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Death in the Sea Understanding Mortality Rainer Froese IFM-GEOMAR (SS 2008) What is Natural Mortality? Proportion of fishes dying from natural causes, such as: • Predation • Disease • Accidents • Old age The M Equation Instantaneous rate of mortality M: Dt / N t = M Where t is the age in years Dt is the number of deaths at age t Nt is the population size at age t The M Equation Probability of survival (lt): lt = e –M t Where M is the instantaneous rate of natural mortality t is the age in years lt ranges from 1.0 at birth to 0.0x at maximum age The M Equation Number of survivors N to age t : Nt = N0 e –M (t) Where N0 is the number of specimens at start age t=0 Nt is the number of specimens at age t M = 0.2 1200 Nt = Nts * exp(-M*(t - ts)) Cohort numbers 1000 800 600 400 200 0 0 5 10 15 Cohort age (years) 20 25 Constant Value of M for Adults (in species with indeterminate growth: fishes, reptiles, invertebrates, ..) • M is typically higher for larvae, juveniles, and very old specimens, but reasonably constant during adult life • This stems from a balance between intrinsic and extrinsic mortality: – Intrinsic mortality increases with age due to wear and tear and accumulation of harmful mutations acting late in life – Extrinsic mortality decreases with size and experience M is Death Rate in Unfished Population In an unfished, stable population – The number of spawners dying per year must equal the number of ‘new’ spawners per year – Every spawner, when it dies, is replaced by one new spawner, the life-time reproductive rate is 1/1 = 1 – If the average duration of reproductive life dr is several years, the annual reproductive rate is α = 1 / dr The P/B ratio is M (Allen 1971) In an unfished, stable population – Biomass B gained by production P must equal biomass lost due to mortality – M is the instantaneous loss in numbers relative to the initial number: Nlost / N = M – If we assume a mean weight per individual, then we have biomass: Blost / B = M – If Blost = P then P / B = M Pauly’s 1980 Equation log M = -0.0066 – 0.279 log L∞ + 0.6543 log K + 0.4634 log T Where L∞ and K are parameters of the von Bertalanffy growth function and T is the mean annual surface temperature in °C Reference: Pauly, D. 1980. On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. J. Cons. Int. Explor. Mer. 39(2):175-192. Jensen’s 1996 Equation M = 1.5 K Where K is a parameter of the von Bertalanffy growth function Reference: Jensen, A.L. 1996. Beverton and Holt life history invariants result from optimal trade-off of reproduction and survival. Canadian Journal of Fisheries and Aquatic Sciences:53:820-822 M = 1.5 K 100 1:1 M observed 10 1 0.1 0.01 0.01 0.1 1 10 100 M = 1.5 K Plot of observed natural mortality M versus estimates from growth coefficient K with M = 1.5 K, for 272 populations of 181 species of fishes. The 1:1 line where observations equal estimates is shown. Robust regression analysis of log observed M versus log(1.5 K) with intercept removed explained 82% of the variance with a slope not significantly different from unity (slope = 0.977, 95% CL = 0.923 – 1.03, n = 272, r2 = 0.8230). Data from FishBase 11/2006 [File: M_Data.xls] Froese’s (in prep.) Equation L∞ = C -0.45 M This is the L∞ – M trade-off, where L∞ is the asymptotic length of the von Bertalanffy growth function and C is an indicator of body plan, environmental tolerance and behavior, i.e., traits that are relatively constant in a given species. If C is known e.g. from other populations of a species, M corresponding to a certain L∞ can be obtained from M = (L∞ / C)-2.2 Hoenig’s 1984 Equation ln M = 1.44 – 0.984 * ln tmax Where tmax is the longevity or maximum age reported for a population Reference: Hoenig, J.M., 1984. Empirical use of longevity data to estimate mortality rates. Fish. Bull. (US) 81(4). Froese’s (in prep.) Equation M = 4.5 / tmax tmax = 4.5 / M Charnov’s 1993 Equation E=1/M Where E is the average life expectancy of adults Reference: Charnov, E.L. 1993. Life history invariants: some explorations of symmetry in evolutionary ecology. Oxford University Press, Oxford, 167 p. Froese’s (in prep.) Equation dr = 1 / M Where dr is the mean duration of the reproductive phase If mortality is doubled then reproduction is shortened by half Life History Summary Fishing Kills Fish Z=M+F Where Z = total mortality rate F = mortality caused my fishing Size at First Capture Matters 140 100 Impact on cohort biomass 120 80 60 80 60 40 Catch of 40 t 40 20 20 0 0 20 40 60 80 Length (cm) 100 120 0 140 Catch (% Bt) Cohort Biomas (t) 100 Size at First Capture Matters Impact on Size-Structure (F=M) 120 Cohort Biomas (t) 100 Lc = 80 cm 80 60 40 Lc = 24 cm 20 0 0 20 40 60 Length (cm) 80 100 120 What You Need to Know Nt = N0 e –M (t) Bt = Nt * Wt Z=M+F Thank You