06 Recruitment (Part 3 & 4)

06 Recruitment (Part 3 & 4) - EXST7025 Recruitment...

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Unformatted text preview: EXST7025 Recruitment models 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 NOTE: NOTE: 41 42 43 NOTE: NOTE: Spring 2002 Page 1 Arcto-Norwegian Cod /*******************************************************/ /* Ricker Recruitment Model */ /* Arcto - Norwegian Cod Example from Ricker */ /*******************************************************/ OPTIONS NOCENTER LS=78 PS=61 nodate nonumber; DATA ONE; INFILE CARDS MISSOVER; TITLE1 'Arcto - Norwegian Cod Example from Ricker'; INPUT YR P R; LABEL P = 'Parental stock'; LABEL R = 'Recruits'; RP = R / P; LABEL RP = 'Recruits / Parent'; PR = P / R; LABEL PR = 'ParentS / Recruit'; LRP = LOG(R / P); LABEL LRP= 'LOG(Recruits / Parent)'; IR = 1 / R; LABEL IR = 'Inverse of Recruits'; IP = 1 / P; LABEL IP = 'Inverse of Parents'; CARDS; The data set WORK.ONE has 24 observations and 8 variables. The DATA statement used 6.48 seconds. ; PROC PRINT; TITLE2 '*** RAW DATA LISTING ***'; The PROCEDURE PRINT printed page 1. The PROCEDURE PRINT used 2.08 seconds. Arcto - Norwegian Cod Example from Ricker *** RAW DATA LISTING *** OBS YR P R RP PR LRP IR IP 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 118 166 180 172 152 139 121 115 139 140 152 120 65 69 88 31 23 21 18 20 16 20 19 13 145 35 20 28 42 55 116 77 98 75 80 168 221 140 43 46 99 145 80 93 81 144 62 10 1.22881 0.21084 0.11111 0.16279 0.27632 0.39568 0.95868 0.66957 0.70504 0.53571 0.52632 1.40000 3.40000 2.02899 0.48864 1.48387 4.30435 6.90476 4.44444 4.65000 5.06250 7.20000 3.26316 0.76923 0.81379 4.74286 9.00000 6.14286 3.61905 2.52727 1.04310 1.49351 1.41837 1.86667 1.90000 0.71429 0.29412 0.49286 2.04651 0.67391 0.23232 0.14483 0.22500 0.21505 0.19753 0.13889 0.30645 1.30000 0.20605 -1.55664 -2.19722 -1.81529 -1.28621 -0.92714 -0.04220 -0.40113 -0.34951 -0.62415 -0.64185 0.33647 1.22378 0.70754 -0.71614 0.39465 1.45963 1.93221 1.49165 1.53687 1.62186 1.97408 1.18270 -0.26236 0.00690 0.02857 0.05000 0.03571 0.02381 0.01818 0.00862 0.01299 0.01020 0.01333 0.01250 0.00595 0.00452 0.00714 0.02326 0.02174 0.01010 0.00690 0.01250 0.01075 0.01235 0.00694 0.01613 0.10000 0.008475 0.006024 0.005556 0.005814 0.006579 0.007194 0.008264 0.008696 0.007194 0.007143 0.006579 0.008333 0.015385 0.014493 0.011364 0.032258 0.043478 0.047619 0.055556 0.050000 0.062500 0.050000 0.052632 0.076923 EXST7025 Recruitment models 45 46 Spring 2002 Page 2 Arcto-Norwegian Cod PROC REG; MODEL LRP=P; TITLE2 '*** RICKER MODEL - LINEARIZED VERSION ***'; OUTPUT OUT=ONE RESIDUAL=E1 PREDICTED=P1; RUN; Arcto - Norwegian Cod Example from Ricker *** RICKER MODEL - LINEARIZED VERSION *** Model: MODEL1 Dependent Variable: LRP Analysis of Variance Source Model Error C Total DF 1 22 23 Root MSE Dep Mean C.V. LOG(Recruits / Parent) Sum of Squares 26.89713 8.07840 34.97553 0.60597 0.13532 447.81179 Parameter Estimates Parameter Variable DF Estimate INTERCEP 1 1.713608 P 1 -0.017893 Mean Square 26.89713 0.36720 F Value 73.249 R-square Adj R-sq 0.7690 0.7585 Standard Error 0.22205220 0.00209062 T for H0: Parameter=0 7.717 -8.559 Prob>F 0.0001 Prob > |T| 0.0001 0.0001 Antilog of intercept exp(1.713608) = 5.548946 48 49 50 PROC NLIN; TITLE2 '*** RICKER MODEL - NONLINEAR VERSION ***'; PARAMETERS B0 = 6.6 B1 = -0.018; MODEL R=B0*P*EXP(B1*P); OUTPUT OUT=ONE RESIDUAL=E2 PREDICTED=P2; Arcto - Norwegian Cod Example from Ricker *** RICKER MODEL - NONLINEAR VERSION *** Non-Linear Least Squares Iterative Phase Dependent Variable R Iter B0 B1 Sum of Squares 6 6.646510 -0.017635 40574.541124 NOTE: Convergence criterion met. Method: DUD Non-Linear Least Squares Summary Statistics Dependent