Exponential – Geometric - Density Independent Population Growth

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

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

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

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

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

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

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.

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

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?

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

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

How important is temporal variation?

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)

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

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

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

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