09 Cohort Analysis

09 Cohort Analysis - EXST7025 : Biological Population...

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Unformatted text preview: EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 1 COHORT ANALYSIS This term was originally applied by Pope to describe a simplified Virtual Population Analysis (VPA), which is a type of life table or demographic analysis. The “Cohort Analysis” technique is an approximation of the VPA, and does not require an iterative solution. The Approximations M ˆ Nt = Nt +1e M + Ct e 2 (derived later) ⎛N ⎞ Ft = ln ⎜ t +1 ⎟ − M ⎝ Nt ⎠ derived from N t +1 = N t e − Ft − M The Development and Assumptions Starting with a true cohort, all hatched in the same season. Nt Age Nt is the Number in the Sea Removals (due to Z and emigration) Nt Nt+1 We will consider a single time interval t t+1 EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 2 Nt Nt+1 t+1 t In order to simplify the assumptions and calculations, Jones (1984) comes up with the following assume that all CATCH (fishing mortality) take place at one instant in time at time t. This is supposed to provide a “good approximation". Nt } catch Nt+1 t t+½ t+1 Schematic Model, we work backwards from D to A A: Nt+1 the endpoint B: Nt +1e M 2 derived from Nt +1 = Nt + 1 e Nt M 2 D C Nt+1 2 B C: add on catch in interval A M 2 Nt +1e + Ct D: Apply mortality as in B for the first half of the interval with catch included M ⎡ ⎤ M 2 2 ⎢ Nt +1e + Ct ⎥ e ⎣ ⎦ Then M ˆ Nt = Nt +1e M + Ct e 2 t t+½ t+1 EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 3 NOTE: mortality affects all individuals (Nt) for half of the interval and survivors for another half interval after the catch. This is the “approximation". It is done without iteration. If the assumption of catch obtained in an instant is not made, the curve must be fitted iteratively. What do we need for Cohort analysis? 1) M for all age groups (usually assumed constant, though it could easily be made variable). 2) Ft for the OLDEST AGE GROUP. The number at time t1 for this group is 0 3) The number captured by size groups (AGE is better if available) One last problem: The analysis starts with the oldest age group and works forward. There are two possible calculations for the oldest age group depending on whether it consists of a particular age grout eg. Age = 8 or if it consists of all ages GREATER THAN some age eg. Age ≥ 8 If the last age is EXACT, then catch equals part of the non-surviving portion Ft / Zt (that part due to fishing mortality). the non-surviving portion ( 1 − e Zt ) ( Ct = ANt = N t 1 − e Zt F )Z t t Nt Ct / A where; A = ( Ft 1 − e Zt ) Zt If, however, the last age includes older ages, then we get a portion of non-survivors which is not all of the non-survivors so, Ct = Ft Nt Zt and, Nt = Ct Z t Ft Cohort Analysis from Jones EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 4 From Jones, R. The effect of changes in exploitation pattern using length composition data (with notes on VPA and cohort analysis). FAO Fisheries Technical Paper 256. 1984. Example from Jones Where M = 0.20 and F4 = 0.4 F5 = 0, so age = 4 is the last group Age 1 2 3 4 5 Numbers caught 50 200 300 250 0 Numbers Attaining each age St [EXP(-Zt)] Zt Ft 1346 831 0.62 0.00000 0.48 0.6 0.28 0.4000 N4 = Ct / A A= ( Ft 1 − e Zt Zt ) = 0.4 (1 − e ) −0.6 = 0.301 0.6 N4 = 250 / 0.301 = 831 M e M = 1.22 and e 2 = 1.11. These remain constant for the analysis. N3 = Nt = Nt +1e + Ct e M M 2 = 831(1.22) + 300(1.11) = 1346 ⎛ N ⎞ ⎛N ⎞ ⎛ 1346 ⎞ F3 = ln ⎜ t ⎟ − M = ln ⎜ 3 ⎟ − M = ln ⎜ ⎟ − 0.2 = 0.28 ⎝ 831 ⎠ ⎝ N4 ⎠ ⎝ N t +1 ⎠ Z3 = Ft + M = 0.28 + 0.2 = 0.48 e Z3 = 0.62 = the non-surviving portion then for age 2 N2 = Nt = Nt +1e + Ct e M M 2 = 1346(1.22) + 200(1.11) = 1865 ⎛N ⎞ ⎛ 1865 ⎞ F2 = ln ⎜ 2 ⎟ − M = ln ⎜ ⎟ − 0.2 = 0.13 ⎝ 1346 ⎠ ⎝ N3 ⎠ Z2 = Ft + M = 0.13 + 0.2 = 0.33 e Z3 = 0.72 = the non-surviving portion Repeat for age 1 The finished table is given below. EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 5 Cohort analysis by age example from Jones 1981, Table 2 Cohort analysis - Age table m =0.2 f4 =0.4 Survival A4 =0.30 Numbers Attaining each age 2334.28 1865.90 1346.71 831.14 Numbers St Age caught [EXP(-Zt)] 1 50 0.79935 2 200 0.72174 3 300 0.61716 4 250 0.00000 5 0 Last age group is equal age with none observed at a larger size Zt Ft 0.2240 0.3261 0.4826 0.6 0.0240 0.1261 0.2826 0.4000 Mean number in the sea 2091.36 1592.22 1068.27 This analysis provides some information that is not otherwise known like age specific values of the fishing mortality. Jones states that the estimate of Ft improves as we move to the younger age groups. Starting Ft value F true Ft Younger Age group Older EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 6 Key equations in the analysis are: The values required for the analysis are M, Ft for the oldest age group and Catch data by age. 1) If all fish of the last age group are caught, then the catch of that age group is the fraction Ft Zt of the NON-surviving portion. The surviving number is given by N t = N t −1e − Z , where the surviving portion is given by e-Z, and the non-surviving portion is A = (1-e-Z). Catch is then Ft 1 − e− Z FA given by Ct = N t = t N t . The number at time t, if the all fish in the interval Zt Zt Z C are either caught or die of natural mortality, is then given by N t = t t Ft A c h If, however, the last age includes older ages, then we get a portion of non-survivors, which is not ALL of the non-survivors. The calculation then omits the value of A. In this case ZC Nt = t t . Ft 2) Once we have found the number entering the oldest age group, we proceed to calculate the F number in all previous age groups as N = G N H t e t +1 M 2 I + C Je K t M 2 M 2 = N t +1e + Ct e . Notice that M M is positive here. 3) There are then a number of ancillary calculations that can be made. a) Annual survival rates: Ratio of number at time t and t+1; St = Nt+1/Nt. b) Total instantaneous mortality rate: since St=e-Zt then Zt = -ln(St) c) Instantaneous rate of Fishing mortality: Zt=Ft+M, so Ft=Zt-M. d) The number ⎯Nt is the number attaining each age (see schematic by Jones below). This is not the same as the mean number in the interval. The mean number in the interval is given by (Nt+1 - Nt)/Zt. See explanation of calculation below. Note that this has all been developed on the premise that we are following a cohort. If we have a single sample, we have something like Nt Age EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 7 Other age based tables from Jones. Cohort analysis by age example from Jones 1981, Table 1 Cohort analysis - Age table Numbers Numbers Attaining Age caught each age 1 867 10106.58 2 1732 7490.08 3 1612 4565.18 4 931 2279.05 5 362 1023.53 6 153 510.44 7 65 279.47 8+ 85 170.00 Last age group is equal age or greater St [EXP(-Zt)] 0.74111 0.60950 0.49923 0.44910 0.49871 0.54751 0.60828 Cohort analysis by age example from Jones 1981, Table 3 Cohort analysis - Age table Numbers Attaining each age 10232.09 7592.83 4649.31 2347.93 1079.92 556.61 317.28 200.95 160.00 Numbers Age caught 1 867 2 1732 3 1612 4 931 5 362 6 153 7 65 8 5 9 80 10+ 0 Last age group is equal age or greater St [EXP(-Zt)] 0.74206 0.61233 