Transcript Population Dynamics Lecture Notes

What is a species?
What is a species?
1) Pre-mating isolating mechanisms.
a) Temporal isolation.
b) Ecological isolation.
c) Behavioral isolation.
d) Mechanical isolation.
2) Post-mating isolating mechanisms.
a) Gametic incompatibility.
b) Zygotic mortality.
c) Hybrid inviability.
d) Hybrid sterility.
e) Hybrid breakdown.
Population Ecology
• Population (N)
– Group of animals, identifiable by species,
place, and time
• Defined by population biology
– Genetic definition would be more specific
• Individuals comprise a population
• Collective effects of individuals
– Natality, mortality, rate of increase
• Most management =
– Populations
– not individuals
Rates
• Natality
– Births (per something)
• Mortality
– Deaths (per something)
• Fecundity
– Ability to reproduce
– Number of eggs
– Female births/adult female
• Productivity
– Number of young produced
• Breeding system, sex and age ratios
– Recruitment (net growth = R)
Definitions
• Age structure
– Number of individuals in different age classes
• Sex ratio
– Male:female
• Buck only deer hunting 1:3
• QDM at Chesapeake Farms 1:1.5
• Some dabbling ducks 10:1
Age Pyramids
Long lived, slow turnover, low productivity, high juvenile survival
Short lived, fast turnover, high productivity, low juvenile survival
Age Pyramids
Long lived, slow turnover, low productivity, high juvenile survival
Short lived, fast turnover, high productivity, low juvenile survival
US population age pyramids
Sex Specific Age Pyramid
males
females
Buck only hunting
Age pyramid
Beavers
Beaver Pop Age Structure
40
35
30
N
25
20
15
10
5
0
1
2
3
4
5
6
Age Class
7
8
9
10
Population Growth – 2 main models
Exponential Growth
Logistic Growth
• Assumes resources
unlimited
• Considers carrying capacity
Population Growth
• Lambda
– Measure of population growth
– Ratio of population sizes – No Units
– >1 population is growing
– <1 population is declining
– Important measure of pop status
N
  t 1
Nt
Demographic Rates
•
•
•
•
•
Birth rate (b)
Death rate (d)
Emigration (e)
Immigration (i)
Realized population growth rate r
r  b  d   i  e
Population Growth
• r
– actual growth rate of population
– birth rate – death rate (Exponential Model)
– (Birth rate + immigration rate) – (death rate +
emigration rate) - > more realistic
Exponential Growth
• Constant per capita rate of increase (r)
– Constant percentage increase
– Ex: 10% per year
• Text
– “ever-increasing rate” per unit time
• Means number added per unit time is ever-increasing
• Population growth model
Year
1
2
3
4
5
6
7
8
9
10
N
N + 1 Recuits R Lambda
100
120
20
120
144
24
144
173
29
173
207
35
207
249
41
249
299
50
299
358
60
358
430
72
430
516
86
516
619
103
r
Year
1
2
3
4
5
6
7
8
9
10
N
N + 1 Recuits R Lambda
100
120
20
1.20
120
144
24
1.20
144
173
29
1.20
173
207
35
1.20
207
249
41
1.20
249
299
50
1.20
299
358
60
1.20
358
430
72
1.20
430
516
86
1.20
516
619
103
1.20
r
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
0.20
• population growth
• Population of 100 individuals (N)
• Each individual can contribute 1/3 (0.33) of an individual
to the population in a given unit of time (r)
• What is rN?
• ΔN/Δt?
• Nt+1 ?
Exponential Growth
• George Reserve example
– Dr. Dale McCullough
• Estimated per capita growth rate for
unencumbered growth (rm)
– New species in optimal habitat
– Maximum per capita growth rate
– Why estimate it?
Problems with Exponential Growth Model
• Assumes unlimited resources for population
growth
– Birth rates and death rates remain constant
When is this true??
Quiz
1) T/F? Both lambda and r increase through
time in the exponential growth model.
2) T/F? Both lambda and r change through time
in the logistic growth model.
3) Humans have a Type I survisorship curve
4) Are feral cat killings of songbirds a type of
compensatory or additive mortality?
Logistic Growth Model
• Why worry about this?
• Fundamental conceptual relationship that
underlies sustained yield harvesting
• NC deer population
– 1.1mm
– Harvest 265,000
• Is that harvest a lot, a few?
• Will the population increase, decline, or what?
• Simple mathematical model
Logistic Growth Model
• Parameters have intuitive biological meaning
– K = carrying capacity
– N = population size
– rm = maximum per capita intrinsic growth rate
(potential)
• Species and habitat specific
– r = realized (actual) per capita growth rate
• For exponential growth r = rm
• Only occurs for small populations for a short time
• McCullough should have estimated rm
Logistic Growth Model
• One specific form of sigmoid growth
–
Growth model
• R = net growth = recruits
• K = carrying capacity
• r = realized growth rate
(K  N)
R  Nrm
K
Logistic Growth Model
• As N approaches K, r = 0
•
(K  N)
0
When N small, then r = rm
K
(K  N)