Variable R Source DF Sum of Squares Mean Square Regression 2 206408.45888 103204.22944 Residual 22 40574.54112 1844.29732 Uncorrected Total 24 246983.00000 (Corrected Total) 23 62707.62500 Parameter B0 B1 Estimate 6.646510339 -0.017634942 Asymptotic Std. Error Asymptotic 95 % Confidence Interval Lower Upper 1.1368204695 4.2889077636 9.0041129139 0.0018777021 -0.0215290271 -0.0137408575 EXST7025 Recruitment models 52 53 Spring 2002 Page 3 Arcto-Norwegian Cod PROC REG; TITLE2 '*** BEVERTON MODEL - FIRST LINEARIZATION ***'; MODEL IR=IP; OUTPUT OUT=ONE RESIDUAL=E3 PREDICTED=P3; RUN; Arcto - Norwegian Cod Example from Ricker *** BEVERTON MODEL - FIRST LINEARIZATION *** Model: MODEL1 Dependent Variable: IR Inverse of Recruits Analysis of Variance Source Model Error C Total Root MSE Dep Mean C.V. Sum of Squares 0.00067 0.00877 0.00944 DF 1 22 23 0.01997 0.01913 104.39221 Parameter Estimates Parameter Variable DF Estimate INTERCEP 1 0.013306 IP 1 0.233693 Mean Square 0.00067 0.00040 F Value 1.668 R-square Adj R-sq 0.0705 0.0282 Standard Error 0.00607857 0.18095493 T for H0: Parameter=0 2.189 1.291 Prob>F 0.2100 Prob > |T| 0.0395 0.2100 inverse of intercept = 1 / 0.013306 = 75.15407 55 56 PROC REG; TITLE2 '*** BEVERTON MODEL - PAULIK LINEARIZATION ***'; MODEL PR=P; OUTPUT OUT=ONE RESIDUAL=E4 PREDICTED=P4; RUN; Arcto - Norwegian Cod Example from Ricker *** BEVERTON MODEL - PAULIK LINEARIZATION *** Model: MODEL1 Dependent Variable: PR ParentS / Recruit Analysis of Variance Source Model Error C Total Root MSE Dep Mean C.V. DF 1 22 23 Sum of Squares 59.02087 50.34548 109.36635 1.51276 1.73122 87.38102 Parameter Estimates Parameter Variable DF Estimate INTERCEP 1 -0.606740 P 1 0.026505 Mean Square 59.02087 2.28843 F Value 25.791 R-square Adj R-sq 0.5397 0.5187 Standard Error 0.55433536 0.00521907 T for H0: Parameter=0 -1.095 5.078 Prob>F 0.0001 inverse of intercept = 1 / 0.026505 = 37.72873 Prob > |T| 0.2856 0.0001 EXST7025 Recruitment models 58 59 60 Spring 2002 Page 4 Arcto-Norwegian Cod PROC NLIN; TITLE2 '*** BEVERTON MODEL - NONLINEAR VERSION ***'; PARAMETERS B0 = 0.0112 B1 = 0.0097; MODEL R = P / (B0*P + B1); OUTPUT OUT=ONE RESIDUAL=E5 Arcto - Norwegian Cod Example from Ricker *** BEVERTON MODEL - NONLINEAR VERSION *** Non-Linear DUD -3 -2 -1 Least Squares DUD Initialization Dependent Variable R B0 B1 Sum of Squares 0.011200 0.009700 62618.585262 0.012320 0.009700 64155.839851 0.011200 0.010670 62621.728931 Non-Linear Least Squares Iterative Phase Dependent Variable R Iter B0 B1 Sum of Squares 0 0.011200 0.009700 62618.585262 1 0.011174 0.009581 62617.650806 2 0.011178 0.009558 62617.649586 3 0.011172 0.009781 62617.625086 4 0.011173 0.009750 62617.623046 5 0.011173 0.009750 62617.623028 NOTE: Convergence criterion met. Non-Linear Least Squares Summary Statistics Dependent Variable R Source DF Sum of Squares Mean Square Regression Residual Uncorrected Total 2 22 24 184365.37697 62617.62303 246983.00000 92182.68849 2846.25559 (Corrected Total) 23 62707.62500 Parameter B0 B1 Estimate Method: DUD Asymptotic Std. Error Asymptotic 95 % Confidence Interval Lower Upper 0.0111727122 0.00207124094 0.00687725565 0.01546816881 0.0097501303 0.06498842110 -.12502653234 0.14452679303 Asymptotic Correlation Matrix Corr B0 B1 -------------------------------------------B0 1 -0.729539818 B1 -0.729539818 1 inverse of intercept = 1 /0.0111727122 = 89.50378 EXST7025 Recruitment models Arcto-Norwegian Cod 225 Spring 2002 Page 5 Fits of the Ricker recruitment curve Arcto - Norwegian Cod I 200 n d e 175 x o 150 f R 125 e c r 100 u i t 75 m e 50 n t 25 0 0 25 50 75 100 125 Index of Parental Stock Level 150 175 200 EXST7025 Recruitment models Arcto-Norwegian Cod Spring 2002 Page 6 Fits of the Beverton-Holt recruitment curve 225 Arcto - Norwegian Cod I n 200 d e x 175 o 150 f R 125 e c r 100 u i t 75 m e n 50 t 25 0 0 25 50 75 100 125 Index of Parental Stock Level 150 175 200 Initial fishery catch per unit of effort starts high, but declines steadily for several months (approximately 60 days). The data is for the fishing grounds of a single fishing village (Quintay). 