0.50501 0.45995 0.51542 0.57001 0.63336 0.79622 m =0.2 f8 =0.2 Zt Ft 0.2996 0.4951 0.6947 0.8005 0.6957 0.6024 0.4971 0.4 0.0996 0.2951 0.4947 0.6005 0.4957 0.4024 0.2971 0.2 m= f8 = 0.2 0.2 Zt Ft 0.2983 0.4905 0.6832 0.7766 0.6628 0.5621 0.4567 0.2279 0.4 0.0983 0.2905 0.4832 0.5766 0.4628 0.3621 0.2567 0.0279 0.2 Mean number in the sea 8733.10 5907.44 3290.82 1568.42 737.47 383.43 220.22 Mean number in the sea 8846.95 6001.24 3368.61 1632.68 789.58 425.79 254.70 179.70 EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 8 Data initially required: Catch at age (Ct) or length class (Cl), natural mortality (M), and fishing mortality for the oldest age group Ft). Calculations for data by ages The number in the oldest age group is calculated as N t = fully fished out and N t = Ct Zt if the oldest group is Ft 1 − e − Z c h Zt Ct if it is not. Ft After the oldest group, numbers at all other ages are: FG H IJ K M M M N t = N t +1e 2 + Ct e 2 = N t +1e M + Ct e 2 Once numbers at each age are obtained, survival is calculated at time t and t+1 as St = Nt+1/Nt. Total instantaneous mortality rate is give as Zt = -ln(St) since St=e-Zt Instantaneous rate of Fishing mortality is Ft=Zt-M since Zt=Ft+M. The average number ⎯Nt (the number attaining each age, which is not the same as the mean number in the interval) is given by ⎯Nt = (Nt+1 - Nt)/Zt. See explanation of calculation below. Modifications for data by lengths. e j Given Lt = L∞ 1 − e − k b t − t0 g , and t = t 0 − FG H IJ K 1 1 − l1 ln , then derive k L∞ FG IJ H K F L − l IJ then the original equation, generalized for t+Δt instead of t+1, Define X = G HL −l K I F N = GN e + C J e , can be rewritten as N = b N X + C g X , where K H Δt = t2 − t1 = ∞ 1 L −l ln ∞ 1 , k L∞ − l2 1 M 2K l ∞ Δt t + Δt Xl = e MΔt 2 2 MΔt 2 MΔt 2 Δt l . The steps in the calculations for tables are: 1) Δt = t2 − t1 = 2) X l = e MΔt 2 FG H 1 L −l ln ∞ 1 k L∞ − l2 IJ K l +Δl l Δl l EXST7025 : Biological Population Statistics Cohort Analysis b James Geaghan Page 9 g 3) N l = N l +Δl X l + CΔl X l 4) Survival is calculated for Δt as SΔt = Nt+Δt/Nt. 5) ZΔt = -ln(SΔt) since St=e-ZΔt 6) Exploitation rate is (FΔt /ZΔt) = (number caught / number dying) = CΔt / (Nt+Δt - Nt) so instantaneous rate of fishing mortality is FΔt = (FΔt/ZΔt)ZΔt = ZΔtCΔt/(Nt+Δt-Nt) 7) Annualized Total mortality at length l: Zl = M / (1 - (FΔt/ZΔt)) = M / (1 - (CΔt/(Nt+Δt-Nt))) 8) Annualized Fishing mortality at length l: Fl = Zl - M 9) The average number for the size class⎯Nl is given by ⎯Nl = (Nl+Δl - Nl)/Zl. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 A B C D E F Cohort analysis by size example from Jones 1981, Table 4 G H Linf = 80 m = 0.2 F = 0.2 Numbers Numbers caught attaining Size Xl *10^6 each age 20 1.044 0.10 50.484 25 1.049 0.47 46.181 30 1.054 3.88 41.535 35 1.061 5.54 33.700 40 1.069 5.37 24.733 45 1.080 4.62 16.618 50 1.095 3.03 9.967 55 1.118 1.68 5.540 60 1.155 1.02 2.929 65 1.225 0.46 1.313 70 0.25 0.500 Last age group is equal size or greater St [EXP(-Zt)] 0.91477 0.89939 0.81138 0.73390 0.67190 0.59975 0.55581 0.52874 0.44841 0.38070 I J K K = 0.2 m/K = 1 F/Z = 0.5 Zt F/Z Fdt 0.0891 0.1060 0.2090 0.3094 0.3976 0.5112 0.5873 0.6373 0.8020 0.9658 0.4 0.023241099 0.10115351 0.495254597 0.617776991 0.661757502 0.694603939 0.684420803 0.643545026 0.631348783 0.565539485 0.5 0.0021 0.0107 0.1035 0.1911 0.2631 0.3551 0.4020 0.4101 0.5064 0.5462 0.2 Mean Z number in sea 0.20475882 21.014 0.222507405 20.882 0.396239369 19.772 0.523254737 17.138 0.591291754 13.724 0.654887293 10.156 0.633755336 6.986 0.561080681 4.653 0.542518215 2.978 