CERT Comité d'évaluation des ressources transfrontalières TRAC Transboundary Resources Assessment Committee Document de travail 2014/30 Working Paper 2014/30 Ne pas citer sans autorisation des auteurs Not to be cited without permission of the authors Estimation of the Intrinsic Rate of Increase for Georges Bank Yellowtail Flounder Loretta O Brien and David McElroy NEFSC, Woods Hole, Ma TRAC Georges Bank Yellowtail Flounder Diagnostic Benchmark April 14-18, 2014 Woods Hole, Ma Ce document est disponible sur l Internet à : This document is available on the Internet at : http://www.mar.dfo-mpo.gc.ca/science/trac/trac.html
Introduction Estimate reproductive potential of GB yellowtail flounder using life-table analysis. Methodology based on the Euler-Lotka equation (Caswell 1989) Apply recently derived GB YT fecundity relationship that includes condition (WP 33) Analysis provides : Intrinsic rate of increase r and instantaneous growth rate λ = exp (r) Net Reproductive rate / reproductive potential Ro
Input Data: Time series : 1973-2013 Population number at age (N); (2013 VPA,Legault 2013) Recruitment (R) = age 1 N Sex ratio (X); assume 50/50 Maturity at age (M); times series average (2013 VPA,Legault 2013) Mean length at age = L a Use NEFSC spring research bottom trawl survey : Closest to spawning time Missing mean length for age 1 for most years, used a 10 year mean for missing years Some missing age 6 or 7, use average of adjacent years Regression fit to observed mean length, fitted mean length applied in life-table
Input Data: Condition Individual weight only available since 1992. Fulton s Condition at length (l) = K l = [Wt (g) / Length (mm) 3 ] *100000 K assumed = 1 for earlier years Mean K at age = K a = Σ (K l /n) Potential Annual Fecundity (F) : combined 2010-2013 (WP 33 McElroy ) Ln (PAF) a = β 0 + β 1ln L a (mm) + β 2 Oocyte diam + β 3 K a oocyte diameter = 450 um (near spawning size) Parameter Estimate SE Intercept -3.3816805 2.7745847 LN (TL) 2.6875112 0.4542765 Mean Oocyte Diameter -0.0023884 0.0005946 K 2.8512876 0.2466295 Total Egg production at age = TEP a = N a *X a *M a *F a Recruitment survival = S = R y-1 / TEP y Average Rct Z = ( Σ -ln [R/TEP] y ) / n Maximum age ~ 11
Population Numbers at ages 1-6+ from 1973-2013
Mean Length (cm) 60.0 50.0 40.0 30.0 20.0 10.0 0.0 GB YT NEFSC Spring - Observed Mean Length 0 1 2 3 4 5 6 7 8 AGE 1973 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Spring Observed Mean length at age 1973-2013
70.00 GB YT NEFSC SPRING -regression of observed mean length 1973 1974 1975 1976 Spring Fitted mean length at age 1973-2013 60.00 1977 1978 1979 1980 Ages 7+ low sample size 1981 1982 50.00 1983 1984 1985 1986 1987 1988 1989 40.00 1990 1991 Mean Length (cm) 1992 1993 1994 1995 30.00 1996 1997 1998 1999 2000 2001 2002 20.00 2003 2004 2005 2006 2007 2008 10.00 2008 2009 2010 2011 2012 2013 0.00 0 2 4 6 8 10 12 age
Fulton s Condition 1.4 Fulton's K (Spring survey -females only) 1.2 1 0.8 0.6 0.4 0.2 1 2 3 4 5 6+ 0 1990 1995 2000 2005 2010 2015
Life-Table Analysis Euler-Lotka equation ; solve to estimate r, intrinsic rate of increase where x=β Σ e -rx l(x) m(x) = 1 x=α l(x) = the probability of surviving to age x m(x) = number of female offspring produced at age x α = age 0 β = maximum age Derive annual instantaneous growth rate : λ = exp (r)
Life-Table Analysis Derive Net reproductive rate: x=β R o = Σ x=α l(x) m(x) where l(x) = the probability of surviving to age x m(x) = number of female offspring produced at age x α = age at maturity β = age at death
Results annual instantaneous growth rate : λ = exp (r) 1.6 Georges Bank Yellowtail Flounder lambda - growth rate 1.4 1.2 1 0.8 0.6 0.4 0.2 M=0.2 M=0.3 M=0.4 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 year
Results influence of K on r rate of increase (r) Georges Bank Yellowtail Flounder M=0.2 0.35 M=0.3 0.3 M=0.4 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 year K has strong effect on trend in r K = 1 in fecundity eqn from 1973-1991 K = observed condition from 1992 rate of increase (r) Georges Bank Yellowtail Flounder M=0.2 0.35 M=0.3 0.3 M=0.4 0.25 0.2 0.15 0.1 0.05 0-0.05-0.1 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 year K = 1 in all years
Results influence of K on Ro net reproductive rate (Ro) 8 7 6 5 4 3 2 Georges Bank Yellowtail Flounder M=0.2 M=0.3 M=0.4 K has strong effect on trend in Ro K = 1 in fecundity eqn from 1973-1991 K = observed condition from 1992 1 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 year 7 Georges Bank Yellowtail Flounder M=0.2 M=0.3 net reproductive rate (Ro) 6 5 4 3 2 1 M=0.4 K = 1 in all years 0 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 year
Summary Population growth (λ ) and net reproduction (Ro) have been declining since 1998 Estimation of r and Ro influenced strongly by inclusion of condition in fecundity estimation If condition is not included, would mistakenly conclude stock is relatively stable even though population numbers are declining. Further work: Sensitivities to varied sex ratio, since YT have dimorphic growth; Females grow/survive to larger size than males Sensitivity and Elasticity (proportional effect) to measure how changes in vital rates i.e. juvenile mortality, survival at age, effect estimation of λ Projections of Spawning Stock Biomass (Monte Carlo Analysis)
Literature Cited Caswell, H. 1989. Matrix Population Models. Construction, Analysis, and Interpretation. Sinauer Assoicates, Inc. Sunderland, Ma. 328 p. Hood, G. M. (2011) PopTools version 3.2.5. URL http://www.poptools.org McElroy, W.D., E.T. Towle, M.J. Wuenschel, and R.S. McBride. 2013. Spatial and annual variation in fecundity of yellowtail flounder in U.S. waters. TRAC GBYT Benchmark WP33 Legault, C. 2013. Stock Assessment of Georges Bank Yellowtail Flounder for 2013. TRAC 2013 WP. 138 p.