Age structured populations Alfred James Lotka (1880-1949) Vito Volterra (1860-1940)

First steps in life tables Mortality rate Survival rate Fecundity Age N f d l 1000 0.15 0.85 10 600 0.09 0.91 20 700 0.1 30 595 1.2 0.25 0.75 40 446 0.6 0.35 0.65 50 290 0.3 0.38 0.62 60 180 0.05 0.42 0.58 70 200 0.55 0.45 80 90 0.95 100 3 1 N0 is the number of newborns. N is the number of females per age cohort. Fecundity f is the average number of offspring per female. d is the mortality rate per cohort. l is the fraction of survivors per cohort.

The pivotal age is the averge age per age cohort class Age t N D d l Number of deaths at each age classage The pivotal age is the averge age per age cohort class Pivotal age Age t N D d l 1000 10 5 850 150 0.150 0.850 20 15 835 0.018 0.982 30 25 812 23 0.028 0.972 40 35 777 0.043 0.957 50 45 737 0.051 0.949 60 55 661 76 0.103 0.897 70 65 551 110 0.166 0.834 80 75 270 281 0.510 0.490 90 85 167 103 0.381 0.619 100 95 7 160 0.958 0.042 105 1 6 0.857 0.143 120 115 1.000 0.000 The basic information needed is the total number of deaths per age cohort. survival mortality 𝑑 𝑥 = 𝐷 𝑥 𝑁 𝑥 𝑙 𝑥 =1− 𝑑 𝑥 = 𝑁 𝑥 𝑁 𝑥−1 Mortality rate survival rate

First steps in life tables Death rate Survival rate Fecundity Age N0 f d l N1 1000 0.15 0.85 1148 10 600 0.09 0.91 850 20 700 0.1 546 30 595 1.2 0.25 0.75 40 446 0.6 0.35 0.65 50 290 0.3 0.38 0.62 60 180 0.05 0.42 0.58 70 200 0.55 0.45 104 80 90 0.95 23 100 3 1 =E27*H27 N1(0) 70 714 268 87 9 =+E28*F28 Population size of each cohort after reproduction Initial age distribution Age distribution of the next generation 𝑁 1 0 = 𝑁 𝑖 × 𝑓 𝑖 𝑁 1 10 = 𝑁 0 (0)×𝑙(0)

If the population is age structured and contains k age classes we get 10 20 30 40 50 60 70 80 90 100 0.1 1.2 0.6 0.3 0.05 0.85 0.91 0.75 0.65 0.62 0.58 0.45 0.25 N0 N1 1000 1148 600 850 700 546 595 446 290 180 200 104 90 50 23 3 Fecundities Survival rates Leslie matrix N0(0) = 1000 N0(1) = 1148 595*0.75=446 The mutiplication of the abundance vector with each row of the Leslie matrix gives the abundance of the next generation.

We have w-1 age classes, w is the maximum age of an individual. Leslie matrix We have w-1 age classes, w is the maximum age of an individual. L is a square matrix. Numbers per age class at time t+1 are the dot product of the Leslie matrix with the abundance vector N at time t The Leslie model is a linear approach. It assumes stable fecundity and mortality rates

At the long run population size increases. Going Excel Age 10 20 30 40 50 60 70 80 90 100 0.1 1.2 0.6 0.3 0.05 0.85 0.91 0.75 0.65 0.62 0.58 0.45 0.25 N0 N1 N2 N3 N4 N5 N6 1000 1148 1132 998 1183 1366 1413 600 850 976 963 848 1005 1161 700 546 774 888 876 772 915 595 464 657 755 744 656 446 348 493 566 558 290 226 321 368 180 140 199 200 104 81 90 47 50 23 12 3 1 Demographic low At the long run population size increases. Diagonal waves in abundances occur. The first age cohort increases fastest.

The effect of age in reproduction 10 20 30 40 50 60 70 80 90 100 0.1 1.2 0.6 0.3 0.05 0.85 0.91 0.75 0.65 0.62 0.58 0.45 0.25 Age 10 20 30 40 50 60 70 80 90 100 0.05 0.3 0.6 1.2 0.1 0.85 0.91 0.75 0.65 0.62 0.58 0.45 0.25 Reproduction in early age contributes more to population size than later reproduction. This is caused by the higher number of females in earlier cohorts.