0.460341028 1.767 dt 0.4351 0.4766 0.5268 0.5889 0.6677 0.7708 0.9116 1.1157 1.4384 2.0273 119.06 Excel calculations: Cells A9-A19 and C9-C19 are observed data, and G3-G5 and I3 are initial parameter estimates. Cell Cell I4: Cell I5: Cell B9: Cell K9: Cell D19: Cell F19: Cell G19: Cell H19: Cell D18: Cell E18: Cell F18: Cell G18: Cell H18: Cell I18: Cell J18: Equation =G4/I3 =G5/(G5+G4) =(($N$3-A9)/($N$3-A10))^($P$4/2) =(1/$P$3)*LN(($N$3-A9)/($N$3-A10)) =C19/I5 =G4+G5 =I5 =G5 =(D19*B18+C18)*B18 =D19/D18 =-LN(E18) =C18/(D18-D19) =F18*G18 =$N$4/(1-G18) =-(D19-D18)/I18 Action copied down to B18 copied down to K18 copied up to D9 copied up to E9 copied up to F9 copied up to GD9 copied up to H9 copied up to I9 copied up to J9 EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 1 Cohort analysis by size example from Jones 1981, Table 5 Linf = 70 m = 0.2 F = 0.4667 Size 10 15 20 25 30 35 40 45 50 55 60 65 Xl 1.018 1.019 1.021 1.024 1.027 1.031 1.037 1.046 1.059 1.084 1.149 Numbers caught *10^3 1 163 1390 4120 4730 3040 1650 827 312 94 10 3 Numbers attaining each age 20246 19553 18662 16530 11746 6529 3191 1376 467 122 17 4 St EXP(-Zt) 0.9657 0.9544 0.8858 0.7105 0.5559 0.4887 0.4311 0.3398 0.2612 0.1404 0.2500 K =0.5 m/K =0.4 F/Z =0.7 Zt F/Z Fdt Z 0.0349 0.0467 0.1213 0.3417 0.5872 0.7159 0.8413 1.0795 1.3425 1.9633 1.3862 0.6667 0.0014 0.1829 0.6522 0.8611 0.9068 0.9107 0.9089 0.9104 0.9034 0.8956 0.7778 0.7000 0.0001 0.0085 0.0791 0.2942 0.5324 0.6520 0.7647 0.9828 1.2128 1.7583 1.0783 0.4667 0.2003 0.2448 0.5751 1.4394 2.1455 2.2393 2.1956 2.2322 2.0700 1.9155 0.9002 0.6667 Mean number in sea 3463 3642 3706 3324 2431 1491 827 407 167 55 14 19526 Last age group is equal size or greater dt 0.1740 0.1906 0.2107 0.2356 0.2671 0.3083 0.3646 0.4463 0.5754 0.8109 1.3863 EXST7025 : Biological Population Statistics Cohort Analysis James Geaghan Page 2 Black drum Cohort analysis : natural mortality = 0.15 - 1986/89 (1989 vertical, 3 year diagonal) Numbers caught Age No attaining each age by cohort 1988 1989 C+0 C+1 C+2 C+3 Survival (EXP(-Zt)) C+0 C+1 C+2 C+0 Zt C+1 Ft 1986 1987 C+2 0 13394 23873 2376 1 733622 382076 441920 975340 8738651 9640015 8439580 2 81595 106216 1028730 90430 6586067 6490793 7546860 3 6088 186013 106799 133861 5132994 5318385 4 1218 14577 17997 8324 4188737 5 8497 12820 17256 5515 3605526 6 11906 2659 8827 812 3110639 7 26148 5594 12205 2749 2776536 2666306 2661787 2651594 0.8519 0.8587 0.8564 0.1602 0.1523 0.1550 8 15460 39672 16047 6748 2296444 2365528 2289721 2279698 0.8544 0.8451 0.8542 0.1573 0.1682 0.1576 9 9715 18056 45068 4870 1961526 1962225 1999223 1955894 0.8561 0.8521 0.8397 0.1554 0.1600 0.1746 10 7207 11224 19134 25429 1688498 1679288 1672151 11 1218 37280 20454 7946 1457744 1446618 1434963 12 64275 26369 5485 10662 1387467 1253562 13 13196 60466 21216 4828 1014510 14 21449 23326 46765 16598 15 14414 19068 7767 16 19185 13452 17 25138 18 Mean number in the sea C+0 C+1 C+2 0.0013 0.0026 0.0002 10675063 9412417 9942730 0 9029100 0.7427 0.7828 0.7713 0.2974 0.2448 0.2596 0.0974 0.0448 0.0596 8439751 7858909 7245213 1 0.20 6509878 0.8075 0.8039 0.6953 0.2138 0.2183 0.3633 0.0138 0.0183 0.1633 4740534 4759259 8596607 2 0.20 5218104 5248013 0.8176 0.7870 0.8002 0.2013 0.2394 0.2229 0.0013 0.0394 0.0229 4229253 5116748 4710732 3 0.20 4197031 4186013 4175587 0.8604 0.8574 0.8567 0.1503 0.1538 0.1546 0.0003 0.0038 0.0046 3823293 3911911 3922624 4 0.15 3604150 3598894 3586238 0.8585 0.8574 0.8562 0.1525 0.1538 0.1552 0.0025 0.0038 0.0052 3315506 3340325 3362319 5 0.15 3095422 3090227 3081588 0.8571 0.8599 0.8580 0.1541 0.1509 0.1531 