NEXT WEEK: Computer sessions all on MONDAY: R AM 7-9 R PM 4-6 F AM 7-9 Lab: last 1/2 of manuscript due Lab VII Life Table for Human Pop Bring calculator! Will complete Homework 8 in lab

1
NEXT WEEKComputer sessions all on MONDAYR
AM 7-9 R PM 4-6 F AM 7-9 Lab
last 1/2 of manuscript due Lab VII Life Table
for Human Pop Bring
calculator! Will complete Homework 8 in lab
2
Ch 14 Population Growth Regulation dN/dt
rN dN/dt rN(K-N)/K
3
Sample Exam ?

• A moth species breeds in late summer and leaves
  only eggs to survive the winter. The adult die
  after laying eggs. One local population of the
  moth increasd from 5000 to 6000 in one year.
• Does this species have overlapping generations?
  Explain.
• What is ? for this population? Show calculations.
• Predict the population size after 3 yrs. Show
  calculations.
• What is one assumption you make in predicting the
  future population size?

4
Objectives

• Age structure
• Life table Population growth
• Growth in unlimited environments
• Geometric growth Nt1 ? Nt
• Exponential growth Nt1 Ntert
• Model assumptions

5
Exponential growth of the human population
6
Population growth can be mimicked by simple
mathematical models of demography.

• Population growth ( ind/unit time)
• recruitment - losses
• Recruitment births and immigration
• Losses death and emigration
• Growth (g) (B I) - (D E)
• Growth (g) (B - D) (in practice)

7
How fast a population grows depends on its age
structure.

• When birth and death rates vary by age, must know
  age structure
• proportion of individuals in each age class

8
Age structure varies greatly among populations
with large implications for population growth.
9
Population Growth (age structure known)

• How fast is a population growing?
• per generation Ro
• instantaneous rate r
• per unit time ?
• What is doubling time?

10
Life Table A Demographic Summary Summary of
vital statistics (births deaths)
by age class Used to determine population
growthSee printout for Life Table for
example
11
Values of ?, r, and Ro indicate whether
population is decreasing, stable, or increasing
Ro lt 1
Ro gt1
Ro 1
12
Life Expectancy How many more years can an
individual of a given age expect to
live?How does death rate change through
time?Both are also derived from life
tableUse Printout for Life Table for
example
13
Survivorship curves note x scale

• death rate constant

plants
14
Sample Exam ?

• In the population of mice we studied, 50 of each
  age class of females survive to the following
  breeding season, at which time they give birth to
  an average of three female offspring. This
  pattern continues to the end of their third
  breeding season, when the survivors all die of
  old age.

15

• Fill in this cohort life table.
• Is the population increasing or decreasing?
• Show formula used.
• How many female offspring does a female mouse
  have in her lifetime?
• At what precise age does a mouse have her first
  child? Show formula used.
• Draw a graph showing the surivorship curve for
  this mouse population. Label axes carefully.
• Explain how you reached your answer.

x nx lx mx lxmx xlxmx
0-1 Etc 1000 1.0 0



0
16
Cohort life table follows fate of individuals
born at same time and followed throughout their
lives.

mx
17
Survival data for a cohort (all born at same
time) depends strongly on environment
population density.
18
What are advantages and disadvantages of a cohort
life table?

• Advantages
• Describes dynamics of an identified cohort
• An accurate representation of that cohort
  behavior
• Disadvantages
• Every individual in cohort must be identified and
  followed through entire life span - can only do
  for sessile species with short life spans
• Information from a given cohort cant be
  extrapolated to the population as a whole or to
  other cohorts living at different times or under
  different conditions

19
Static life table based on individuals of known
age censused at a single time.
20
Static life table avoids problem of
variation in environment can be constructed
in one day (or season)
n 608
21
E.g. exponential population growth
? 1.04
22
Two models of population growth with unlimited
resources

• Geometric growth
• Individuals added at
• one time of year
• (seasonal reproduction)
• Uses difference equations
• Exponential growth
• individuals added to population continuously
  (overlapping generations)
• Uses differential equations
• Both assume no age-specific birth /death rates

23
Difference model for geometric growth with
finite amount of time

• ?N/ ?t rate of ? (bN - dN) gN,
• where bN finite rate of birth or
• per capita birth rate/unit of time
• g b-d, gN finite rate of growth