The effect of the initial age composition disappears over time Age composition approaches an equilibrium although the whole population might go extinct. Population growth or decline is often exponential Age 10 20 30 40 50 60 70 80 90 100 0.1 1.2 0.6 0.3 0.05 0.25 0.91 0.85 0.75 0.65 0.62 0.58 0.45 High early death rates cause fast population extinction and would need high fecundities for population survival

This vector is the eigenvector U of the matrix. Does the Leslie approach predict a stationary point where population abundances doesn’t change any more? 𝑵 𝑡+1 = 𝑅𝑵 𝑡 𝑵 𝑡+1 = 𝑳𝑵 𝑡 = 𝑵 𝑡 𝑵 𝑡+1 = 𝑅 𝑡 𝑵 0 𝑵 𝑡+1 = 𝑳 𝑡 𝑵 0 We’re looking for the stable state vector that doesn’t change when multiplied with the Leslie matrix. This vector is the eigenvector U of the matrix. Eigenvectors are only defined for square matrices. 𝑳𝑼=𝜆𝑼 𝑵 𝑡+1 = 𝑳𝑵 𝑡 =𝜆 𝑵 𝑡 =𝑅 𝑵 𝑡 Important properties: Eventually all age classes grow or shrink at the same rate Initial growth depends on the age structure Early reproduction contributes more to population growth than late reproduction The largest eigenvalue l of a Leslie matrix denotes the long-term average net reproduction rate. The right (dominant) eigenvector contains the stable state age distribution.

Eggs Larva 1 Larva 2 Larva 3 Imago Eggs Larva 1 Larva 2 Larva 3 Imago Leslie matrices in insect populatons Age Eggs Larva 1 Larva 2 Larva 3 Imago 2000 0.25 0.15 0.1 Largest eigenvalue r = l = 1.02 2000 female eggs per individual are cause a steady population increase. This relates to 4000 eggs when including males. Leslie matrices deal with effective populations sizes. Age Eggs Larva 1 Larva 2 Larva 3 Imago 200 0.25 0.15 0.1 Largest eigenvalue r = l = 0.65 The population steadily declines. 𝑵 1 = 𝑳𝑵 0 N0 Eggs Larva 1 Larva 2 Larva 3 Imago 100000 11250 1265.6 25000 2813 2812.5 3750 421.9 421.875 421.88 563 562.5 63.28 63.2813 63.281 56 56.25 6.32813 6.3281 The diagonal matrix elements show how many individuals survive.

Eggs Larva 1 Larva 2 Larva 3 Imago U U Stable age distribution Age Eggs Larva 1 Larva 2 Larva 3 Imago 2000 0.25 0.15 0.1 𝑵 𝑡+1 = 𝑳𝑵 𝑡 =𝜆 𝑵 𝑡 The largest eigenvalue l of a Leslie matrix denotes the long-term net population growth rate R. The right (dominant) eigenvector contains the stable state age distribution. U 0.970859 0.237064 0.034732 0.005088 0.000497 U Nstable 0.970859 0.777782 0.237064 0.189919 0.034732 0.027825 0.005088 0.004077 0.000497 0.000398 Sum 1.248241 1 Age Eggs Larva 1 Larva 2 Larva 3 Imago For the population to survive the number of first instars has to be 0.189919/0.000398 = 477 time larger than the number of imagines. U = l = 1.02 Stable age class distribution

Eggs Larva 1 Larva 2 Larva 3 Imago U U Remaining in the same age class Age Eggs Larva 1 Larva 2 Larva 3 Imago 0.10 2000 0.25 0.15 0.05 0.1 0.5 The probability that an egg survives and remaines in the egg state is 0.10 The probability that an imago survives and reproduces in the next generation is 0.5. This is the case in biannual insects (for instance some Carabus) 𝑵 𝑡+1 = 𝑳𝑵 𝑡 =𝜆 𝑵 𝑡 Largest eigenvalue R = l = 1.21 l = 1.21 l = 1.02 U 0.972869 0.229415 0.029662 0.003835 0.00054 U Nstable 0.972869 0.786907 0.229415 0.185563 0.029662 0.023992 0.003835 0.003102 0.00054 0.000437 Sum 1.236321 1 Age Eggs Larva 1 Larva 2 Larva 3 Imago Nstable 0.777782 0.189919 0.027825 0.004077 0.000398 U= Without staying the same Stable age class distribution

