Estimation of Mortality

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Presentation transcript:

Estimation of Mortality Recruitment Natural Mortality Population Numbers Fishing Mortality Immigration Emigration

Recall Discrete Population Growth Model Suppose that … Nt+1 = (1+r)Nt where r = b-d+i-e Suppose that … … population is closed … following the same group of fish Thus, r = -d but d is usually replaced with A A is the annual mortality rate Solve for A Solve Nt+1 = (1-A)Nt for A Catch Curves

Mortality Rate Concept For example, N1 = 1000 and N2 = 850. What is the mortality rate? What is the survival rate? S is an annual survival rate Note that A+S = 1 Such that S=1-A or A=1-S Catch Curves

Instantaneous Mortality Rate (Z) Similarly examine continuous model … r = -d but replace d with Z such that Nt+1 = Nte-Z solve for Z thus, Z is an instantaneous mortality rate Note that S=e-Z and A = 1-e-Z Catch Curves

Two Problems Population sizes are not usually “seen.” Z can be computed from CPEs Recall that Ct = qftNt Algebraically show that Z=log(CPEt)-log(CPEt+1) Catches or CPEs are subject to variability Catches are samples; Z is, thus, a statistic. If a cohort is followed over time, individual estimates of Z can be made and averaged. Solve catch equation for Nt, plug into continuous popn growth model, solve for z Catch Curves

Example Calculations Calculate Z from each time step of … IDEAL REAL t Nt Ct Ct* 0 1000 200 211 1 800 160 159 2 640 128 126 3 512 102 104 4 410 82 81 5 328 66 64 Calculate Z from each time step of … population sizes. idealistic catches. realistic catches. composite (average) of realistic catches. Catch Curves

Catch Curve Longitudinal Cross-sectional Catch-at-age for a single cohort of fish. Cross-sectional Catch-at-age in a single year (across many cohorts of fish). Catch Curves

Longitudinal vs. Cross-Sectional Catch-at-age across several capture years. Capture Year Age 2009 2010 2011 2012 2013 2014 2015 2016 0 200 200 200 200 200 200 200 200 1 160 160 160 160 160 160 160 160 2 128 128 128 128 128 128 128 128 3 102 102 102 102 102 102 102 102 4 82 82 82 82 82 82 82 82 5 66 66 66 66 66 66 66 66 What is the cross-sectional catch-at-age for 2012? What is the longitudinal catch-at-age for the 2010 year-class? Longitudinal=cross-sectional if Z and N0 are constant across time and cohorts. Catch Curves

Catch Curve Model Recall: CPEt = qNt and Nt = N0e-Zt Substitute second into first … CPEt = qN0e-Zt Can this be linearized? What is estimate of Z? 1 2 3 4 5 80 120 160 200 Age / Time CPE Catch Curves

Catch Curve Characteristics 2 4 6 8 10 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Age / Time log(CPE) Asc Dome Descending -Z 1 Fit regression of log(CPE) on age only for ages on descending limb. Catch Curves

Catch Curve Assumptions Population closed to immigration and emigration. Z is constant. q is constant “Sample” is unbiased regarding any age-group (i.e., be careful of selective gears) Accurate ages Follow a cohort (if longitudinal CC used) Recruitment on descending limb is constant (if cross-sectional CC used) Catch Curves