DemographyDemographyFuture population is a result of the interplay between internal characteristics and processes and extrinsic forces and processes.We will begin by focusing on the internal factors - the demographics.Future Population SizePredicting future population size requires understanding the internalworkings of the population.We learn about these “workings” by observing changes in populations and looking for principles which would allow us to predict them.Population ChangeSpeed of population change is referred to as rate of increase.Change results from interplay between fecundity rate, mortality rate, and composition (age distribution and sex ratio).Population ChangeFuture population size can be predicted from current population size and its rate of change:Nt+1 = Nt x (rate of change)Nt+1= Nt λλ = finite rate of increaseλ = Nt+1 / NtIf population is constant λ = 1.0Instantaneous Rate of ChangeSometimes its better to express the finite rate of increase as an instanteous (or exponential) rate of increase (r): λ = Nt+1 / Nt = erln λ = loge λ = rIf population is constant then r = 0Future Population SizeIf rate of change is constant thenNt = N0 ertln Nt = ln N0 + rtEquation for a straight line:y = a + bxExamples:Estimating rate of increaseFinite rate of increase: λ = Nt+1 / Nt Instantaneous rate of increase: Regress ln Nt on tr = (ln Nt - ln N0 ) / tRate of changeWhat really determines the rate of change (increase or decrease) in a population?Births?Deaths?Immigrants?Emigrants?Rate of changeChange results from balance between rates of increase and decreaseIncrease: Births + ImmigrantsDecrease: Deaths + Emigrants∆N = (B + I) - (D + E)Immigrants and Emigrants are samll in number and depend on other populations so we’ll simplify it by ignoring them (for a while).Deaths = MortalityWhat do mortality rates depend on?Age? Population size? Resources?Summarize mortality in a life tableLife tablex = age (years)fx = nx = Survival frequencylx = Survivorshipdx = Mortalityqx = Mortality rateCohort Life TableSurvival of male Song Sparrows hatched in 1976 on Mandarte Island, BCAge fx0 1151 252 193 124 25 16 0Static Life Table (Catch Curve)Ages of male reindeer, South Georgia IslandAge fx0 781 402 193 144 105 5Constructing a Life Table Standardize to l0 = 1.0Male reindeer, South Georgia IslandAge fx fx / f0 lx dx 0 78 78/78 1.000 0.4871 40 40/78 0.513 0.269 2 19 19/78 0.244 0.0643 14 14/78 0.179 0.0514 10 10/78 0.128 0.6415 5 5/78 0.064 0.064Constructing a Life TableMale reindeer, South Georgia IslandAge fx lx dx qx 0 78 1.000 0.487 0.4871 40 0.513 0.269 0.5252 19 0.244 0.064 0.2633 14 0.179 0.051 0.2864 10 0.128 0.641 0.5005 5 0.064 0.064 1.000Graphing a Life Table: SurvivorshipSurvivorship0 1 2 3 4 500.20.40.60.81lxxGraphing a Life Table: Mortality RateMortality Rate0 1 2 3 4 500.20.40.60.81qxxExamples of SurvivorshipExpanded Life Table - Females Add fecundity (mx=female offspring per female age x)Age(x) lx mx lxmx lxmxx0 1.000 0.0 0.0 0.01 0.8 0.6 0.48 0.482 0.7 1.0 0.7 1.43 0.6 1.0 0.6 1.84 0.0 0 0 0Sum 1.78 3.68 (fitness)Population CharacteristicsR0 = Net Reproductive Rate (Fitness) No. daughters generation t+1 = -------------------------------- No. females generation t Σ lx mx 1.78 = ------------- = -------- = 1.78 l0 1.0FitnessFitness defined as Σ lx mx for a genotypeThis fundamental formula is at the heart of both demographic and genetic analyses for a populationIt is very clever because it combines both survival( lx ) and fecundity(mx)Population CharacteristicsG = Mean length of a generation = Mean period elapsing between birth of parents and birth of offspring Σ lx mx x 3.68 = ------------- = ------- = 2.07 Σ lx mx 1.78Population Characteristics ln (R0) ln (1.78)r = ---------- = -------- = 0.28 G 2.07λ = er = e0.28 = 1.321 = Σ e-rxlxmxNote: First 2 equations are approximations, but solving this last equation iteratively is exactly correct.So What?What is value of this?λ and r are good descriptions for either a simple population or a more complex population.Our simple models to predict (or analyze) population growth might work surprisingly well.Nt+1= Nt λFinite Growth Rateλ = 1.32 t Nt0 101 132 173 234 305 40Finite Growth Rate ModelN0 1 2 3 4 5 6 7 8 9 10050100150200Nt tContinuous Growth ModelNt = N0 ertln Nt = ln N0 + rtPlot of ln Nt vs. t is a straight lineslope of line is rContinuous Growth Modelln N0 5 1000.511.522.5ln Nt tIs this correct?What assumptions must be true?Rate of increase (λ or r) is constant.Stable age distribution in the population. Stable Age DistributionLotka (1922) showed that a population that is subject to a constant schedule of birth and death rates will gradually approach a fixed or stable age distribution, whatever the initial age distribution may have been, and will maintain this age distribution indefinitely.Stable Age DistributionCx = proportion of population in age category x to x+1Mertz (1970) showed that Cx = λ-x l x / Σ λ-x l xEven if a population starts out with a different age distribution it will attain this stable age distribution after a whileStable Age DistributionAge(x) lx λ-x λ-xlx Cx0 1.000 1.0 1.0 0.441 0.8 0.758 0.606 0.272 0.7 0.574 0.402 0.183 0.6 0.435 0.261 0.114 0.0 0.329 0 0Sum 2.269Is the rate of increase constant?If resources are unlimited, such as introduction of species to new area OR if population remains at fairly similar numbers over time (i.e. r=0 or λ=1.0 approximately), but even here random environmental variation will produce random changes in r (or ).Typically resources are limited and more animals consume more resources, reducing amount available for each