1. The Basics Populations rarely have a constant size Intrinsic Factors BIRTH IMMIGRATION DEATH EMIGRATION Extrinsic factors Predation Weather N t+1 = N t + B + D + E + I Populations grow IF (B + I) > (D + E) Populations shrink IF (D + E) > (B + I) Diagrammatic Life-Tables…. What is a population? Assume E = I

2. Adults N t Adults N t+1 Seeds N t.f Seedlings N t.f.g f g e p BIRTH SURVIVAL N t+1 = (N t.p) + (N t.f.g.e) Adults M F 2.3 Adults M F 2.5 Pods 18.25 Eggs 200.75 Instar I 15.86 Instar II 11.42 Instar III 8.91 Instar IV 6.77 P=0 7.3 11 0.079 0.72 0.78 0.76 0.69 t = 0 t = 1 t = 0 t = 1

3. Adults M F 5 Adults M F 8.2 Eggs 50 1 mo Nestlings 42 3 mo Fledglings 29.8 10 0.71 0.1 0.5 0.84 Overlapping Generations: Discrete Breeding a0a0 a1a1 a2a2 a3a3 anan t1t1 a0a0 a1a1 a2a2 a3a3 anan t3t3 a0a0 a1a1 a2a2 a3a3 anan t2t2 p 01 p 12 p 23 Birth NB: Different age groups have different probabilities of surviving from one time interval to the next, and different age groups produce different numbers of offspring t1t1 t2t2 p 01 p 12 p 23 Birth NB – ALL Adults or Females?

4. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0

5. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed

6. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed q age specific mortality, calculated as d x / l x : data cannot be summed

7. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed q age specific mortality, calculated as d x / l x : data cannot be summed p age specific survivorship, calculated as 1 - q x (or a x+1 / a x ): cannot be summed

8. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed q age specific mortality, calculated as d x / l x : data cannot be summed p age specific survivorship, calculated as 1 - q x (or a x+1 / a x ): cannot be summed k killing power – reflects stage specific mortality and can be summed K

9. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed q age specific mortality, calculated as d x / l x : data cannot be summed p age specific survivorship, calculated as 1 - q x (or a x+1 / a x ): cannot be summed k killing power – reflects stage specific mortality and can be summed K F Total number offspring per age/stage class

10. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed q age specific mortality, calculated as d x / l x : data cannot be summed p age specific survivorship, calculated as 1 - q x (or a x+1 / a x ): cannot be summed k killing power – reflects stage specific mortality and can be summed K F Total number offspring per age/stage class m mean number offspring per individual a, F x / a x

11. Conventional Life-Tables Best studied from Cohort – Define Subscript x refers to age/stage class a refers to actual numbers counted – case specific l refers to proportions wrt t 0 – allows comparisons between cases: l x = a x / a 0 d refers to standardised mortality, calculated as l x – l x+1 : data can be summed q age specific mortality, calculated as d x / l x : data cannot be summed p age specific survivorship, calculated as 1 - q x (or a x+1 / a x ): cannot be summed k killing power – reflects stage specific mortality and can be summed K F Total number offspring per age/stage class m mean number offspring per individual a, F x / a x lm number of offspring per original individual REAL DATA

12. Σ l x m x = R 0 = ΣF x / a 0 = Basic Reproductive rate R 0 = mean number of offspring produced per original individual by the end of the cohort It indicates the mean number of offspring produced (on average) by an individual over the course of its life, AND, in the case of species with non-overlapping generations, it is also the multiplication factor that converts an original population size into a new population size – ONE GENERATION later Σ l x m x = R 0 = 0.51 N 0. R 0 = 44000. 0.51 = 22400 = N T Generation time R 0 is a predictor that can be used to project populations into the future – in terms of generations

13. For populations with overlapping generations, we must tackle the problem in a roundabout manner Fundamental Reproductive Rate (R) = N t+1 / N t IF N t = 10, N t+1 = 20: R = 20 / 10 = 2 Populations will increase in size if R >1 Populations will decrease in size if R < 1 Populations will remain the same size if R = 1 R combines birth of new individuals with the survival of existing individuals Population size at t+1 = N t.R Population size at t+2 = N t.R.R Population size at t+3 = N t.R.R.R N t = N 0.R t R 0 ONLY reflects the birth of new individuals (survival = 0)