1600 1400 Leslie model Catch per boat day 1200 1000 800 600 400 200 0 0 40000 80000 120000 Cumulative catch 160000 Diagnosis : Stock depletion Declines over the short period are due to fishing mortality Indicated Analysis : Leslie model N1 = N0 - C1 where N is number and C is catch N2 = N0 - C1 - C2 N3 = N0 - C1 - C2 - C3 Ut = U0 - q∑Ci where U is catch per unit of effort and qN - U Assumptions: 1) negligible natural mortality over a short period 2) negligible recruitment over the period 3) balanced immigration and emigration Prognosis: CPUE continues to decline as stock fished out. Prediction based on Leslie model : 319602 locos available 200000 Result of initial analysis : CPUE declined, but stabilized and continued beyond the expected limit (102 days) 1600 1400 Leslie model Catch per boat day 1200 1000 800 600 400 200 0 0 50000 100000 150000 200000 250000 300000 350000 Cumulative catch Initial fishery catch per unit of effort starts high, but declines steadily for several months, but levels off 1600 1400 Delury model Catch per boat day 1200 1000 800 600 400 200 0 0 50 100 150 200 250 300 350 400 450 Cumulative effort 500 550 600 650 700 Diagnosis : Stock depletion - Declines over the short period are due to fishing mortality Analysis : Delury model U t = U 0e− qΣft Assumptions: same as the Leslie model Prognosis: CPUE declines as stock fished to non-commercial limits Prediction based on Leslie model : 476693 locos available Result of initial analysis : Fishing continues beyond expected limit and the Expected Delury limit passed in 190 days of 273 day season 1600 1400 Delury model Catch per boat day 1200 1000 800 600 400 200 0 0 200 400 600 800 1000 1200 1400 1600 Cumulative effort Initial fishery catch per unit of effort starts high and declines, but stabilizes Diagnosis : Initial stock depletion due to fishing mortality, but apparent recruitment maintains fishery Analysis : No model available Assumptions: No longer a short period, so natural mortality NOT negligible and recruitment NOT negligible balanced immigration and emigration still assumed consideration of growth not needed (?), work with numbers Prognosis: Need new model Fish Stock Natural Mortality (m) Nt = N 0e− mt = N 0e− M One of the most difficult parameters to measure Important parameter, but may be minor influence Mortality relationship between number and time Fishing Mortality (f) Leslie Model U t = U 0 + qΣCt Delury Model U t = U 0e − qΣft log(U t ) = log(U 0 ) + qΣft Total mortality = - Nt = N 0e − qf − mt = N 0e − Zt Mortality Curves 1000 Mortality over time 900 800 700 N t 600 Natural mortality 500 400 Fishing mortality 300 200 Total mortality 100 0 0 5 10 15 20 25 30 Time Initial fishery catch per unit of effort starts high and declines, but then remains steadily 1600 1400 Delury model Catch per boat day 1200 1000 800 600 400 200 0 0 200 400 600 800 1000 1200 1400 1600 Cumulative effort Diagnosis : Stock model with constant recruitment There are very few models which behave well with this type of recruitment, and none designed specifically for continuous recruitment. Analysis : Do it yourself Assumptions: Constant recruitment over the study period Assume a separate and independent stock, with negligible or equal immigration and emigration Prognosis: catch equals recruitment in stable CPUE fishery. If catch exceeds recruitment then CPUE declines. The new model should