High mortality, high fecundity r strategist species Sensitivity analysis Age Eggs Larva 1 Larva 2 Larva 3 Imago 2000 0.25 0.15 0.1 Age Eggs Larva 1 Larva 2 Larva 3 Imago 2.5 0.95 0.91 0.93 l = 1.02 l = 1.14 High mortality, high fecundity r strategist species Low mortality, low fecundity K strategist species l > 1 → effective population size increases How robust is l with respect to changes in survival and fecundity rates? Age Eggs Larva 1 Larva 2 Larva 3 Imago 1.4 0.95 0.91 0.93 l = 1.01 The lowest possible fecundity is 1.4 female eggs per female.

Increasing mortality rates until the population stops increasing Sensitivity analysis Age Eggs Larva 1 Larva 2 Larva 3 Imago 2.5 0.95 0.91 0.93 l = 1.14 Increasing mortality rates until the population stops increasing Age Eggs Larva 1 Larva 2 Larva 3 Imago 2.5 0.86 0.819 0.837 0.855 l = 1.05 Mortality rates might be 10% higher to remain effective population sizes still increasing.

Average number alive in a cohort Cumulative number alive in a cohort Survivorship tables Number of death Death rate Average number alive in a cohort Cumulative number alive in a cohort Survival rate Average life expectation Age N 1000 10 850 20 835 30 812 40 777 50 737 60 661 70 551 80 270 90 167 100 7 110 1 120 D 150 15 23 35 40 76 110 281 103 160 6 1 l 0.85 0.98 0.97 0.96 0.95 0.90 0.83 0.49 0.62 0.04 0.14 0.00 +H25/H24 d 0.15 0.02 0.03 0.04 0.05 0.10 0.17 0.51 0.38 0.96 0.86 1.00 +J25/H24 L 925 843 824 795 757 699 606 411 219 87 4 1   +(H25+H24)/2 SL 6168 5243 4401 3577 2783 2026 1327 721 310 92 5 1   +SUMA(L$24:L24) e 61.7 52.7 44.1 35.8 27.5 20.1 13.1 11.5 5.5 6.4 5.0   +M24/H24*$G$14 𝐿 𝑥 = 𝑁 𝑥 +𝑁(𝑥+1) 2 Σ𝐿= 𝑥 𝑚𝑎𝑥 𝐿(𝑥) 𝑒 𝑥 = Σ 𝐿 𝑥 𝑁 𝑥 𝑘 k = length of cohort (10 years)

The female life table of Polish women 2012 (GUS 2013) Age N D l d L SL e 100000 99787 8095890 80.9589 1 99574 426 0.99574 0.00426 99563 7996103 80.30312 2 99552 21 0.99978 0.00021 99544.5 7896540 79.32076 3 99537 15 0.99985 0.00015 99531 7796996 78.33264 4 99525 11 0.99988 0.00011 99520 7697465 77.34202 5 99515 10 0.99990 0.00010 99510.5 7597945 76.34974 6 99506 0.99991 99501 7498434 75.3566 7 99496 9 0.00009 99491.5 7398933 74.36413 8 99487 99483 7299442 73.37081 99479 0.99992 99474.5 7199959 72.37667 99470 0.00008 99465.5 7100484 71.38317 99461 99456.5 7001019 70.38958 12 99452 99446 6901562 69.39591 13 99440 99433.5 6802116 68.40422 14 99427 0.99987 0.00013 99419.5 6702683 67.4131 99412 16 0.00016 99402.5 6603263 66.4232 99393 18 0.99981 0.00018 99383 6503861 65.4358 17 99373 0.99980 99361.5 6404478 64.44887 99350 23 0.99977 0.00023 99338 6305116 63.46367 19 99326 24 0.99976 0.00024 99314 6205778 62.47889 20 99302 99290 6106464 61.49387 Average life expectancy at birth 𝐿 𝑥 = 𝑁 𝑥 +𝑁(𝑥+1) 2 Σ𝐿= 𝑥 𝑚𝑎𝑥 𝐿(𝑥) 𝑒 𝑥 = Σ 𝐿 𝑥 𝑁 𝑥 𝑘 98 4317 1524 0.739086 0.260914 3710 8889.5 2.059185 99 3103 1214 0.718786 0.281214 2634 5179.5 1.669191 100 2165 938 0.697712 0.302288 1814 2545.5 1.175751 >100 1463 702 0.675751 0.324249 731.5 0.5 >120 1

