1. Exponential – Geometric - Density Independent Population Growth

2. Exponential/Geometric/Density Independent Population Growth
How do we come by Nhat (and hence N of models)?
Census
Quadrats
Mark-recapture (capture-recapture)
Mark-resight
Distance Sampling
Index

3. Exponential Population Growth
Why model?
Too expensive to directly estimate Nhat every year
Explore consequences of what we believe true
Increase our understanding of a system
Sensitivity analysis (now called elasticity analysis)
Identify weak links in data
Evaluate alternative actions in a relative manner

4. Exponential Population Growth
Population Model Limitations
Projections into future - sketchier the farther you go
Begin to believe
Often a lack of data
Lack of variance estimates

5. Exponential Model Assumptions
Density independence (b and d are constant)
No age, sex, or other class (e.g., size) structure
No environmental (spatial and temporal) variation
No demographic stochasticity
No sampling variation
No genetic structure (no individual heterogeneity)
N represents a perfect census

6. Discrete versus Continuous Models
Discrete Breeders
- Annual breeders - Coyotes, ungulates, many birds
- Birth Pulse - Fish
Continuous Breeders
- Humans, elephants, kangaroos, …

7. Discrete versus Continuous Models
Distinction mathematically relevant only at high growth rates
- r > 0.20
Notation Issues (variety of nomenclature)
r – instantaneous growth rate, continuous breeders, continuous time
R – finite birth rate, discrete time, discrete breeders
l – finite population growth rate, discrete time, discrete breeders
Excel Example

8. Stochasticity, Heterogeneity, and Variation
Variation (math)
Marked difference or deviation from the normal or recognized form, function, or structure.
Stochastic (stat)
Involving or containing a rand variable(s): stochastic calculus.
Involving chance or probability: a stochastic stimulation.
Of, relating to, or characterized by conjecture; conjectural.
Heterogeneous
Consisting of dissimilar elements or parts; not homogeneous.
Completely different; incongruous.

9. Stochasticity, Heterogeneity, and Variation
Variation – spatial, temporal, individual, demographic, sampling
Stochasticity – including variation in a model, usually done as a stochastic process

10. Why Include Stochasticity?
Paraphrase of Renshaw (1991):
Why would we not accept the weight of one Scotchman as
representative of the entire population of Scotland, but will accept the time trace of a population as representative of the population's dynamics?

11. Temporal Variation Annual variation Catastrophes
such as typical annual differences in weather
Catastrophes
such as severe storms or fire 50
100
150
200
250
300
350
1970
1975
1980
1985
1990
1995
Year
Snowfall (cm)
Geo Mean

12. Temporal Variation Annual variation
such as typical annual differences in weather
Low snowfall – high survival
High snowfall – low survival 50
100
150
200
250
300
350
1970
1975
1980
1985
1990
1995
Year
Snowfall (cm)
Geo Mean

13. How important is temporal variation?

14. Modeling Temporal Heterogeneity
If possible, take advantage of historical data to find extremes
Model vial rates (b, d, i, e) or R, l as random variables
Normal or beta distribution commonly used 50
100
150
200
250
300
350
1970
1975
1980
1985
1990
1995
Year
Snowfall (cm)

15. Add Temporal Stochasticity (R)
Parametric Bootstrap
Normal distribution
Beta distribution
Uniform distribution
Non-parametric bootstrap

16. Normal Distribution ~N(0.1,0.08) Data Table Year R 1985 0.031 1986
0.103
1987
0.177
1988
-0.134
1989
-0.060
1990
-0.091
1991
-0.194
1992
0.052
1993
-0.082
1994
0.165
1995
0.091
1996
-0.174
1997
0.110
1998
0.192
1999
-0.051
2000
0.137
2001
0.035
2002
-0.006
2003
-0.172
~N(0.1,0.08)
0.1

17. Uniform Distribution Mean = 0.1 ~Unif(-0.09,0.29) Data Table Year R
1985
0.031
1986
0.103
1987
0.177
1988
-0.134
1989
-0.060
1990
-0.091
1991
-0.194
1992
0.052
1993
-0.082
1994
0.165
1995
0.091
1996
-0.174
1997
0.110
1998
0.192
1999
-0.051
2000
0.137
2001
0.035
2002
-0.006
2003
-0.172
Mean = 0.1
~Unif(-0.09,0.29)
Conservative
0.1

18. Adding Temporal Stochasticity
Uniform distribution
Normal distribution
Beta distribution, poisson …
Non-parametric bootstrap
Which one you choose very important assumption!
Canned programs???
Excel example