0.0041 0.0009 0.0031 2909538 2839486 2872214 6 0.15 0.0102 0.0023 0.0050 2637002 2416143 2451456 7 0.15 0.0073 0.0182 0.0076 2074633 2273802 2072203 8 0.15 0.0054 0.0100 0.0246 1727871 1775845 1960813 9 0.15 1678935 0.8567 0.8545 0.8500 0.1546 0.1572 0.1624 0.0046 0.0072 0.0124 1530245 1545709 1585844 10 0.15 1421482 0.8599 0.8367 0.8474 0.1509 0.1782 0.1655 0.0009 0.0282 0.0155 1239329 1432997 1328390 11 0.15 1210529 1216109 0.8177 0.8411 0.8565 0.2012 0.1729 0.1549 0.0512 0.0229 0.0049 1426596 1122998 979889 12 0.15 1134573 1054487 1036823 0.8486 0.8112 0.8420 0.1641 0.2092 0.1719 0.0141 0.0592 0.0219 841662 1173698 912966 13 0.15 903226 860954 920439 887923 0.8386 0.8355 0.8135 0.1759 0.1796 0.2063 0.0259 0.0296 0.0563 777350 755229 915441 14 0.15 35093 729651 757514 719390 748843 0.8423 0.8373 0.8506 0.1715 0.1775 0.1617 0.0215 0.0275 0.0117 675098 723228 630513 15 0.15 23238 2522 784550 614644 634308 611978 0.8380 0.8404 0.8267 0.1767 0.1739 0.1903 0.0267 0.0239 0.0403 705749 544774 610410 16 0.15 108483 10479 17757 555609 657469 516550 524395 0.8187 0.7076 0.8418 0.2000 0.3458 0.1721 0.0500 0.1958 0.0221 412604 787507 334598 17 0.15 4907 45503 36663 8489 418648 454895 465244 434877 0.8498 0.7679 0.7875 0.1627 0.2641 0.2388 0.0127 0.1141 0.0888 282315 474122 443763 18 0.15 19 14414 21050 9564 25437 386074 355782 349317 366426 0.8260 0.8058 0.8353 0.1911 0.2159 0.1800 0.0411 0.0659 0.0300 343476 353387 294272 19 0.15 20 24020 26804 7389 12113 366526 318925 286695 291787 0.7999 0.7827 0.8367 0.2233 0.2450 0.1782 0.0733 0.0950 0.0282 338398 319724 215896 20 0.15 21 21413 36277 11758 2732 241020 293188 249634 239905 0.7782 0.7459 0.8170 0.2507 0.2931 0.2021 0.1007 0.1431 0.0521 213433 297535 182449 21 0.15 22 14341 3221 22822 9147 191160 187582 218693 203954 0.7911 0.8447 0.7638 0.2343 0.1687 0.2693 0.0843 0.0187 0.1193 176964 129036 228827 22 0.15 23 10932 6792 5519 13491 150652 151228 158465 167058 0.7933 0.8190 0.8283 0.2314 0.1996 0.1883 0.0814 0.0496 0.0383 151132 132870 132031 23 0.15 24 24165 8669 6910 13784 135605 119525 123863 131272 0.6953 0.7934 0.8089 0.3633 0.2314 0.2120 0.2133 0.0814 0.0620 153019 91468 87660 24 0.15 25 7171 10573 4154 9131 125629 94298 94834 100199 0.8077 0.7566 0.8200 0.2135 0.2788 0.1984 0.0635 0.1288 0.0484 105951 100656 74856 25 0.15 26+ 46726 40591 28541 31108 116814 101477 71353 0.1 0.1 13214 11789143 10334510 11030793 11042772 0.8177 0.8166 0.8185 0.2013 0.2026 0.2002 77770 S 0.1 1986 1987 1988 Age M 0.2 26+ 0.15 EXST7025 : Biological Population Statistics Cohort Analysis Black drum Cohort analysis : natural mortality = 0.10 - 1986/89 Numbers caught No attaining each age by Age 1986 1987 1988 1989 C+0 C+1 C+2 0 13394 23873 2376 13214 5862213 4407580 5103864 1 733622 382076 441920 975340 3886091 4787455 3587020 2 81595 106216 1028730 90430 2612743 2517854 3573920 3 6088 186013 106799 133861 1878725 2065303 1965336 4 1218 14577 17997 8324 1523753 1532661 1522615 5 8497 12820 17256 5515 1378023 1377591 1372943 6 11906 2659 8827 812 1252784 1238805 1234301 7 26148 5594 12205 2749 1222913 1122241 1118387 8 15460 39672 16047 6748 1016745 1081665 1010124 9 9715 18056 45068 4870 903384 905283 940993 10 7207 11224 19134 25429 816854 808175 801958 11 1218 37280 20454 7946 739659 732265 720590 12 64275 26369 5485 10662 792299 668113 627118 13 13196 60466 21216 4828 543750 655762 579451 14 21449 23326 46765 16598 517210 479453 535841 15 14414 19068 7767 35093 421978 447587 411638 16 19185 13452 23238 2522 524179 368111 386856 17 25138 108483 10479 17757 358319 456047 320285 18 4907 45503 36663 8489 264909 300308 