Background knowledge expected
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Transcript of Background knowledge expected
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Background knowledge expected
Population growth models/equations
exponential and geometric
logistic
Refer to
204 or 304 notes
Molles Ecology Ch’s 10 and 11
Krebs Ecology Ch 11
Gotelli - Primer of Ecology (on reserve)
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Habitat loss Pollution Overexploitation Exotic spp
Small fragmented isolated popn’s
InbreedingGenetic Variation
Reduced N Demographic stochasticity
Env variation
CatastrophesGenetic processes
Stochastic processes
The ecology of small populations
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How do ecological processes impact small populations?
Stochasticity and population growth
Allee effects and population growth
Outline for this weeks lectures
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Immigration +Emigration -
Birth (Natality) +
Death (Mortality)
-
Nt+1 = Nt +B-D+I-E
Population Nt
Demography has four components
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Exponential population growth(population well below carrying capacity, continuous
reproduction closed pop’n)
Change in population at any time
dN = (b-d) N = r N where r =instantaneous rate of increasedt
∆t
∆N
Cumulative change in population Nt = N0ert
N0 initial popn size,
Nt pop’n size at time t
e is a constant, base of natural logs
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Trajectories of exponential population growth
r > 0r = 0r < 0
N
t
Trend
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Geometric population growth(population well below carrying capacity, seasonal reproduction)
Nt+1 = Nt +B-D+I-E
∆N = Nt+1 - Nt
= Nt +B-D+I-E - Nt
= B-D+I-E
Simplify - assume population is closed; I and E = 0
∆N = B-D
If B and D constant, pop’n changes by rd = discrete growth factor
Nt+1 = Nt +rd Nt
= Nt (1+ rd) Let 1+ rd = , the finite rate of increase
Nt+1 = Nt
Nt = t N0
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DISCRETE vs CONTINUOUS POP’N GROWTH
Reduce the time interval between the teeth and the
Discrete model converges on continuous model
= er or Ln () = r
Following are equivalent r > 0 > 1
r = 0 = 1
r< 0 < 1
Trend
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Geometric population growth(population well below carrying capacity, seasonal reproduction)
Nt+1 = (1+rdt) Nt
= (1+rdt) (1+rdt-1) Nt-1
= (1+rdt) (1+rdt-1) (1+rdt-2) Nt-2
= (1+rdt) (1+rdt-1) (1+rdt-2) (1+rdt-3) Nt-3
Add dataNt-3= 10rdt = 0.02
rdt-1 = - 0.02rdt-2 = 0.01rdt-3 = - 0.01
What is the average growth rate?
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Geometric population growth(population well below carrying capacity, seasonal reproduction)
What is average growth rate?
= (1+0.02) + (1-0.02) + (1+0.01) + (1-0.01) = 14
Arithmetic mean
Predict Nt+1 given Nt-3 was 10
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Geometric population growth(population well below carrying capacity, seasonal reproduction)
What is average growth rate?
Geometric mean = [(1+0.02) (1-0.02) (1+0.01) (1-0.01)]1/4 = 0.999875
KEYPOINTLong term growth is determined by the geometric not the arithmetic meanGeometric mean is always less than the arithmetic mean
Calculate Nt+1 using geometric mean
Nt+1 = 4 x 10
(0.999875)4 x10 = 9.95
Nt+1 = (1+0.02) (1-0.02) (1+0.01) (1-0.01) 10= 9.95
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DETERMINISTIC POPULATION GROWTH
For a given No, r or rd and t The outcome is determined
Eastern North Pacific Gray whales Annual mortality rates est’d at 0.089Annual birth rates est’d at 0.13
rd=0.13-0.89 = 0.041 = 1.04
1967 shore surveys N = 10,000
Estimated numbers in 1968 N1= N0 = ?
Estimated numbers in 1990 N23= 23 N0 = (1.04)23.
