Chapter 9: Population GrowthPopulation has a tendency to exponentially grow and go towards a carrying capacity

Usually populations overshoot carrying capacity and then die off and then go back up to reach the capacity

Many different types of growth: exponential growth, linear growth, boom and bust growth, sigmoidal growth (exponential in the beginning then start to reach carrying capacity)

o Others fluctuate around carrying capacity- Theoretical framework that governs the dynamics of population growth; helpful for understanding but does not actually reflect what is going on in nature

a lot of times in a population model, you create a theoretical framework and put it in the model and check nature and see if your model is explaining what nature is doing

o if not, keep fixing modelo nature is not constant, systematic and random

Cheat sheet:N(t) = individuals in a population at time tNx = number of ind. Alive in any age classIx = probability at birth surviving to a given agePopulation growth

Open system, closed system Birth and death Immigration and emigration; in a closed system only thing that

matters is birth and death ONLY ADDRESS CLOSED POPULATION

First example: Hydra in an aquarium Some time (t) and look at initial individuals

o Initially 100, 40 new, 10 that are dying, and calculate what is the population after a single day

o (100) + (40-10) = 130o How do you calculate day 2? You have to create a RATE

Birth and Death rates The actual number of birth and deaths depends on population size Take initial birth and divide by population; 40/100 = 0.4 This is your birth rate

Death rate: Take 10/100 = 0.1 = this is the death rate If we assume birth and death rate are constant, they can be used to

predict the growth of a population over tine, regardless of population size

N(t+1) = N(t) + b*N(t) – d*N(t)So day 2:

(130) + (0.4)(130) – (0.1)(130) = 169How do populations change form one stage to the next?

N(t+1) – N(t) = (b – d)N(t) Usually populations change more rapidly from one step to the next Between day 1 and day 0 was a difference of 30; day 2 and day 1

was a difference of 39; day 3 and day 2 is 50.7Rate of change in population over time

Assumption: b and d are constant Lump them together and say r = b – d So N/T = rN(t) The growth rate of the population is the slope = exponential growth

rate

o slope is shallow in the beginning, steep later r is the instantaneous per capita rate of growth using derivatives

N = N0e^rt When r = 0, b = d

o Population size does not change When r > 0, b > d

o Population size increases exponentially When r <0, b < d

o Population size decreases exponentiallyExponentially growth is continuously accelerating or decelerating based on the population size you start withWhen does exponential growth happen?

Populations that usually live in favorably environments, are at low population densities, can adapt well

Need to worry about when: invasive species (no natural predators)Exponential Growth

How do you figure out r? how to find birth rate and death rate?o Are they always going to be the same? How do you figure it

out for bacteria, insects, small vertebrates, large vertebrates?Life Table

A life table is an age-specific account of mortality Used to examine systematic patterns of mortality and survivorship

within populations Life tables follow a cohort, from birth to death

o Animals are relatively short life span so easy to do

SurvivorshipIx = probability at birth of surviving to a given age

Right now: only worry about death, not birth Always Nx/N0; proportion of the original number

Age-specific mortality dx A whole number, difference between the number of indivduals alive

for the next age class How many in the population die from one age to the next? How many die? Dx = N0 – N1 Use this to calculate another rate

Age-specific mortality rate qx Qx = dx/nx Rate of mortality = # of ind died/ # of ind alive Limited to a particular age group Always just compare to age right before it At the end it’ll always be 1 since everyone is dying

Life table for insects Life table makes sense for vertebrates to use age Sometimes can’t use age; in insects, which only live to one year, it

wouldn’t make sense to use ageo A life table for insects? Use life cycles/different stages of lifeo Gypsy moth: broken down by eggs, instars, prepupae, pupae,

and then adultsWatch podcast 10-10:06

Mortality Age Mortality with trying to be born Once you’re born, mortality goes down until about the age of 13 From 13-20, mortality increases and is steadily increasing to end of

lifeSurvivorship curves

More steep more likely to die More flat more likely to survive How is this useful? Compare males to females; see that females live

longer in squrriels and survivorship curve is a little less steepThree types:

Type 1 (strongly convex) – mortality likely late life (humans) Type 2 ( straight) – mortality constant throughout life (plants) Type 3 (concave) – mortality likely early life (oysters)

Birth Rate is Age-Specific Birth rates = births per individual x 1000/time Can be improved by only considering females (They’re the ones that

can have the offspring); female age In sexually dimorphic species, population is only a function of

females in the populationDetermining the birth rate for females by age class (age-specific birth rate)

Population increase is a function of the number of females in the population

Bx = mean number of females born to each female in an age group Gross reproductive rate = sum across all age classes of how

many females they will have in their life time Only count females that will replace and keep reproducing

Fecundity Table Shows fertility Survivorship and multiply by number of babies

o Probability of making it and if she makes it, how many number of babies will she have

o 1 year old squirrel = 0.3(2) = 0.6 Net reproductive rate = the average number of females that are

produced by a newborn female during her lifetime R0 = SUM(Ixbx) across all age classes

If R0 = 1; population is stableIf R0 > 1 = population is growingIf R0 < 1 = population is decreasing

In the gray squirrel example: gross reproductive rate = 10; net reproductive rate = 1.4

Can use this information to project into the future; we’re not assuming everyone is the same in the population and not everyone ahs the same likelihood of reproducing at a certain age

o Or surviving to that ageA population projection table uses Sx

Qx = proportion of individuals that die before reaching the next age class hence

o Sx = 1 – qx Probability that will survive to the next age class

Use this to calculate R0 Multiple number of individuals by survivorship of getting to the next

age, then of those surviving multiple probability of having offspring and how many

o Calculate total offspring = add to survivors = new population #

Stable Age Distribution

what percentage of that population does it account for divide the number in each age class by the total population size for

that year N(t) the proportion of each group remains the same year after year even

if the population is increasing = finite multiplication rate (lambda)

N(t+1)/N(t)If lambda > 0 = increasing at same rateIf lambda = 0 = stableIf lambda < 0 = decreasing

The general equation to predict for any year in the future: N(t) = N(0)λ^t

As long as birth rates aren’t changing, everything is based on initial conditions

If what you calculated is not exact in population prediction table; its because stable age distribution doesn’t become stable until a certain ageGeometric Growth Rate

Finite/discrete rate; age 1 to age 2, age 2 to age 3 Exponential = continuous Relate lambda to r; set equations equal to each other λ^t = e^rt λ = e^r ln λ = r r = for continuous; λ is for discrete

Estimate r from λ does not assume that all individuals in the population are identicalVariety of factors lead to Extinction

Extinction is possible if the resource base does not recover in time for survivors to reproduce

Stochastic processes can also influence population dynamics In reality, birth rates and death rates are not constant!

Demographic Stochasticity – variation in birth rates and death rates occurring in populations from year to year

the inevitable variability ~ GENETIC DRIFTEnvironmental stochasticitiy – high temperatures, no rain, flooding, big natural disasters

Populations that are declining in size can eventually become extinct Low genetic variability

Largest amount of extinction/critically endangered species in the tropics due to temperature changes and deforestation

Species that migrate are more susceptible to extinction Leading causes of population declines and extinctions: cutting

forests, draining wetlands, creating dams and reservoirs

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11/10/15 9:31 AM