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)
The ecology of small populations
Habitat loss Pollution Overexploitation Exotic spp
Small
fragmented
isolated popn’s
Inbreeding
Genetic Variation
Genetic processes
Reduced N
Env variation
Demographic
stochasticity
Catastrophes
Stochastic processes
Outline for this weeks lectures
How do ecological processes impact
small populations?
Stochasticity and population growth
Allee effects and population growth
Demography has four components
Birth (Natality) +
Immigration +
Emigration Population Nt
Death (Mortality) Nt+1 = Nt +B-D+I-E
Exponential population growth
(population well below carrying capacity, continuous
reproduction
closed pop’n)
∆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 pop’n size at time t
e is a constant, base of natural logs
Trajectories of exponential population growth
N
Trend
r>0
r=0
r<0
t
Geometric population growth
(population well below carrying capacity, seasonal reproduction)
Nt+1
∆N
= Nt +B-D+I-E
= 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)
Nt+1
=  Nt
Nt
=  t N0
Let 1+ rd = , the finite rate of increase
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
Trend
Following are equivalent
r>0 >1
r=0 =1
r< 0  < 1
Geometric population growth
(population well below carrying capacity, seasonal reproduction)
Nt+1
Add data
= (1+rdt)
= (1+rdt)
= (1+rdt)
= (1+rdt)
rdt
rdt-1
rdt-2
rdt-3
Nt
(1+rdt-1) Nt-1
(1+rdt-1) (1+rdt-2) Nt-2
(1+rdt-1) (1+rdt-2) (1+rdt-3) Nt-3
= 0.02
= - 0.02
= 0.01
= - 0.01
What is the average growth rate?
Nt-3= 10
Geometric population growth
(population well below carrying capacity, seasonal reproduction)
What is average growth rate?
Arithmetic mean
= (1+0.02) + (1-0.02) + (1+0.01) + (1-0.01) = 1
4
Predict Nt+1 given Nt-3 was 10
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
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
KEYPOINT
Long term growth is determined by the geometric not the
arithmetic mean
Geometric mean is always less than the arithmetic mean
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 0.089
Annual 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
DETERMINISTIC POPULATION GROWTH
For a given No, r or rd and t
The outcome is determined
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 US
Pacific Gray Whales
were delisted in 1994
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
Variance in r = 2e
=
(∑r)2
∑r2 -
N
N
Bad 0
b-d
Good
Population growth
+ environmental stochasticity
Deterministic
1+r= 1.06, 2e = 0
Expected
Ln N
1+r= 1.06, 2e = 0.05
t
Expected rate of increase is r- 2e/2
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
Global climate change (GCC) is expected to
----> changes in mean environmental conditions
----> increases in variance (ie env. stochasticity)
Data from Negev
mean
drought < 41 mm
Pre-GCC
Post-GCC
Mean rainfall is the same BUT
Variance and drought frequency is greater in “post GCC”
Simulating impact on populations via rainfall impact on RS
Variance in rainfall
Low
High
Number of quasi-extinctions
= times pop’n falls below 40
Simulating impact on populations adding impact on survival
CONC’n
Environmental stochasticity can influence extinction risk
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
---> so population sizes can not be predicted
precisely
Demographic stochasticity
Demographic stochasticity
Dusky seaside sparrow
subspecies
non-migratory
salt marshes of southern Florida
decline DDT
flooding habitat for mosquito control
Habitat loss - highway construction
1975
six left
All male
Dec 1990 declared extinct
Extinction rates of birds as a function of
population size over an 80-year period
60
%
Extinction
10 breeding pairs – 39% went extinct
10-100 pairs – 10% went extinct
1000>pairs – none went extinct
*
30
*
*
*
0
*
1
10
100
*
*
1000
*
10,000
Population Size (no. pairs)
Jones and Diamond. 1976. Condor 78:526-549
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
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
Partitioning variance
Species
Swallow
Dipper
Great tit
Brown bear
2d
0.18
0.27
0.57
0.16
2e
0.024
0.21
0.079
0.003
in large populations N >> 2d / 2e
Environmental stochasticity is more important
Demographic stochasticity can be ignored
Ncrit = 10 * 2d / 2e (approx Ncrit = 100)
Stochasticity and population growth
N0= 50
 = 1.03
N*
Unstable eqm
below which pop’n
moves to
extinction
Simulations -  = 1.03, 2e = 0.04, 2d = 1.0
N* =
2d /4
r - (2e /2)
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
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