show organisms` mortality and reproduction as a function of age examination of a population during one segment (time interval) - segment = group of individuals of different cohorts - designed for long-lived organisms ASSUMPTIONS: - birth-rate and survival-rate are constant over time - population does not grow DRAWBACKS: confuses age-specific changes in e.g. mortality with temporal variation Static (vertical) life-tables Cervus elaphus Sx- number of survivors of a given age Dx- number of dead lx- standardised number of survivors qx- age specific mortality Lowe (1969) 0S S l x x = x x x S D q = x Sx Dx lx px qx mx 1 129 15 1.000 0.884 0.116 0.000 2 114 1 0.884 0.991 0.009 0.000 3 113 32 0.876 0.717 0.283 0.310 4 81 3 0.628 0.963 0.037 0.280 5 78 19 0.605 0.756 0.244 0.300 6 59 -6 0.457 1.102 -0.102 0.400 7 65 10 0.504 0.846 0.154 0.480 8 55 30 0.426 0.455 0.545 0.360 9 25 16 0.194 0.360 0.640 0.450 10 9 1 0.070 0.889 0.111 0.290 11 8 1 0.062 0.875 0.125 0.280 12 7 5 0.054 0.286 0.714 0.290 13 2 1 0.016 0.500 0.500 0.280 14 1 -3 0.008 4.000 -3.000 0.280 15 4 2 0.031 0.500 0.500 0.290 16 2 2 0.016 0.000 1.000 0.280 examination of a population in a cohort = a group of individuals born at the same period followed from birth to death provide reliable information designed for short-lived organisms only females are included Cohort (horizontal) life-table Vulpes vulpes x Sx Dx lx px qx mx 0 250 50 1.000 0.800 0.200 0.000 1 200 120 0.800 0.400 0.600 0.000 2 80 50 0.320 0.375 0.625 2.000 3 30 15 0.120 0.500 0.500 2.100 4 15 9 0.060 0.400 0.600 2.300 5 6 6 0.024 0.000 1.000 2.400 6 0 0 0.000 survival and reproduction depend on stage / size rather than age age-distribution is of no interest used for invertebrates (insects, invertebrates) time spent in a stage / size can differ Lymantria dispar Campbell (1981) x Sx Dx lx px qx mx Egg 450 68 1.000 0.849 0.151 0 Larva I 382 67 0.849 0.825 0.175 0 Larva II 315 158 0.700 0.498 0.502 0 Larva III 157 118 0.349 0.248 0.752 0 Larva IV 39 7 0.087 0.821 0.179 0 Larva V 32 9 0.071 0.719 0.281 0 Larva VI 23 1 0.051 0.957 0.043 0 Pre-pupa 22 4 0.049 0.818 0.182 0 Pupa 18 2 0.040 0.889 0.111 0 Adult 16 16 0.036 0.000 1.000 185 display change in survival by plotting ln(lx) against age (x) logarithmic transformation allows to compare survival based on different population size sheep mortality increases with age survivorship of lapwing (Vanellus) is independent of age Pearls (1928) classified hypothetical age-specific mortality: Type I .. mortality is concentrated at the end of life span (humans) Type II .. mortality (qx) is constant over age (seeds), Type III .. mortality is highest in the beginning of life (invertebrates, fish, reptiles) ln(Survivorship) 0 Type I Type II Type III 1 Time fecundity - potential number of offspring fertility - real number of offspring semelparous .. reproducing once a life iteroparous .. reproducing several times during life birth pulse .. discrete reproduction (seasonal reproduction) birth flow .. continuous reproduction Numberofoffsprings 0 Time reproductivepre-reproductive post-reproductive 0 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 100 120 140 Time [Days] Fecundity Triaeris stenapsis Geospiza scandens number.ofbirths/individual 0 0.4 age 16 Cervus elaphus Odocoileus numberofbirths/individual 0 6age 0.8 model of Leslie (1945) uses parameters (survival and fecundity) from life-tables where populations are composed of individuals of different age, stage or size with specific births and deaths used for modelling of density-independent processes (exponential growth) Nx,t .. no. of organisms in age x and time t Gx .. probability of persistence in the same size/stage ANN tt =+1 number of individuals in the first age class number of individuals in the remaining age classes combined into one matrix formula: = + ++== n x ttxtxt FNFNFNN 0 1,10,0,1,0 ... xtxtx pNN ,1,1 =++ N1 N2 N3 N4 Age-structured p12 p23 p34 m4 m3 m2 m1 = × + + + + 1,3 1,2 1,1 1,0 ,3 ,2 ,1 ,0 34 23 12 4321 000 000 000 t t t t t t t t N N N N N N N N p p p FFFF each column in A specifies fate of an organism in a specific age: 3rd column: organism in age 2 produces F2 offspring and goes to age 3 with probability p23 A is always a square matrix Nt is always one column matrix = a vector transition matrix A age distribution vectors Nt fertilities (F) and survivals (p) depend on whether population has discrete or continuous reproduction - for populations with discrete pulses post-reproductive survivals and fertilities are - for populations with continuous reproduction post-reproductive survivals and fertilities are x x x S S p 1+ = xxx mpF = + + - + xx xx x SS SS p 1 1 ( )( ) 4 1 11 ++ = xxx x mpmS F Egg Larva Pupa Imago Stage-structured p2 p3p1 m4 000 000 000 000 34 23 12 4 p p p m only imagoes reproduce thus m1,2,3 = 0 no imago survives to another reproduction period: p4 = 0 Size-structured Tiny Small Medium Large p12 p23 p34 m4 m3 G11 G22 G33 G44 4434 3323 2212 43211 00 00 00 Gp Gp Gp FFFG model of Lefkovitch (1965) uses 3 parameters (mortality, fecundity and persistence) F1 = 0 m2 addition / subtraction multiplication by a vector by a scalar Matrix operations determinant eigenvalue () 1 = 2.41 2 = -0.41a acbb 2 42 2,1 -- = = + 1510 73 85 41 75 32 =× 2115 96 3 75 32 = ×+× ×+× = × 55 23 5745 5342 5 4 75 32 23472 74 32 =×-×= 12)425.0()0()2( 025.0 42 025.0 42 2 --=×--×-= - - = t 0t ANN = ANN 12 tt = ANN 23 tt = ttt NAAANN 2 12 ==+ parameters are constant over time and independent of population density follows constant exponential growth after initial damped oscillations Population density of the true bugs Coreus marginatus was recorded for 10 years. Here are the densities: 160, 172, 188, 154, 176, 185, 168, 194, 170, 169 Does population increase or decrease? What is the average population growth (R)? Project population for another 10 years using R and N0 = 90. Simulate population growth for the next 20 years using observed finite-growth rates. bug<-c(160, 172, 188, 154, 176, 185, 168, 194, 170, 169) plot(bug,type="b") lambda<-bug[-1]/bug[-10] lambda plot(lambda) R<-prod(lambda)^0.1 R time<-1:10 Nt<-90*R^time plot(time,Nt,type="b") sim<-sample(lambda,20,replace=T) years<-20 N<-numeric(years+1) N[1]<-100 for(t in 1:years) N[t+1]<-{ N[t]*sim[t]} plot(0:years,N,type="b") Population density of the mite Acarus siro was recorded every 3 days during 28 days. The following densities were found: 165, 145, 139, 125, 105, 101, 88, 81, 73, 69 What is the intrinsic rate of increase (r) and what was the initial density ? How long it takes for a population to decrease to half size? Project population growth for another 5 weeks using estimated r and N0 = 69. What would be the estimated rate if you know the initial and final density?