Introduction to Population Biology – BDC222 Prof Mark J Gibbons: Rm 4.102, Core 2, Life Sciences Building, UWC Tel :

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

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 Eggs Instar I Instar II Instar III 8.91 Instar IV 6.77 P= t = 0 t = 1 t = 0 t = 1

Adults M F 5 Adults M F 4 Eggs 50 1 mo Nestlings 42 3 mo Fledglings 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?

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

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

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

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

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

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

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

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

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 = = = N T Generation time

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 cannot be used to predict population sizes at one time interval to another, if the populations have overlapping generations – to do that we need to calculate R

N t = N 0.R t Overlapping generations N T = N 0.R 0 Non-overlapping generations 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 – already know R 0 X = age class How do you calculate R from a life-table? - Indirectly lnR = r = lnR 0 / T = intrinsic rate of natural increase ln = natural log – Calculated in MSExcel as =LN(cell address)

T = Σxl x m x / R 0 T can be calculated from the cohort life tables – already know R 0 X = age class lnR = r = lnR 0 / T = intrinsic rate of natural increase ln = natural log – Calculated in MSExcel as =LN(cell address) R calculated from r by raising e (base of natural logs) to power r: =exp(r)

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

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 = = e.g. convert daily survival (p) = 0.99, to annual survival Adjusted = Observed ts/to = /1 = =

INSTANTANEOUS MORTALITY rates = Log e (FINITE SURVIVAL rates) ALWAYS negative Finite Mortality Rate = 1 – Finite Survival rate: (q = 1 – p) Finite Mortality Rate = 1.0 – e Instantaneous Mortality Rate MUST SPECIFY TIME UNITS E.G. IF FINITE SURVIVAL (p) = 0.35, then INSTANEAOUS MORTALITY (Z) =

STATIC LIFE-TABLES RAW Data

General Smoothing

Selected Smoothing

WHAT DO LIFE TABLES TELL US? Allow us to make generalisations - Survivorship

Allow us to make generalisations - Fecundity

Allow us to build models of populations…..

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

Adding Fecundity Copy Down

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 )

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 = / ( ) =

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 = / =

N t+1 = N t.(Survival Rate).(1 + Birth Rate) N t+1 / N t = R = e r = (Survival Rate).(1 + Birth Rate) B = S = At Stable-Age R = x ( ) = 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

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

All the calculations that we have hitherto done concern populations displaying, pulsed births - where reproduction is concentrated at a single point when individuals leave an age class. In birth flow populations, there is a constant addition of individuals to an age class and constant leaving. Furthermore, reproduction is spread across an age class, so that individuals at the end of age class may produce a different number of offspring to those at the start of an age class. Birth Flow vs Birth Pulse First of all we must assume that all reproduction occurs at the mid-point of an age-class. The m x values are appropriate for the end of an x class - not at the middle - need get average. To get at average of m x and m x+1 = (m x-1 + m x )/2

NEXT - need to consider survival from period from x-1 through x to x+1 i.e. p x = [(l x + l x+1 )/2] / [(l x + l x-1 )/2] = [(l x + l x+1 )] / [(l x + l x-1 )] and so we then adjust m x * as: m x * = ((m x-1 + p x m x )/2 This is the basic life table (birth pulse) that we have constructed so far: projections are based on m. MUST ADJUST m

p x = [(l x + l x+1 )/2] / [(l x + l x-1 )/2] = [(l 1 + l 2 )] / [(l 1 + l 0 )] = p 1 m x * as: m x * = ((m x-1 + p x m x )/2) = m 1 * These values of m x now get used in projections of your population NOTE - px values used in the life tables, and calculations therein, do not change. I.e. px (birth-flow) = px (birth-pulse) and the revised px values above are only used to calculate mx values.

The difference between these results and those calculated using pulse-flow models may appear inconsequential, but it is not!

Pulse Flow Stable R

Unitary or Modular Organisms….

I Module Type I = genet Population = genets and ramets Modular organisms often branched.. Predation does not lead to death.. Cloning… II Module II = ramet