309456 19 14414 21050 9564 25437 262202 235032 228446 20 24020 26804 7389 12113 266658 223539 192642 21 21413 36277 11758 2732 169798 218434 176770 22 14341 3221 22822 9147 137614 133271 163139 23 10932 6792 5519 13491 110882 110876 117525 24 24165 8669 6910 13784 107409 89931 93864 25 7171 10573 4154 9131 97258 74202 73127 26+ 46726 40591 28541 31108 93451 81182 57083 James Geaghan Page 3 (1989 vertical, 3 year diagonal) cohort Survival (EXP(-Zt)) C+3 C+0 C+1 C+2 C+0 5115842 0.8166 0.8138 0.8183 0.2025 4176540 0.6479 0.7465 0.7072 0.4340 2536938 0.7904 0.7805 0.5582 0.2351 1995245 0.8157 0.7372 0.7695 0.2036 1512446 0.9040 0.8957 0.8935 0.1008 1360599 0.8989 0.8959 0.8928 0.1065 1225876 0.8957 0.9027 0.8980 0.1100 1108446 0.8844 0.9000 0.8944 0.1227 1000349 0.8903 0.8699 0.8897 0.1161 898734 0.8946 0.8858 0.8592 0.1114 808576 0.8964 0.8916 0.8821 0.1093 707441 0.9032 0.8564 0.8778 0.1017 632560 0.8276 0.8672 0.8965 0.1891 562223 0.8817 0.8171 0.8700 0.1258 504128 0.8653 0.8585 0.8218 0.1446 440365 0.8723 0.8643 0.8868 0.1366 365077 0.8700 0.8700 0.8476 0.1392 327936 0.8381 0.6785 0.8737 0.1766 279838 0.8872 0.7607 0.7921 0.1197 245132 0.8525 0.8196 0.8650 0.1595 197609 0.8191 0.7907 0.8683 0.1995 167281 0.7848 0.7468 0.8415 0.2422 148764 0.8057 0.8818 0.7717 0.2160 125906 0.8110 0.8465 0.8601 0.2094 101091 0.6908 0.8131 0.8348 0.3699 78359 0.8347 0.7692 0.8508 0.1807 62216 Black drum Cohort analysis : natural mortality = 0.20 - 1986/89 (1989 vertical, 3 year diagonal) Numbers caught No attaining each age by cohort Survival (EXP(-Zt)) Age 1986 1987 1988 1989 C+0 C+1 C+2 C+3 C+0 C+1 C+2 C+0 0 13394 23873 2376 13214 29948682 28494049 29190333 29202311 0.8183 0.8179 0.8186 0.2005 1 733622 382076 441920 975340 23606424 24507788 23307353 23896873 0.7906 0.8046 0.8015 0.2349 2 81595 106216 1028730 90430 18759164 18663497 19719563 18682581 0.8147 0.8135 0.7715 0.2048 3 6088 186013 106799 133861 15100768 15284874 15184271 15214179 0.8183 0.8077 0.8123 0.2004 4 1218 14577 17997 8324 12350415 12357954 12345884 12335194 0.8186 0.8176 0.8174 0.2001 5 8497 12820 17256 5515 10113051 10110563 10104647 10091671 0.8179 0.8175 0.8171 0.2009 6 11906 2659 8827 812 8288804 8272178 8266229 8257371 0.8174 0.8184 0.8177 0.2016 7 26148 5594 12205 2749 6896409 6775526 6770280 6759829 0.8152 0.8179 0.8170 0.2042 8 15460 39672 16047 6748 5549043 5622642 5542270 5531993 0.8162 0.8123 0.8161 0.2031 9 9715 18056 45068 4870 4529909 4529183 4567533 4523107 0.8167 0.8151 0.8098 0.2024 10 7207 11224 19134 25429 3709724 3699985 3691844 3698800 0.8169 0.8159 0.8140 0.2021 11 1218 37280 20454 7946 3046098 3030744 3019136 3005313 0.8183 0.8076 0.8126 0.2004 12 64275 26369 5485 10662 2637444 2492832 2447631 2453352 0.7966 0.8091 0.8167 0.2273 13 13196 60466 21216 4828 1972266 2101198 2017099 1998987 0.8126 0.7926 0.8092 0.2074 14 21449 23326 46765 16598 1650150 1602814 1665604 1632264 0.8069 0.8055 0.7933 0.2145 15 14414 19068 7767 35093 1301266 1331620 1291167 1321366 0.8087 0.8057 0.8132 0.2123 16 19185 13452 23238 2522 1237423 1052344 1072985 1050090 0.8047 0.8071 0.7991 0.2173 17 25138 108483 10479 17757 889409 995757 849415 857459 0.7931 0.7201 0.8075 0.2317 18 4907 45503 36663 8489 668291 705441 717097 685960 0.8120 0.7603 0.7724 0.2081 19 14414 21050 9564 25437 576464 542711 536393 553935 0.7961 0.7836 0.8025 0.2280 20 24020 26804 7389 12113 511493 458927 425287 430508 0.7762 0.7658 0.8030 0.2533 21 21413 36277 11758 2732 341011 397041 351484 341510 0.7619 0.7360 0.7884 0.2719 22 14341 3221 22822 9147 262495 259821 292245 277132 0.7692 0.8075 0.7480 0.2623 23 10932 6792 5519 