Polish survivorship curve 2012 Type I Type I, high survivorship of young individuals: large mammals, birds Type II, survivorship independent of age, seed banks Type III, low survivorship of young individuals, fish, many insects Type II Type III Polish mortality rates 2012 Newborns New motocycle and car drivers

Average life expectancy at birth in Poland 81 years Women 8 years 72 years Men Average life expectancy at age 60 in Poland 84 years 5 years 78 years

Reproduction life tables Pivotal age Survival rate Number offspring Birth rate R Age t N0 D l 1000 10 5 850 150 0.85 20 15 835 0.98 30 25 812 23 0.97 40 35 777 0.96 50 45 737 0.95 60 55 661 76 0.90 70 65 551 110 0.83 80 75 270 281 0.49 90 85 167 103 0.62 100 95 7 160 0.04 105 1 6 0.14 120 115 0.00 Sum +I25/I24 B b 0.000 20 0.024 515 0.634 342 0.440 59 0.080 2 0.003 =L25/I25 lb 0.000 0.024 0.617 0.421 0.076 0.003 1.140 +M25*K25 lbt 0.0 0.4 15.4 14.7 3.4 0.1 34.096 29.9 +N25*H25 𝐺= 𝑙 𝑖 𝑏 𝑖 𝑡 𝑖 𝑙 𝑖 𝑏 𝑖 The mean generation length is the mean period elapsing between the birth of parents and the birth of offspring. It is the weighted mean of pivotal age weighted by the number of offspring. 𝑅 0 = 𝑅 𝑖 = 𝑙 𝑖 𝑏 𝑖 Net reproduction rate

Reproductive value at age x Species Body mass (kg) Litter size Age of maturity (yr) Life expectancy (yr) Life expectation at maturity (yr) Reproductive value at maturity Generation length (yr) Castor canadensis 18 6.6 2 1.52 2.22 5.63 4.87 Clethreonomys glareolus 0.025 5 0.11 0.16 0.48 7.9 0.33 Peromyscus leucopus 0.02 0.15 0.21 0.28 4.52 0.27 P. maniculatus 3.6 0.23 0.43 5.04 0.35 Sciurus carolinensis 0.6 2.9 1 1.37 2.17 5.95 2.07 Spermophilus armatus 5.3 1.38 1.72 1.78 S. beldingi 0.25 7.4 1.3 5.89 1.56 S. lateralis 0.155 5.2 1.47 2.12 5.08 2.45 S. parrylii 0.7 7.3 1.28 1.71 6.17 1.59 Tamias striatus 0.1 4.2 1.24 1.63 6.84 Tamiasciurus hudsonicus 0.189 4 1.5 4.9 1.95 Ochotona princeps 0.13 2.8 2.33 6.51 Sylvilagus floridanus 1.25 1.48 2.62 1.29 Lutra canadensis 7.2 3 2.88 3.79 5.07 Lynx rufus 7.5 2.48 3.48 2.87 Mephitis mephitis 2.25 6 1.33 1.9 5.71 Taxidea taxus 7.15 1.45 Equus burchelli 270 3.84 7.95 8.74 Aepycerus melampus 44 3.44 4.8 2.42 4.36 Cervus elaphus 175 3.85 1.73 5.7 Connochaetes taurinus 165 4.79 2.56 6.29 Hemitragus jemlahicus 100 3.97 4.71 5.43 Hippopotamus amphibicus 2390 10 7.62 16.4 3.98 19.82 Kobus defassa 200 3.35 5.87 2.94 Ovis canadensis 55 3.81 5.48 2.74 6.52 Phacochoerus aethiopicus 87 1.6 2.82 6.76 4.28 Sus scrofa 85 1.79 1.91 4.82 3.15 Syncerus caffer 490 4.47 2.41 6.98 Loxodonta africana 4000 15 17.9 19.1 2.24 25.8 Life history data and body size Reproductive value at age x 𝑅𝑉 𝑥 = 𝑓 𝑥 + 𝑦=𝑥+1 𝑚𝑎𝑥 𝑙 𝑦 𝑙 𝑥 𝑓 𝑦 𝑙 𝑦 𝑏 𝑦 𝑥−𝑦 Life history data are allometrically related to body size. Data from Millar and Zammuto 1983, Ecology 64: 631

