02 Mortality

02 Mortality - Fish Stock Natural Mortality (m) Nt = N...

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Unformatted text preview: 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 POPULATION STATISTICS - many standard relationships exist which are used extensively in working with biological populations. We will generally follow the notation used by Ricker, though there will be exceptions. First imagine a population. The first step in working with a biological population is defining a statistical population. There are often assumptions made about the population, depending on the model used. Typical assumptions made are; 1) All fish are equally susceptible to the fishing gear. 2) The population is closed (no immigration or emigration) or the number of immigrants equals the number of emigrants. 3) Stable age structure - the pattern observed for various cohorts at one time is the same as for one cohort over time. BASIC POPULATION RELATIONSHIPS (numbers of individuals) Nt = the number of individuals in the population at time t if the fish are randomly distributed and the nets are set at random the catch in the net will average some fraction of the total population (Nt ). q = fraction of total population caught by 1 application of effort. called the catchability coefficient this is often a weak link in calculations because the assumptions are often not met. (ie. fishing is not random) and the value may not be a constant (ie. seasonal variation, etc.) CPUEt = Ut = the average catch in 1 unit of effort at time t CPUE> = Ut = q*Nt = Ct ft where; Ct = the total catch at time t ft = the total effort at time t EXST025 - Biological Population Statistics INSTANTANEOUS RATES - there are many applications and advantages in using instantaneous rates. It describes losses and gains which are proportional instead of linear. MORTALITY is usually described as an instantaneous rate. eg. suppose 50% of a population dies off in a time interval of 1 (the time starts at t" and ends at t# ) define dt = ˜ t = t# - t" N! as the population size at time 0, for example 1000 individuals then N! = 1000 Nt as the population size after “t" intervals. r is the rate of loss, for example -0.50 or -50% (negative since it is a loss) r = -0.50 this results in a loss of 500 in the first time interval. as a linear equation; N" = N! - 500*t = 1000 - 500*1 = 500 N# = N! - 500*t = 1000 - 500*2 = 0 so as a linear equation there is a 50% loss in the first time period, but a 100% loss in the second. if there is a constant 50% reduction each time we expect the losses to go 1000, 500, 250, 125, 62, 31, 16, 8, 4, 2, 1, 0. to follow this pattern we need an exponential equation. Nt = N! ert the r in this equation is NOT the same as the previous r, the r in this equation is an instantaneous rate. Page 2 EXST025 - Biological Population Statistics To calculate instantaneous rates it we can use Nt N = ert we know that N" /N! = 0.50 for our example, and if t = 1 (for only 1 time interval), then ert = er = 0.50, so N r = ln(0.50) = ln( N!t ) taking logs we get r = ln(0.50) = -0.693147 The value or r = -0.693147 can be used in the equation Nt = N! ert to solve for Nt at any time t eg. N! = N! e-0.693147*0 = 1000*1 = 1000 N" = N! e-0.693147*1 = 1000*0.5 = 500 N# = N! e-0.693147*2 = 1000*0.25 = 250 then at time (t) Nt 0 1000 . 1 500 . 2 250 . 3 125 . 4 62 5 . 5 31 25 . 6 15 625 . 7 7 8125 . 8 3 90625 . 9 1 953125 . 10 0 976562 . 11 0 488281 . 12 0 244141 . Page 3 EXST025 - Biological Population Statistics NOTE that the whole expression “ert " acts like a proportion such that Nt N! = ert the proportion “ert " is the surviving portion and is called S S = ert A = 1 - ert = the non-surviving portion NOTE: The two times do not have to be time 0 and some other time, they can be any two times, for example t# and t" = ˜ t. Then Nt# = Nt" er(t# -t" ) = Nt" er(˜t) Previously we defined CPUE (catch per unit effort) as Ut = qNt which is fine for 1 application of a unit of effort, however, q is also a proportion of the catch, so as soon as the application of effort starts to reduce the population we must consider q as an instantaneous rate. q is not a function of time, it is a function of effort (f). Then, Nt = N! eqf> Page 4 EXST025 - Biological Population Statistics Page 5 EXPONENTIAL GROWTH (AND DECAY) MODEL - NATURAL MORTALITY Nt = N! ert where; t is time (in units which correspond to units of r) N! is number at time zero Nt is number at time t r is the rate of growth in units of N t (eg. indiv. day = 1 day , otherwise it could be in pounds kg week , month , etc.) the direction of the curve is determined by + r growth - r mortality or decay (usually indicated as “m" instead of “r") in theory r = births deaths = b - d where; b is the rate of birth (per individual) d is the rate of death so the change in the number of individuals (N) is given by ˜ N = bN - dN the “intrinsic" rate of growth is the maximum possible rate of growth ( in the absence of deaths) rmax = b , where d = 0 (or an absolute minimum) Instantaneous rates can be easily converted