Background knowledge expected Population growth models ...
Transcript of Background knowledge expected Population growth models ...
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)
Habitat loss Pollution Overexploitation Exotic spp
Small fragmented isolated popn’s
Inbreeding Genetic Variation Reduced N Demographic
stochasticity
Env variation
Catastrophes Genetic processes
Stochastic processes
The ecology of small populations
How do ecological processes impact small populations?
Stochasticity and population growth
Allee effects and population growth
Outline for this weeks lectures
Immigration + Emigration -
Birth (Natality) +
Death (Mortality) -
Nt+1 = Nt +B-D+I-E
Population Nt
Demography has four components
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 increase dt
!t
!N
Cumulative change in population Nt = N0e
rt
N0 initial popn size,
Nt pop’n size at time t
e is a constant, base of natural logs
Trajectories of exponential population growth
r > 0 r = 0 r < 0
N
t
Trend
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
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
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 data Nt-3= 10
rdt = 0.02 rdt-1 = - 0.02 rdt-2 = 0.01 rdt-3 = - 0.01
What is the average growth rate?
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) = 1 4
Arithmetic mean
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
KEYPOINT Long term growth is determined by the geometric not the arithmetic mean Geometric 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
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
Mean r
\
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
b-d Bad 0 Good
Variance in r = "2e = "r2 -
("r)2
N N
Population growth + environmental stochasticity
Ln N
t
Deterministic 1+r= 1.06, "2
e = 0
1+r= 1.06, "2e = 0.05
Expected
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)
mean drought < 41 mm
Pre-GCC Post-GCC Mean rainfall is the same BUT
Variance and drought frequency is greater in “post GCC”
Data from Negev
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!
0
30
60
1 10 100 1000 10,000
* * *
* *
* * *
10 breeding pairs – 39% went extinct
10-100 pairs – 10% went extinct
1000>pairs – none went extinct
*
Population Size (no. pairs)
%
Extinction
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
Environmental stochasticity is typically independent of N
Long term data from Great tits in Whytham Wood, UK
Partitioning variance
Species "2d "
2e
Swallow 0.18 0.024 Dipper 0.27 0.21 Great tit 0.57 0.079 Brown bear 0.16 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
Simulations - ! = 1.03, "2e = 0.04, "
2d = 1.0
N* = "2d /4
r - ("2e /2)
N* Unstable eqm below which pop’n moves to
extinction
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
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