10,000 = 24,462
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DETERMINISTIC POPULATION GROWTH
For a given No, r or rd and t The outcome is determined
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Population growth in eastern Pacific Gray Whales
- fitted a geometric growth curve between 1967-1980
- shore based surveys showed increases till mid 90’s
In USPacific Gray Whales were delisted in 1994
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Mean r
\
SO what about variability in r due to good and bad years?ENVIRONMENTAL STOCHASTICITY
leads to uncertainty in racts on all individuals in same way
b-dBad 0 Good
Variance in r = 2e = ∑r2 -
(∑r)2
NN
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Population growth + environmental stochasticity
Ln N
t
Deterministic1+r= 1.06, 2
e = 0
1+r= 1.06, 2e = 0.05
Expected
Expected rate of increase is r- 2e/2
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Predicting the effects of greater environmental stochasticity
Onager (200kg)
Israel - extirpated early 1900’s
- reintroduced 1982-7
- currently N > 100
RS varies with Annual rainfall
Survival lower in droughts
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Global climate change (GCC) is expected to
----> changes in mean environmental conditions
----> increases in variance (ie env.
stochasticity)
meandrought < 41 mm
Pre-GCC Post-GCC
Mean rainfall is the same BUT
Variance and drought frequency is greater in “post GCC”
Data from Negev
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Simulating impact on populations via rainfall impact on RS Variance in rainfall
Low High
Number of quasi-extinctions
= times pop’n falls below 40
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Simulating impact on populations adding impact on survival
CONC’nEnvironmental stochasticity can influence extinction risk
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But what about variability due to chance events that act on individuals
Chance events can impactthe breeding performanceoffspring sex ratioand death of individuals
---> so population sizes can not be predicted precisely
Demographic stochasticity
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Demographic stochasticity
Dusky seaside sparrowsubspeciesnon-migratorysalt marshes of southern Florida
decline DDTflooding habitat for mosquito controlHabitat loss - highway construction
1975 six left
All male
Dec 1990 declared extinct
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Extinction rates of birds as a function of population size over an 80-year period
0
30
60
1 10 100 1000 10,000
** *
**
***
10 breeding pairs – 39% went extinct10-100 pairs – 10% went extinct1000>pairs – none went extinct
*
Population Size (no. pairs)
% Extinction
Jones and Diamond. 1976. Condor 78:526-549
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random variation in the fitness of individuals (2
d)
produces random fluctuations in population growth rate that are inversely proportional to N
demographic stochasticity = 2d/N
expected rate of increase is r - 2d/2N
Demographic stochasticity is density dependant
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How does population size influence stochastic processes?
Demographic stochasticity varies with N
Environmental stochasticity is typically independent of N
Long term data fromGreat tits in Whytham Wood, UK
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Partitioning variance
Species 2d 2
e Swallow 0.18 0.024Dipper 0.27 0.21Great tit 0.57 0.079Brown bear 0.16 0.003
in large populations N >> 2d / 2
e
Environmental stochasticity is more importantDemographic stochasticity can be ignored
Ncrit = 10 * 2d / 2
e (approx Ncrit = 100)
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Stochasticity and population growth
N0= 50 = 1.03
Simulations - = 1.03, 2e = 0.04, 2
d = 1.0
N* = 2d /4
r - (2e /2)
N* Unstable eqm below which pop’n moves to
extinction
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Environmental stochasticity-fluctuations in repro rate and probability of mortality imposed by good and bad years-act on all individuals in similar way-Strong affect on in all populations
Demographic stochasticity-chance events in reproduction (sex ratio,rs) or survival acting on individuals- strong affect on in small populations
Catastrophes -unpredictable events that have large effects on population size (eg drought, flood, hurricanes)-extreme form of environmental stochasticity
SUMMARY so far
Stochasticity can lead to extinctions even when the mean population growth rate is positive
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Key points
Population growth is not deterministic
Stochasticity adds uncertainty
Stochasticity is expected to reduce population growth
Demographic stochasticity is density dependant and less important when N is large
Stochasticity can lead to extinctions even when growth rates are, on average, positive