consider both fishing and natural mortality and also include recruitment. Growth (increase in biomass only) need not be considered when working with numbers. One derivation of the Delury model from the Leslie model depends on the observation that eX is approximately equal to 1+X. The approximation is especially good for small negative X values. X -0.001 -0.01 -0.1 eX 0.999 0.99005 0.904837 1+X 0.999 0.99 0.9 The change in the population from any time t to any time t+1 can be expressed as Nt +1 = Nt e− qft − m + rt where ft is the fishing mortality during the time period, m is the natural mortality during any time period and r is the recruitment during the time period. Note that natural mortality and a constant recruitment term become part of the model. Reversing the Delury approximation mentioned above N t +1 = N t (1 − qft − m) + rt N t +1 = N t − qft N t − mN t + rt when multiplied through by q, and expressed as a cumulative values t −1 t −1 i =1 i =0 U t = U 0 − q ∑ Ci − m∑ U i + (qr )t High multicolinearity when fitted as cumulative values. All models are simplified. This one is perhaps over-simplified. In this case we need a better understanding of the fishery and the basic biology. The stock is still not clearly defined. How much do locos move? How much does one stock depend on another for recruitment? The parameter q still needs investigation. It will differ from site to site, but can it be considered constant within a site? How much does it vary between fishermen over time? Recruitment is not well understood and must be improved. Can it remain a constant indefinitely? Should it be adjusted seasonally for growth rate? Does it vary over the annual cycle. Recruitment is not dependent of adult stock levels in the model, and it probably should be. How much lag is needed? This relationship is not clear. Where do recruits come from? How far do they travel? Is natural mortality really a constant? Growth How is it affected by population levels? Does it seasonally effects recruitment? How variable is the period of time for juvenile growth to recruitment? Geaghan, J. P. and J. C. Castilla. 1988. Population dynamics of the Loco (Concholepas concholepas) in central Chile. Invest. Pesq. (Chile). 34:21-31. Geaghan, J. P. and J. C. Castilla. 1986. Use of catch and effort data for parameter estimates for the "Loco" (Concholepas concholepas) fishery of central Chile. Jamieson, G. S. and N. Bourne [ed.]. N. Pac. Workshop on stock assessment and management of invertebrates. Can. Spec. Publ. Fish. Aquat. Sci. 92:168-174. 1600 Modified model Catch per boat day (mean of 20) 1400 1200 Quintay U0 q m qr r No = 530.203 = -0.0010335 = -0.00040157 = -1.78955 = 1731.54 = 513016.93 1000 800 600 400 200 0 01JUN83 04OCT83 06FEB84 10JUN84 13OCT84 15FEB85 20JUN85 Quintay U0 = 530.203 q =-0.0010335 m =-0.0004016 qr = 1.78955 r = 1731.54 No = 513017 1800 Catch per boat day (mean of 50) 1600 Cruz Grande 1400 El Quisco 183.1195 -0.000964 -0.0004016 1.38575 1437.50 189958 1200 1000 Quintay 800 Hornos 600 Cruz Grande Hornos 1006.7968 400 U0 = 1545.844 q =-0.000726 -0.001339 m =-0.000531 -0.0005315 200 qr = 3.89749 3.836682 r = 5370.00 2865.44 751930 0 No = 2129878 JAN80 JAN81 JAN82 JAN83 El Quisco JAN84 JAN85 JAN86 JAN87 JAN88 Cushing recruitment (term coined by Kimura, 1984). This is a single parameter model, which may be it’s only attraction. Bi k Ri R1 B1 r where: R1 and B1 are virgin population recruitment and biomass respectively, Bi-k is biomass in the ith year – k (k is the age at recruitment, giving the lag period.) Ri is recruitment in the ith year r is a shape parameter r = 0 (highly resilient) Bi k B1 r 0 < r< r= 1 1 o (n s re n ie il y) c Bi k B1 ...
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