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
Chs 10 and 11 Krebs Ecology Ch 11 Gotelli
- Primer of Ecology (on reserve)
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The ecology of small populations

Habitat loss Pollution Overexploitation Exotic spp
Small fragmented isolated popns
Env variation
Inbreeding Genetic Variation
Demographic stochasticity
Reduced N
Genetic processes
Catastrophes
Stochastic processes
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Outline for this weeks lectures
How do ecological processes impact small
populations? Stochasticity and population growth
Allee effects and population growth
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Demography has four components
Birth (Natality)
Emigration -
Immigration
Population Nt
Death (Mortality) -
Nt1 Nt B-DI-E
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Exponential population growth
(population well below carrying capacity,
continuous reproduction closed popn)
?N
?t
Change in population at any time dN (b-d) N
r N where r instantaneous rate of
increase dt
Cumulative change in population Nt N0ert
N0 initial popn size, Nt
popn size at time t e is a constant,
base of natural logs
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Trajectories of exponential population growth
N
Trend
r gt 0 r 0 r lt 0
t
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Geometric population growth
(population well below carrying capacity,
seasonal reproduction)
Nt1 Nt B-DI-E ?N Nt1 - Nt Nt
B-DI-E - Nt B-DI-E Simplify - assume
population is closed I and E 0 ?N B-D If
B and D constant, popn changes by rd discrete
growth factor Nt1 Nt rd Nt Nt (1 rd)
Let 1 rd ?, the finite rate of
increase Nt1 ? Nt Nt ?t N0
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DISCRETE vs CONTINUOUS POPN GROWTH
Reduce the time interval between the teeth and
the Discrete model converges on continuous
model ? er or Ln (?) r Following are
equivalent r gt 0 ? gt 1 r 0 ?
1 rlt 0 ? lt 1
Trend
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Geometric population growth
(population well below carrying capacity,
seasonal reproduction)
Nt1 (1rdt) Nt (1rdt) (1rdt-1) Nt-1
(1rdt) (1rdt-1) (1rdt-2) Nt-2 (1rdt)
(1rdt-1) (1rdt-2) (1rdt-3) Nt-3
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?
Arithmetic mean
Predict Nt1 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 (10.02) (1-0.02) (10.01)
(1-0.01)1/4 0.999875
Calculate Nt1 using geometric mean Nt1 ?4 x
10 (0.999875)4 x10 9.95
Nt1 (10.02) (1-0.02) (10.01) (1-0.01)
10 9.95
KEYPOINT Long term growth is determined by the
geometric not the arithmetic mean Geometric mean
is always less than the arithmetic mean
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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 estd at 0.089 Annual birth rates
estd at 0.13 rd0.13-0.89 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
  90s

In US Pacific Gray Whales were delisted in 1994

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SO what about variability in r due to good and
bad years? ENVIRONMENTAL STOCHASTICITY leads to
uncertainty in r acts on all individuals in same
way
\
Mean r
(?r)2
?r2 -
Variance in r ?2e
N
N
Bad 0 Good
b-d
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Population growth environmental stochasticity
Deterministic 1r 1.06, ?2e 0
Expected
Ln N
1r 1.06, ?2e 0.05
t
Expected rate of increase is r- ?2e/2
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Predicting the effects of greater environmental
stochasticity
Onager (200kg) Israel - extirpated early 1900s
- reintroduced 1982-7 - currently N gt 100
RS varies with Annual rainfall Survival lower
in droughts
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Global climate change (GCC) is expected to ----gt
changes in mean environmental conditions ----gt
increases in variance (ie env. stochasticity)
Data from Negev
mean drought lt 41 mm
Pre-GCC Post-GCC
Mean rainfall is the same BUT Variance and
drought frequency is greater in post GCC
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Simulating impact on populations via rainfall
impact on RS
Variance in rainfall Low High
Number of quasi-extinctions times popn
falls below 40
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Simulating impact on populations adding impact on
survival
CONCn Environmental stochasticity can influence
extinction risk
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But what about variability due to chance events
that act on individuals

Chance events can impact the breeding
performance offspring sex ratio and death of
individuals ---gt so population sizes can not be
predicted precisely
Demographic stochasticity
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Demographic stochasticity

• Dusky seaside sparrow
• subspecies
• non-migratory
• salt marshes of southern Florida
• decline DDT
• flooding habitat for mosquito control
• Habitat loss - highway construction
• 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
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10 breeding pairs 39 went extinct 10-100 pairs
10 went extinct 1000gtpairs none went extinct

Extinction
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0



1 10 100 1000 10,000
Population Size (no. pairs)
Jones and Diamond. 1976. Condor 78526-549
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random variation in the fitness of individuals
(?2d) 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
Long term data from Great tits in Whytham Wood, UK
Environmental stochasticity is typically
independent of N
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Partitioning variance
Species ?2d ?2e Swallow 0.18 0.024 Dippe
r 0.27 0.21 Great tit 0.57 0.079 Brown
bear 0.16 0.003
in large populations N gtgt ?2d /
?2e Environmental stochasticity is more
important Demographic stochasticity can be
ignored Ncrit 10 ?2d / ?2e (approx Ncrit
100)
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Stochasticity and population growth
N0 50 ? 1.03
N Unstable eqm below which popn moves to
extinction
Simulations - ? 1.03, ?2e 0.04, ?2d
1.0 N ?2d /4 r - (?2e /2)
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SUMMARY so far

• 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

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