13491 200681 201937 209809 218620 0.7694 0.7882 0.7949 0.2621 24 24165 8669 6910 13784 168715 154412 159186 166783 0.6891 0.7679 0.7794 0.3723 25 7171 10573 4154 9131 156658 116267 118578 124078 0.7773 0.7364 0.7870 0.2519 26+ 46726 40591 28541 31108 140177 121772 85624 93325 Zt Ft C+1 0.2060 0.2923 0.2477 0.3048 0.1100 0.1098 0.1023 0.1053 0.1393 0.1212 0.1147 0.1550 0.1424 0.2020 0.1525 0.1458 0.1392 0.3878 0.2735 0.1989 0.2347 0.2919 0.1257 0.1666 0.2069 0.2623 C+2 0.2005 0.3464 0.5829 0.2619 0.1125 0.1133 0.1075 0.1115 0.1168 0.1517 0.1254 0.1303 0.1092 0.1393 0.1962 0.1200 0.1652 0.1350 0.2330 0.1450 0.1412 0.1725 0.2591 0.1506 0.1805 0.1616 C+0 0.0025 0.2340 0.0351 0.0036 0.0008 0.0065 0.0100 0.0227 0.0161 0.0114 0.0093 0.0017 0.0891 0.0258 0.0446 0.0366 0.0392 0.0766 0.0197 0.0595 0.0995 0.1422 0.1160 0.1094 0.2699 0.0807 S 0.1 Zt C+1 0.2009 0.2174 0.2063 0.2135 0.2013 0.2014 0.2004 0.2009 0.2078 0.2044 0.2034 0.2137 0.2118 0.2323 0.2162 0.2160 0.2142 0.3283 0.2740 0.2438 0.2667 0.3064 0.2138 0.2379 0.2641 0.3059 C+2 0.2001 0.2212 0.2594 0.2078 0.2016 0.2019 0.2012 0.2020 0.2032 0.2110 0.2057 0.2075 0.2025 0.2117 0.2315 0.2067 0.2242 0.2137 0.2582 0.2199 0.2194 0.2377 0.2903 0.2295 0.2492 0.2395 C+0 0.0005 0.0349 0.0048 0.0004 0.0001 0.0009 0.0016 0.0042 0.0031 0.0024 0.0021 0.0004 0.0273 0.0074 0.0145 0.0123 0.0173 0.0317 0.0081 0.0280 0.0533 0.0719 0.0623 0.0621 0.1723 0.0519 Start F= C+1 0.0060 0.0923 0.0477 0.1048 0.0100 0.0098 0.0023 0.0053 0.0393 0.0212 0.0147 0.0550 0.0424 0.1020 0.0525 0.0458 0.0392 0.2878 0.1735 0.0989 0.1347 0.1919 0.0257 0.0666 0.1069 0.1623 0.1 C+2 0.0005 0.1464 0.3829 0.0619 0.0125 0.0133 0.0075 0.0115 0.0168 0.0517 0.0254 0.0303 0.0092 0.0393 0.0962 0.0200 0.0652 0.0350 0.1330 0.0450 0.0412 0.0725 0.1591 0.0506 0.0805 0.0616 0.1 Ft C+1 0.0009 0.0174 0.0063 0.0135 0.0013 0.0014 0.0004 0.0009 0.0078 0.0044 0.0034 0.0137 0.0118 0.0323 0.0162 0.0160 0.0142 0.1283 0.0740 0.0438 0.0667 0.1064 0.0138 0.0379 0.0641 0.1059 0.1 C+2 0.0001 0.0212 0.0594 0.0078 0.0016 0.0019 0.0012 0.0020 0.0032 0.0110 0.0057 0.0075 0.0025 0.0117 0.0315 0.0067 0.0242 0.0137 0.0582 0.0199 0.0194 0.0377 0.0903 0.0295 0.0492 0.0395 0.1 Mean number in the sea 1986 1987 1988 Age M 5298190 4045078 4571386 0 0.2 3894862 3454482 2989193 1 0.20 1488508 1502312 4292469 2 0.20 1343814 2107332 1758637 3 0.20 1356014 1481777 1503088 4 0.10 1267119 1304167 1338551 5 0.10 1224357 1129388 1180390 6 0.10 1245263 988429 1040638 7 0.10 896238 1131104 895664 8 0.10 742410 805693 1032547 9 0.10 726677 752412 811965 10 0.10 556029 817156 684130 11 0.10 912556 592578 433734 12 0.10 407716 760432 477634 13 0.10 422326 411363 579160 14 0.10 400702 451764 346354 15 0.10 463453 325326 400786 16 0.10 238427 602496 166239 17 0.10 141904 341317 305510 18 0.10 230882 253142 184150 19 0.10 248712 241210 130797 20 0.10 153241 231976 117493 21 0.10 131752 77593 183473 22 0.10 119896 97352 94045 23 0.10 130289 65932 60835 24 0.10 80846 86090 54867 25 0.10 26+ 0.10 Mean 1986 27136621 22027535 15523391 13232702 11143078 9140087 7527181 6293975 4981785 4030220 3332471 2670350 2504432 1699674 1442725 1175903 1105851 707565 508181 509880 463009 297347 236235 190527 177497 132238 0.1 number in the sea 1987 1988 Age M 25868797 26401284 0 0.2 21338123 20609716 1 0.20 15545445 20130397 2 0.20 14179153 13745370 3 0.20 11210016 11214526 4 0.20 9157266 9171873 5 0.20 7470574 7492967 6 0.20 6093802 6118662 7 0.20 5153971 4978384 8 0.20 4066236 4218683 9 0.20 3341649 3369532 10 0.20 2814410 2730768 11 0.20 2221820 2095301 12 0.20 2003965 1770447 13 0.20 1411554 1559167 14 0.20 1221791 1138846 15 0.20 928592 986236 16 0.20 1071759 628665 17 0.20 684078 660257 18 0.20 509389 459334 19 0.20 434655 338917 20 0.20 383801 272304 21 0.20 195093 287205 22 0.20 176036 177171 23 0.20 121272 118814 24 0.20 116156 95725 25 0.20 0.1 EXST7025 : Biological Population Statistics Cohort Analysis Interpretation of Jones' Model James Geaghan Page 1 There are 3 particularly important outputs from the model. 