24
Projection model of geometric growth (to predict
future population size)

• Nt1 Nt gNt
• (1 g)Nt Let ? (lambda) (1
  g), then
• Nt1 ? Nt
• ? Nt1 /Nt
• Proportional ?, as opposed to finite ?, as above
• Proportional rate of ? / time
• ? finite rate of increase, proportional/unit
  time

25
Geometric growth over many time intervals

• N1 ? N0
• N2 ? N1 ? ? N0
• N3 ? N2 ? ? ? N0
• Nt ?t N0
• Populations grow by multiplication rather than
  addition (like compounding interest)
• So if know ? and N0, can find Nt

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Example of geometric growth (Nt ?t N0)

• Let ? 1.12 (12 per unit time) N0 100
• N1 1.12 x 100
  112
• N2 (1.12 x 1.12) 100 125
• N3 (1.12 x 1.12 x 1.12) 100
  140
• N4 (1.12 x 1.12 x 1.12 x 1.12) 100 157

27
Geometric growth
?? gt 1 and g gt 0
N
N0
?? 1 and g 0
?? lt 1 and g lt 0
time

28
Differential equation model of exponential
growthdN/dt rN

• rate of contribution number
• change of each of
• in individual X individuals
• population to population in the
• size growth population

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dN / dt r N

• Instantaneous rate of birth and death
• r difference between birth (b) and death (d)
• r (b - d) so r is analogous to g, but
  instantaneous rates
• rates averaged over individuals (i.e. per capita
  rates)
• r intrinsic rate of increase

30
Exponential growth Nt N0 ert
r gt 0
r 0
r lt 0

• Continuously accelerating curve of increase
• Slope varies directly with population size (N)

31
Exponential and geometric growth are related

• Nt N0 ert
• Nt / N0 ert
• If t 1, then ert ?
• N1 / N0 ? er
• ? ln ? r

32
The two models describe the same data equally
well.
Exponential
TIME
33
Environmental conditions influence r, the
intrinsic rate of increase.
34
Population growth rate depends on the value of r
r is environmental- and species-specific.
35
Value of r is unique to each set of
environmental conditions that influenced birth
and death ratesbut have some general
expectations of pattern High rmax for
organisms in disturbed habitatsLow rmax for
organisms in more stable habitats
36
Rates of population growth are directly related
to body size.

• Population growth
• increases directly with the natural log of net
  reproductive rate (lnRo)
• increases inversely with mean generation time
• Mean generation time
• Increases directly with body size

37
Rates of population growth and rmax are directly
related to body size.

• Body Size Ro T r
• small 2 0.1 6.93
• medium 2 1.0 0.69
• large 2 10 0.0693

6.9 .69 .069
if Ro2
Generation time decreases w/ increase in r T
increases w/ decrease in r
r
0.1 1 10
T
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Assumptions of the model

• 1. Population changes as proportion of current
  
• population size (? per capita)
• ? x individuals --gt? in population
• 2. Constant rate of ? constant birth and death
• rates
• 3. No resource limits
• 4. All individuals are the same (no age or size
  
• structure)

39
Sample Exam ?

• A moth species breeds in late summer and leaves
  only eggs to survive the winter. The adult die
  after laying eggs. One local population of the
  moth increasd from 5000 to 6000 in one year.
• Does this species have overlapping generations?
  Explain.
• What is ? for this population? Show calculations.
• Predict the population size after 3 yrs. Show
  calculations.
• What is one assumption you make in predicting the
  future population size?

40
Sample Exam ?

• In the population of mice we studied, 50 of each
  age class of females survive to the following
  breeding season, at which time they give birth to
  an average of three female offspring. This
  pattern continues to the end of their third
  breeding season, when the survivors all die of
  old age.

41

• Fill in this cohort life table.
• Is the population increasing or decreasing?
• Show formula used.
• How many female offspring does a female mouse
  have in her lifetime?
• At what precise age does a mouse have her first
  child? Show formula used.
• Draw a graph showing the surivorship curve for
  this mouse population. Label axes carefully.
• Explain how you reached your answer.

x nx lx mx lxmx xlxmx
0-1 Etc 1000 1.0 0



0
42
Objectives

• Age structure
• Life table Population growth
• Growth in unlimited environments
• Geometric growth Nt1 ? Nt
• Exponential growth Nt1 Ntert
• Model assumptions

43
Vocabulary