The characteristic life expectancy The Weibull distribution is particularly used in the analysis of life expectancies and mortality rates 𝑓 𝛼,𝛽,𝑡 =𝛼𝛽 𝑡 𝛽−1 𝑒 −𝛼 𝑡 𝛽 𝑓 𝛼,𝛽 𝑑𝑡=𝐹 𝛼,𝛽,𝑡 =1− 𝑒 −𝛼 𝑡 𝛽 a=1 b=0.1 b=0.5 b=1.0 b=2.0 b=3.0

We interpret the time t as the time to death. 𝑓 𝛼,𝛽,𝑡 =𝛼𝛽 𝑡 𝛽−1 𝑒 −𝛼 𝑡 𝛽 We interpret the time t as the time to death. b > 1: Probability of death increases with time b = 1: Probability of death is constant over time b < 1: Probability of death decreases with time 𝛼= 1 𝑇 𝛽 𝑓 𝛽,𝑡 = 𝛽 𝑇 𝑡 𝑇 𝛽−1 𝑒 − 𝑡 𝑇 𝛽 𝑓 𝛽 𝑑𝑡=𝐹 𝛽,𝑡 =1− 𝑒 − 𝑡 𝑇 𝛽 The two parameter Weibull probability density function Characteristic life expectancy T 𝐹 𝛽,𝑡 =1− 𝑒 − 𝑡 𝑇 𝛽 ; t = T 2.2 𝐹 𝛽,𝑡 =1− 𝑒 −1 ≈0.632 The characteristic life expectancy T is the age at which 63.2% of the population already died. F is the cumulative number of deaths.

How to estimate the characteristic life expectancy? 𝐹=1− 𝑒 − 𝑡 𝑇 𝛽 ln⁡(1−𝐹)=− 𝑡 𝑇 𝛽 ln − ln 1−𝐹 =𝛽 ln 𝑡 −𝛽ln⁡(𝑇) Linear function Y = bX + C 𝑇= 𝑒 𝐶 −𝛽 Age t N0 D SD F ln(-ln(1-F)) ln(t) 1000 10 5 630 370 0.37 -0.772 1.609 20 15 420 210 580 0.58 -0.142 2.708 30 25 250 170 750 0.75 0.327 3.219 40 35 110 140 890 0.89 0.792 3.555 50 45 60 940 0.94 1.034 3.807 55 34 26 966 0.966 1.218 4.007 70 65 19 985 0.985 1.435 4.174 80 75 995 0.995 1.667 4.317 90 85 3 2 997 0.997 1.759 4.443 100 95 1 999 0.999 1.933 4.554 105 +U25/S$13 b = 0.95 𝑇= 𝑒 −2.54 −0.95 =14.4 C = -2.54 Type III survivorship curve

The female life table of Polish women 2012 (GUS 2013) Age N D F ln(-ln(1-F)) ln(age) 100000 1 99574 426 0.00426 -5.456 0.000 2 99552 21 0.00447 -5.408 0.693 3 99537 15 0.00462 -5.375 1.099 4 99525 11 0.00473 -5.351 1.386 5 99515 10 0.00483 -5.330 1.609 6 99506 0.00493 -5.310 1.792 7 99496 9 0.00502 -5.292 1.946 8 99487 0.00511 -5.274 2.079 99479 0.0052 -5.256 2.197 99470 0.00528 -5.241 2.303 Mortalities at younger age do not follow a Weibull distribution 86 41622 3983 0.58378 -0.132 4.454 87 37586 4037 0.62415 -0.022 4.466 88 33550 4035 0.6645 0.088 4.477 89 29573 3978 0.70428 0.197 4.489 90 25710 3863 0.74291 0.306 4.500 91 22020 3690 0.77981 0.414 4.511 92 18551 3469 0.8145 0.522 4.522 93 15352 3199 0.84649 0.628 4.533 94 12461 2891 0.8754 0.734 4.543 95 9906 2556 0.90096 0.838 4.554 96 7699 2207 0.92303 0.942 4.564 𝑇= 𝑒 𝐶 −𝛽 T = 86.8 years The characteristic life expectancy of Polish woman in 2012 was 87 years

The female life table of Polish women 2012 (GUS 2013) Maximum mortality 87 Mortality of newborns