14. N t = N 0.R t Overlapping generations N T = N 0.R 0 Non-overlapping generations lnR = r = lnR 0 / T = intrinsic rate of natural increase N T = N 0.R T IF t = T, then R 0 = R T lnR 0 = T.lnR Can now link R 0 and R: T = Σxl x m x / R 0 T can be calculated from the cohort life tables X = age class

16. Other statistics that you can calculate from basic life tables Life Expectancy – average length of time that an individual of age x can expect to live L average number of surviving individuals in consecutive stage/age classes: (a x + a x+1 ) / 2 T cumulative L: Σ L x i n e life expectancy: T x / a x NB. Units of e must be the same as those of x Thus if x is measured in intervals of 3 months, then e must be multiplied by 3 to give life expectancy in terms of months Can also calculate T and L using l x values T and L are confusing – call them Bob (L) and Margaret (T)

17. A note on finite and instantaneous rates The values of p, q hitherto collected are FINITE rates: units of time those of x expressed in the life-tables (months, days, three-months etc) They have limited value in comparisons unless same units used [Adjusted FINITE] = [Observed FINITE] ts/to Where ts = Standardised time interval (e.g. 30 days, 1 day, 365 days, 12 months etc) to = Observed time interval To convert FINITE rates at one scale to (adjusted) finite rates at another: e.g. convert annual survival (p) = 0.5, to monthly survival Adjusted = Observed ts/to = 0.5 1/12 = 0.5 0.083 = 0.944 e.g. convert daily survival (p) = 0.99, to annual survival Adjusted = Observed ts/to = 0.99 365/1 = 0.99 365 = 0.0255

18. INSTANTANEOUS MORTALITY rates = Log e (FINITE SURVIVAL rates) ALWAYS negative Finite Mortality Rate = 1 – Finite Survival rate Finite Mortality Rate = 1.0 – e Instantaneous Mortality Rate MUST SPECIFY TIME UNITS

19. Projecting Populations into the future: Basic Model Building KEY PIECES of INFORMATION: p and m Rearrange Life Table WHY? Dealing first with survivorship Copy Formula Down and Across Table quickly fills up with 0s

20. 54256.42 Adding Fecundity Copy Down

21. NB – R eventually stabilises R = (N t+1 ) / N t Converting NUMBERS of each age class to PROPORTIONS (of the TOTAL) generates the age- structure of the population. NOTE, when R stabilises, so too does the age-structure, and this is known as the stable-age distribution of the population, and proportions represent TERMS (c x )

22. Because the terms of the stable age distribution are fixed at constant R, we can partition r (lnR) into birth and death per individual N t+1 = N t.(Survival Rate) + N t. (Survival Rate).(Birth Rate) N t+1 = N t.(Survival Rate).(1 + Birth Rate) No Births = No a 0 Calculating Birth Rate First Divide by No Individuals producing them: Σa x 1 n e.g. B = 35648277 / (1685933 + 80401 + 0) = 20.1821

23. Calculating Survival Rate Σa x 1 n Survivors: Total number of individuals at time t, older than 0: Survival Rate: No Survivors at time t, divided by total population size at time t-1 e.g. Survival Rate (t 4 ) = No survivors (t 4 ) / total population size (t 3 ) S = 348069 / 1452894 = 0.2396

24. N t+1 = N t.(Survival Rate).(1 + Birth Rate) N t+1 / N t = R = e r = (Survival Rate).(1 + Birth Rate) B = 20.1821 S = 0.2396 At Stable-Age R = 0.2396 x (20.1821 + 1) = 5.07 Annual Survival Rate for an individual in the population is in the range p 0, p 1, p 2, but NOT the average Annual Birth Rate for an individual in the population is between m 1 and m 2, but NOT the average NOTE

25. Reproductive Value (v x ) – a measure of present and future contributions by the different age classes of a population to R v x is calculated as the number of offspring produced by an individual age x and older, divided by the number of individuals age x right now v x * = [(v x+1.l x+1 ) / (l x.R)] v x * = residual reproductive value v x = m x + v x * This expression can ONLY be used to calculate v x * IF the time intervals used in the life-table are equal. To calculate v x * work backwards in the life-table, because v x * = 0 in the last year of life Copy upwards

26. STATIC LIFE TABLES………