1
(K  N)
K
r  rm
K


R  Nr
Logistic Growth Model
(K  N)
r  rm
K
Density-dependent growth
(K  N)
R  Nr  Nrm
K
Year Recruits
Residual
1
3
9
2
4
12
3
5
16
4
7
21
5
9
28
6
12
37
..
..
..
20
11
371
21
8
382
22
6
390
23
3
396
24
1
399
25
0
400
N r
N+ 1
0.333
12
0.333
16
0.313
21
0.333
28
0.321
37
0.324
49
..
..
0.030
382
0.021
390
0.015
396
0.008
399
0.003
400
0.000
400
Year
1
2
3
4
5
6
..
20
21
22
23
24
25
Recruits
3
4
5
7
9
12
..
..
11
8
6
3
1
0
Residual N
9
12
16
21
28
37
..
..
371
382
390
396
399
400
r
0.333
0.333
0.313
0.333
0.321
0.324
N+
12
16
21
28
37
49
0.030 382
0.021 390
0.015 396
0.008 399
0.003 400
0.000 400
(K  N)
r  rm
K
dN/dt = rN(K-N)/K
r vs population size
population size versus time
(K  N )
R  Nr  Nrm
K
# Recruits vs population size
(K  N)
r  rm
K
dN/dt = rN(K-N)/K