to different units of time suppose growth = 10% per year annual rate b = 0.1 monthly rate b = 0.008333 daily rate b = 0.000274 all give the same increase t=1 t = 12 t = 365 EXST025 - Biological Population Statistics Page 6 The exponential DECAY model is asymptotic on 0.0. Yi = "! e-"" Xi In order to get the model to be asymptotic on something other than 0.0, we must add a constant. Yi = "# "! e"" Xi This model is now NONLINEAR. It will be discussed later. CALCULATION OF HALF-LIFE OR DOUBLING TIMES Nt = N! e-rt then, to calculate the time (t " ) at which N! is reduced by # N! 2 1 2 = N! e = e " # -rt " # -rt " # log( " ) = -rt " # # and t " = # log( " ) # -r = -0.6931472 -r = 0.6931472 r doubling times are given by 2N! = N! e-rt‡# and t‡# = 0.6931472 r so halving and doubling take the same time, and the sign on the instantaneous rate determines which occurs EXST025 - Biological Population Statistics Page 7 DERIVATION of the Exponential Growth model r is a proportionality constant -- this is important Y = bX Y proportional to X assume that the change in N in one time unit is proportional to the number at the start of that time unit eg. ˜N ˜t = + rN where, ˜ N is the change in N ˜ t is 1 time unit r is a proportionality constant N is the number at the start of the time period then; ˜t ˜N = 1 rN integrate over time (one unit, such as one year) t = 1 r ' ˜N N = 1 r ln(N) + c = t ln(N) + ln(c') = rt N c' = ert N = c'' ert where, solving for c at t = 0 N! = c ert! = c er 0 = c = N! c'' = N when t = 0, or N! (for biological application) eg. Nt = c ert = N! ert for any 1 interval of time we can also use Nt = Nt-1 er EXST025 - Biological Population Statistics Page 8 as a STATISTICAL MODEL let Nt = N! ert %> log(Nt ) = log(N! ) + rt + log(%> ) a semilog model in simple linear regression form, Y = b! + b" X where Y = ln(Nt ) X=t "! = intercept = log(N! ) "" = r = slope Note: assume that log(%> ) is distributed NID (0,5 # ) all assumptions apply to transformed model original model has multiplicative error term OTHERWISE: NONLINEAR where Nt = N! ert + %> Shape of the Curves; when r p exponential growth curve r p exponential decay or mortality curve 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 as a statistical model -- “X" (the independent variable) does not have to be “time". We will see other uses. EXST025 - Biological Population Statistics Page 9 TOTAL MORTALITY Assume that natural mortality is a function of time only and is given by Nt = N! emt Assume that the fishing mortality is a function of only effort and is given by Nt = N! eqft As instantaneous rates these can be easily combined into a single model Nt = N! eqft +mt where both “q" and “m" would be negative numbers (for mortality) Very often the time unit will be 1 year so we often see this simplified as Nt = N! eqf+m The total fishing mortality (qf) is also often represented as F and the total natural mortality as M. where; F = qf M = mt then Nt = N! eF+M or Nt = N! e(F+M)t This can be expressed as a simplified total mortality where; Z = F+M or Z> = qf> + m> or Zt = qf> + mt then Nt = N! eZ or Nt = N ! e Z > EXST7025 : Biological Population Statistics Mortality James Geaghan Page 1 Cohort of Flies (S. nudiseta). From Poole, RW. 1974. An Introduction to Quantitative Ecology. McGraw-Hill, NY. Days Number log(number) Predicted Y Residuals Standardized exp(Pred) 0 100 4.61 4.71 -0.1 -0.24 110.65 2.5 75 4.32 4.64 -0.32 -0.74 103.11 7.5 64 4.16 4.49 -0.34 -0.79 89.53 12.5 64 4.16 4.35 -0.19 -0.46 77.74 17.5 64 4.16 4.21 -0.05 -0.12 67.5 22.5 64 4.16 4.07 0.09 0.21 58.61 27.5 48 3.87 3.93 -0.06 -0.14 50.89 32.5 46 3.83 3.79 0.04 0.09 44.19 37.5 44 3.78 3.65 0.14 0.32 38.37 42.5 42 3.74 3.51 0.23 0.54 33.31 47.5 34 3.53 3.36 0.16 0.38 28.93 52.5 34 3.53 3.22 0.3 0.71 25.12 57.5 32 3.47 3.08 0.38 0.9 21.81 62.5 28 3.33 2.94 0.39 0.92 18.94 67.5 20 3 2.8 0.2 0.46 16.44 72.5 20 3 2.66 0.34 0.79 14.28 77.5 18 2.89 2.52 0.37 0.87 12.4 82.5 14 2.64 2.38 0.26 0.62 10.76 87.5 6 1.79 2.23 -0.44 -1.04 9.35 92.5 2 0.69 2.09 -1.4 -3.28 8.11 Regression Statistics (from Excel) Multiple R 0.89 R Square 0.8 Adjusted R2 0.79 Standard Error 0.43 Observations 20 Analysis of Variance df Sum of Squares Mean Square F Significance F Regression 1 13.08 13.08 71.64 1.08803E­07 Residual 18 3.29 0.18 Total 19 16.36 Coefficients Standard Error Intercept anitlog Intercept x1 t Statistic P­value Lower 95% Upper 95% 110.65 75.94 160.77 4.71 0.18 26.39 1.9584E­16 4.33 5.08 ­0.03 0 ­8.46 7.18267E­08 ­0.04 ­0.02 EXST7025 : Biological Population Statistics Mortality James Geaghan Page 2 120 Mortality of cohort of flies Number 100 80 60 40 92.5 87.5 82.5 77.5 72.5 67.5 62.5 57.5 52.5 47.5 42.5 37.5 32.5 27.5 22.5 17.5 12.5 7.5 0 0 2.5 20 Days 0.40 Residual 0.20 0.00 -0.20 -0.40 -0.60 -0.80 -1.00 -1.20 -1.40 -1.60 0 10 20 30 40 50 60 70 80 90 100 Days ...
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This note was uploaded on 12/29/2011 for the course EXST 7025 taught by Professor Geaghan,j during the Spring '08 term at LSU.

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