1) The estimates of F are age (or size) specific. Up until this point, we always assumed that F was constant for a population. The Z value is simply F+Z. Recall that Jones states that the estiamte of Ft improves as we move from the oldest to the youngest ages. It would be possible to add a variable value of Mt (different for each age). This would be easy to incorporate into the equations, but this is not estimated by this model. Independent estimates would be needed. It is possible that part of the changing F observed for this model is actually due to changing M. 2) The “Mean Number in the Sea" is an estimate of the mean standing stock. Note that it is much smaller than the “Numbers Attaining a Size". 3) The “Numbers Attaining a Size" includes all individuals that grow into an age or size class during a time interval. Some of these stay in the interval, Some grow THROUGH the interval, Some are caught (F) Some die of natural mortality For equilibruim: N1dt = Zndt + N2dt n = (N1 - N2) / Z Nt Stock Size N1 Zndt N2 dt n N 1 dt dt L1 L2 N 2 dt Length Jones’ Figure 14: Caption - The numbers, n, in a length interval (shaded), and the changes in a short period Δt, due to growth into and out of the interval, and due to deaths. Therefore, the number attaining a size can be much larger than the number actually in the size class at a given time. Jones considers the “Numbers attaining a Size" to be Recruits. EXST7025 : Biological Population Statistics James Geaghan Cohort Analysis Page 2 Length Frequency techniques for finding TOTAL MORTALITY (or Natural mortality in the absence of fishing). Nt or Freq Length The descending arm is a function of time. If we actually had the age, instead of the length, this is a simple exponential decay curve with slope equal to Z. If we fortunate enough to obtain measurements of a virgin stock, we could estimate M. However, we have already seen a formula to change LENGTHS to AGES, based on the von Bertalanffy growth curve (a similar transformation can be derived for other growth curves). from t= 1 ⎛ L∞ − Lt ln ⎜ K ⎝ L∞ ⎞ ⎟ + t0 ⎠ We previously derived Δt = t2 − t1 = 1 ⎛ L∞ − l1 ⎞ ln ⎜ ⎟ K ⎝ L∞ − l2 ⎠ EXST7025 : Biological Population Statistics James Geaghan Cohort Analysis Page 3 There are two approaches to fitting Mortality estimate to the descending arm of the length frequencies. One is to calculate from the peak a Δt for each length. The peak then becomes an arbitrary t0, and we have a series of Δt's to fit N t +Δt = N 0 e Z Δt or N t +Δt = N t e Z Δt And since Δt = t2 − t1 = Then N t +Δt = N t e Z 1 ⎛ L∞ − l1 ⎞ ln ⎜ ⎟ K ⎝ L∞ − l2 ⎠ 1 ⎛ L∞ −l1 ⎞ ln ⎜ ⎟ K ⎝ L∞ −l2 ⎠ The second approach is due to vanSickle (Oecologia 27, 1977). He develops a relationship (generic) describing mortality as Z = f(lt) * ln[f(Nt)] - f(growth) and proposes the following estimates to estimate the functions f (lt ) = Δl which can be estimated by the mean growth rate per year and Δt lnf(Nt) = b1 is the slope of the length frequencies fitted as ln(Number) on length ( ≈ f(growth) the Brody growth coefficient (an instantaneous rate) There has been much additional length frequency work Δl ) Δt ...
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