(K  N )
R  Nr  Nrm
K
(K  N)
r  rm
K
dN/dt = rN(K-N)/K

(K  N )
R  Nr  Nrm
K
Density Dependent Growth
Fundamental relationship
that underlies sigmoid
growth. As N increases,
per capita growth r
decreases.
Density-dependent factors
vs density-independent factors
Density Dependent Growth
• Combined effects of natality and mortality
– Births decline as N increases above a certain point
– Deaths increase as N increases above a certain point
Density Dependent Growth
• Residual population (N)
– Population size which produces the recruits (R)
– Pre-recruitment population
• Stock population
• Birth pulse population
– Births occur about the same time
• Deer in spring
Sustained Yield
• Inflection point (I)
– Sigmoid curve slope changes from positive to
negative
– Peak hump-shaped SY (or R) curve
• Maximum R per unit time
– Point of MSY (K/2)
Population Growth
George Reserve Deer
SY  R
R
rh
N
r per capita
growth, h is per
capita harvest rate
Hump-shaped, not
bell-shaped
George Reserve Deer
R  Nr
R  SY
SY  Nr
1
1
MSY  K  rm
2
2
MSY occurs
at the
inflection
point I
George Reserve Deer
SY
h
N
R
r
N
Theoretically,
sustainable
harvests range from
0-90%;
MSY about 50%
George Reserve Deer
R  SY
Right side of MSY (I)
stable
negative
feedback
between N
and R
George Reserve Deer
R  SY
Left side of MSY (I)
unstable
Positive
feedback
between N
and R
Logistic Growth Assumptions
•
•
•
•
All individuals the same
No time lags
Obviously, overly simplistic
Does provide conceptual bases for
management.
Population Models
• Forces thinking
– Conceptual value
• Requires data
– What needs to be known?
– How are those data acquired?
• Predict future conditions
– Assess management alternatives
NC Deer
NC deer
population
1.1mm
Harvest 265,000
Can this model
suggest anything
about the harvest
level in NC?
NC Deer
NC deer
population
1.1mm
Harvest 265,000
SY
265,000
h
 30%
N
1,100,000  265,000
Density Dependent Factors
• Density dependent (proportional)
– Mortality
– Natality
• Density independent
– Asian openbill storks example
• Compensatory mortality and natality
A population of Spotted Fritillary butterflies
exhibits logistic growth. If the carrying capacity
is 500 butterflies and r = 0.1
individuals/(individuals x month), what is the
maximum population growth rate for the
population?
(Hint: maximum population growth rate occurs
when N = K/2).
In the question you're given the following
information:
K = 500 r = 0.1 maximum population growth at
K/2
Therefore, the maximum population size = K/2 =
500/2 = 250
dN/dt = rN[1-N/K] - this is the logistic growth
equation
dN/dt = (0.1)(250) [1 - (250)/500)]
dN/dt = 12.5 individuals/month
Isle Royale Lessons
Moos
e
Wolves
Isle Royale Lessons
• Predator/prey dynamic balance?
• Populations fluctuate due to a myriad of
factors
– Food, disease, weather, competition, genetics,
random events, etc.
• Disequilibrium
– No such thing as
the “balance of nature”
Mating
• Sex ratio and breeding systems
– Monogamous
• Balanced sex ratio
– Ducks -- sexually dimorphic
» Sexes w/ different susceptibility to predation, hunting
– Canada geese -- monomorphic
– Polygynous
• Manage for a preponderance of females
– Pheasants, turkeys -- dimorphic
– Ruffed grouse, quail -- monomorphic
– Promiscuous
• Deer
– To grow, unbalanced sex ratio
– QDM, balanced sex ratio
Age-Specific Birth Rates
Age-specific natality (female young/female)
Natality
Immature
Adults
AGE
Age-Specific Natality
• Deer reproduction Table 5-2
– PA dense, IA sparse
– Fawns pregnant only in Iowa
• Fawns only breed when populations are low
– Corpora lutea per doe (ovulation sites)
• Less in PA (1.6) than in IA (2.23)
– Fetuses/pregnant doe
• Less in PA (1.4) than in IA (2.1)
• George Reserve rm = 0.956
Additive vs. Compensatory
• Additive mortality
– As more mortality factors are added (e.g. hunting) survival
decreases
• Compensatory mortality
– As more mortality factors are added, survival remains the
same (up to a point).
– Rationale to justify hunting
• Would have died anyway, why not from hunting?
• In terms of N remaining constant, could be
compensation in natality, mortality, both
Additive vs. Compensatory
Compensation
Survival
rate
Additive
Harvest rate
Number of survivors
Survivorship Curves
Percent of maximum life span
Survivorship Curves
BioEd Online
Survivorship Curves
Life Tables
• Actuarial tables
Life Tables
Table 5.4
x Lx
Dx
qx = dx/lx
Ex
1 1000
54
54/1000=0.054
7.1
2 100054=946
145
145/946=0.153
------
12/801=.015
7.7
3 946-145= 12
801
Life Tables
• life tables.xls Methods to calculate
• Birth rates and death rates constant for appropriate time (life
span)
– Age distribution (Sx) must be stable
– Sx is the proportion of the number born that are alive at a given age
fx/f0
• Mark individuals at birth and record age at death (lx)
• Calculate number dying in a particular interval
• Know number alive at age x and x+1 (lx)
• Know age distribution and rate of increase
– lx = product of Sx and rate of increase, i.e., number born
• What to estimate?
– N might be enough
– Demographic rates more diagnostic
Life Tables
• Take home message
– Need constant schedules of mortality and natality
so the age distribution stabilizes
– Nearly impossible to meet these conditions for
wild populations
– So, actually constructing a life table for a wild
population is not likely to be possible
– BUT, life tables are of great conceptual value in
modeling populations
Population Data
• Two problems in estimating N
– First observability
• Proportion of animals seen p is observability
• C = count
C  pN
C
N
p
Estimating N
• Count 43 salamanders and you know you
observe 10%, then
C
N
p
43
N
 430
0.1
Population Data
• Problems in estimating N
– Second sampling
• Too expensive in time and money to count everywhere all the
time.
Population Index
Population Index = assume p is constant
Used to make comparisons over time or space
C
N
p
C1
N1 
p
C2
N2 
p
N1  C1
N 2  C2
Unfortunately, probably rarely true.
HIP and Duck Stamps
• Migratory Bird Harvest Information System
– HIP (Harvest Information Program) certification on
hunting license
– Used to sample hunters of doves, woodcock, and other
webless migratory birds
• Duck Stamps
–
–
–
–
All duck, geese, swan hunters purchase
1934 drawn by “Ding” Darling
$750mm for refuges ($.98/$1.00)
Used to sample hunters
BBS
• Breeding Bird Survey
– Volunteers
– About 4,000 routes in US and Canada
– 50 stops on roads at 1/2 mile intervals
– Record birds seen and heard w/i 1/4 mi
– Began 1966
– Over 40 years of trend data
– BBS
Bird Banding
• Amateur and professionals
• Federal bird banding lab
– Early 1900’s
– # bands, color, petagial
tags, collars, etc.
– Migration patterns,
distributions, survival,
behavior, philopatry
Patuxent Wildlife Res. Center
• 1936
• USGS
• Patuxent
•
•
•
•
BBL, BBS, zoo curators, scientists, toxicologists
Whooping cranes
Video
Ultralight
Metapopulations
• Subpopulations of varying sizes somewhat isolated
from each other
• Genetic exchange within subpopulations > between
them
• Subpopulations might wink in and out of existence
– Unoccupied patches still important
• Dispersal and recolonization are critically important
• Habitat fragmentation might exacerbate
• Model