish derived by fishery g in year t, Et, g is the fishing effort of fishery g in year t, qg is the catchability of fishery g, and Sa, g is the fishing gear selectivity of fish at age a derived by fishery g.

Dynamic age-structured model � Recruitment ◦ The Beverton and Holt stock-recruitment relationship which is re-parameterized in terms of the "steepness" of the stock-recruitment relationship. where ε is normally distributed process error, and σ2 is variance of process error in recruitment.

Dynamic age-structured model � Selectivity ◦ Selectivity of longline (SLL) is assumed to be logistic curve ◦ Selectivity of purse-seine (SPS) is assumed to be double dome-shaped curve.

Dynamic age-structured model � Selectivity ◦ The total age-specific fishing mortality scaled to a maximum of one is used to represent longline and purse-seine combined selectivity in the equilibrium model to estimate MSY based quantities. �Gear-combined selectivity (SLL+PS) in 2010 is used to make comparison with assumptions of LL and PS selectivity.

Dynamic age-structured model � Yield � Catch rate (index of relative abundance)

Knife-edged selectivity assumption When age at first capture is increased to a specific level (retains large amount of small fish), MSY will occur at a very high value of fishing mortality.

Double dome-shaped selectivity assumption When SD of age is smaller than a specific level (fishes are caught at a narrow age/size range), MSY will occur at a very high value of fishing mortality.