Population Dynamics Fundamental Equation: N (t+1) = N (t) + B – D + I – E N (t+1) - N (t) = B...
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Population Dynamics Fundamental Equation: N (t+1) = N (t) + B – D + I – E N (t+1) - N (t) = B – D + I – E = N = B – D + I – E I E D B
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Transcript of Population Dynamics Fundamental Equation: N (t+1) = N (t) + B – D + I – E N (t+1) - N (t) = B...
Population Dynamics
Fundamental Equation:
N(t+1) = N(t) + B – D + I – E
N(t+1) - N(t) = B – D + I – E
= N = B – D + I – E
I
ED
B
Estimating Patterns of Survival
• Three main methods of estimation:– Cohort life table
• .
Estimating Patterns of Survival
• Three main methods of estimation:– Static life table
• .
Estimating Patterns of Survival• Three main methods of estimation:
– Age distribution• Calculate difference in proportion of individuals in
each age class.• .
Cohort vs Static Life Tables
High Survival Among the Young• Murie collected Dall
Sheep skulls, Ovis dalli– Major Assumption:
Proportion of skulls in each age class represented typical proportion of individuals dying at that age
• Reasonable given sample size of 608
High Survival Among the Young
– Constructed survivorship curve
• Discovered
bi-modal mortality– <1 yr
– 9-13 yrs
Survivorship Curves• Type I:
– Dall Sheep
• Type II:
– American Robins
• Type III:
• .– Sea Turtles
Survivorship Curves Plot
Log10lx vs. X
Dall sheep (Ovis dalli)
Life Table
Static life table for Dall Sheep
x nx dx lx S1000
0 752 129 1.000 1000
1 623 114 0.828 828
2 509 113 0.677 677
3 396 81 0.527 527
4 315 78 0.419 419
5 237 59 0.315 315
6 178 65 0.237 237
7 113 55 0.150 150
8 58 25 0.077 77
9 33 9 0.044 44
10 24 8 0.032 32
11 16 7 0.021 21
12 9 2 0.012 12
13 7 1 0.009 9
14 6 4 0.008 8
15 2 2 0.003 3
total 752
x = age class
nx = number alive
dx = number dead
lx = proportion
surviving
S1000 = # per 1000
alive
Ovis dalli dalli
Static life table for Dall Sheep
x nx dx lx S1000
0 752 129 1.000 1000
1 623 114 0.828 828
2 509 113 0.677 677
3 396 81 0.527 527
4 315 78 0.419 419
5 237 59 0.315 315
6 178 65 0.237 237
7 113 55 0.150 150
8 58 25 0.077 77
9 33 9 0.044 44
10 24 8 0.032 32
11 16 7 0.021 21
12 9 2 0.012 12
13 7 1 0.009 9
14 6 4 0.008 8
15 2 2 0.003 3
total 752
Age class x = 0 = newborns = 100% survive
Age class x = 1 only 623 in this
age class = 752-129
prop surviving (l1) = 623/752 = 0.828
Age class x = 2 only 509 survive
= 623-114 prop surviving (l2) =
509/752 = 0.677
Age Distribution
• Age distribution of a population reflects its history of survival, reproduction, and growth potential
• Miller published data on age distribution of white oak (Quercus alba)– Determined relationship between age and trunk
diameter– Age distribution biased towards young trees.
• Sufficient reproduction for replacement– Stable population
Age Distribution
Age Distribution
• Rio Grande Cottonwood populations (Populus deltoides wislizenii) are declining– Old trees not being replaced– Reproduction depends on seasonal floods
• Prepare seed bed
• Keep nursery areas moist
– Because floods are absent, there are now fewer germination areas
Dynamic Population in a Variable Climate
• Grant and Grant studied Galapagos Finches.– Drought in 1977 resulted in no recruitment
• Gap in age distribution
• Additional droughts in 1984 and 1985
• Reproductive output driven by exceptional year in 1983– Responsiveness of population age structure to environmental
variation
Age Structure
Creation of Stable Age Distribution
3
2
1
Age
1st Gen. 2nd Gen. 3rd Gen.
Not Stable Not Stable Stable
1
65
34
20%
30%
50%
10
35
55
10
35
55
Rates of Population Change
• Birth Rate:
• Fecundity Schedule:
Frequency of Reproduction in Populations
Time
Nu
mb
er
of o
ffspr
ing
Discrete,
non-overlapping
Discrete,
overlapping
Continuous
generation
Estimating Rates for an Annual Plant
• P. drummondii– Ro = Net reproductive rate; Average number of seeds
produced by an individual in a population during its lifetime
– Ro=Σlxmx
• X= Age interval in days
• lx = % pop. surviving to each age (x)
• mx= Average number seeds produced by each individual in each age category
Estimating Rates for an Annual Plant
• Because P. drummondii has non-overlapping generations, can estimate growth rate– Geometric Rate of Increase (λ):
• λ =N t+1 / Nt
• N t+1 = Size of population at future time
• Nt = Size of population at some earlier time
Estimating Rates when Generations Overlap
• Common Mud Turtle
(K. subrubrum)– About half turtles nest each yr– Average generation time:
T = Σ xlxmx / Ro
– X= Age in years
– Per Capita Rate of Increase:
r = ln Ro / T
– ln = Base natural logarithms
Fecundity (Fertility) Schedule
Life Table Calculations
0+2.95+3.06+1.52+0.26 = 7.70
X(lx)(m
x)
(lx)(m
x)
Generation Time
(1*2.95)(2 *3.06)3*1.52)(4 *0.26)7.70
14.677.70
1.905
7.70Sum = 14.67
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