Ch 14: Population Growth Regulation dN/dt = rN dN/dt = rN(K-N)/K

1
Ch 14 Population Growth Regulation dN/dt
rN dN/dt rN(K-N)/K

• BRING to
• LECTURE
• PRINT of
• THIS PPT
• 2) Pg. 79 in
• Manual

2
Objectives

• Population Structure
• Population Dynamics
• Growth in unlimited environment
• Geometric growth Nt1 ? Nt
• Exponential growth Nt1 Ntert
• 
  dN/dt rN
• Model assumptions
• Growth in limiting environment
• Logistic growth dN/dt rN (K - N)/ K
• D-D birth and death rates
• Model assumptions

3
Population all individuals of a species in
an area

• Subpopulations in different habitat patches
• What are structures (traits) of populations?
• Size (abundance)
• Age structure
• Sex ratio
• Distribution (range)
• Density (/unit area)
• Dispersion (spacing)
• Genetic structure

4
Draw two graphs of population growth showing

• Growth with unlimited resources
• Growth with limited resources
• Label axes.
• Indicate carrying capacity (K).
• 3) What are equations representing both types of
  growth
• A) exponential?
• B) logistic?

5
Population growth predicted by the exponential
(J) vs. logistic (S) model.
6
Population growth can be mimicked by simple
mathematical models of demography.

• Population growth ( ind/unit time)
• recruitment - losses
• Recruitment
• Losses
• Growth (g)
• Growth (g) (B - D) (in practice)

7
Two models of population growth with unlimited
resources

• Geometric growth
• Individuals added at
• one time of year
• (seasonal reproduction)
• Uses
• Exponential growth
• individuals added to population continuously
  (overlapping generations)
• Uses
• Both assume

8
Difference model for geometric growth with
finite amount of time

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

9
Projection model of geometric growth (to predict
future population size)

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

10
Geometric growth over many time intervals

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

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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 125
• N3 140
• N4 157

12
Geometric growth
?? gt 1 and g gt 0
N
N0
?? 1 and g 0
?? lt 1 and g lt 0
time

13
Values of ?, r, and Ro indicate whether
population is
Ro lt 1
Ro gt1
Ro 1
14
Differential equation model of exponential
growth

• 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

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

16
E.g. exponential population growth
? 1.04
17
Exponential growth Nt
r gt 0
r 0
r lt 0

• Continuously accelerating curve of increase
• Slope varies directly
• (N) (gets steeper as size increases).

18
Environmental conditions and species influence r,
the intrinsic rate of increase.
19
Population growth rate depends on the value of
is environmental- and species-specific.
20
Value of r is unique to each set of
environmental conditions that influenced birth
and death rates

• but have some general expectations of pattern
• High rmax for organisms in
  habitats
• Low rmax for organisms in
  habitats

21
Rates of population growth are directly related
to body size.

• Population growth
• increases inversely with
• Mean generation time
• Increases directly with

22
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
• 3. No resource limits
• 4. All individuals are the same (
  )

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Sample Exam ? Problem Set 2-1 (pg. 79)

• A moth species breeds in late summer and leaves
  only eggs to survive the winter. The adult dies
  after laying eggs. One local population of the
  moth increased 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?

24
Review Problem Set 1 Geometric Growth
ModelExponential Growth Model Select correct
formula
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Objectives

• Growth in unlimited environment
• Geometric growth Nt1 ? Nt
• Exponential growth Nt1 Ntert
• 
  dN/dt rN
• Model assumptions
• Growth in limiting environment
• Logistic growth dN/dt rN (K - N)/ K
• D-D birth and death rates
• Model assumptions

26
Populations have the potential to increase
rapidlyuntil balanced by extrinsic factors.
27
Population growth rate

• Intrinsic Population Reduction in
• growth X size X growth
  rate
• rate at due to crowding
• N close
• to 0

28
Population growth predicted by the
model.
K
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Assumptions of the exponential model

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

30
Population growth rates become
aspopulation size increases.

• Assumption of constant birth and death rates is
  violated.
• Birth and/or death rates must change as pop. size
  changes.

31
Population equilibrium is reached when
Those rates can change with density (
).
32
Density-dependent factors .
33
Habitat quality affects reproductive variables
affected ( is lowered).
34
Reproductive variables are
35
Population size is regulated by density-dependent
factors affecting birth and/or death rates.
36
1) Density-dependence in plants first decreases
growth.Size hierarchy develops.
skewed
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2) Density-dependence secondly increases some
components of reproduction decreases others
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3) Density-dependence thirdly decreases
survival. Intraspecific competition causes
self-thinning.
Biomass (g)
Density of surviving plants
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r (intrinsic rate of increase) decreases as a
.

• Population growth is
  .

rm
slope rm/K
r
r0
K
N
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Logistic equation

• Describes a population that experiences
• density-dependence.
• Population size stabilizes at K
• dN/dt
• where rm maximum rate of increase w/o
• resource limitation
• intrinsic rate of increase
• carrying capacity
• environmental break
  (resistance)
• proportion of unused
  resources

41
Logistic ( ) growth occurs when
the population reaches a resource limit.

• at K/2 separates accelerating
  and decelerating phases of population growth
  point of

42
Logistic curve incorporates influences of
per capita growth rate and
population size.
Specific
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Assumptions of logistic model

• Population growth is proportional to the
  remaining resources (linear response)
• All individuals can be represented by an average
  (no change in age structure)
• Continuous resource renewal (constant E)
• Instantaneous responses to crowding
• .
• K and r are specific to particular organisms in a
  particular environment.

44
Review Logistic Growth Model

• Problem Set 2-3 (see pg. 80)