1. FW364 Ecological Problem Solving
Population Variation
October 7, 2013

2. Continue to make population growth models more realistic
Outline for Today
Continue to make population growth models more realistic
Before the midterm:
Introduced the concept of stochastic variation
Discussed demographic stochasticity
Objectives for Today:
Introduce risk curves
Discuss environmental stochasticity
Text (optional reading):
Chapter 2

3. Stochastic Variation
Two types of stochastic variation that we can include in models:
Demographic stochasticity
Environmental stochasticity
Demographic stochasticity:
Variation that arises as a consequence of working with individuals
Strongest in small populations
Environmental stochasticity:
Unpredictable change in the environment in time and space
Often changes due to weather

4. Demographic Stochasticity Example
Example from Textbook: Muskox population growth
Addresses how demographic stochasticity affects our conclusions regarding muskox population size between consecutive time periods
31 muskox in 1936, with λ = 1.15
N1937 = N1936 λ
N1937 = 31 * 1.15
N1937 = 35.7 muskox
With deterministic equation, Nt+1 = Nt λ, we predict:
In textbook,
authors ran replicate trials of the model with demographic stochasticity
(note: just one time step in this model, )

5. Demographic Stochasticity Example
Example from Textbook: Muskox population growth
sum
new population size
reproduce?
yes no
stop
survive?
Population:
In textbook,
authors ran replicate trials of the model with demographic stochasticity
(note: just one time step in this model, )
 Used the stochastic simulation method we discussed on Monday
Completed four sets of trials: 10, 100, 1,000, and 10,000 trials
using same starting population size (N1936) and λ for each trial

6. Demographic Stochasticity Example
Each set of trials represents a distribution of outcomes (i.e., population sizes in 1937),
for which a mean and variation can be determined
The more trials that are run, the more the distribution looks like a bell-shaped curve
The greater the number of trials,
the more confidence we can have in
the mean and variation of outcomes
Fig. 2.1

7. Demographic Stochasticity Example
Most common population size:
35 muskox
Lowest population size
(occurred in set of 10,000 trials):
26 muskox
Highest population size
(occurred in set of 10,000 trials):
47 muskox
Fig. 2.1

8. Demographic Stochasticity Example Important Application:
Can use these data to predict chance (risk) of a population reaching certain critical sizes (Nc) in the future
We’ll use just the best data
(simulation with 10,000 trials)
to build a table of the number of trials (frequency) that resulted in each population size
Fig. 2.1

9. Risk Table
What is the chance the population size will be 31 muskox or smaller in 1937?
Population sizes (x-axis)
Height of bars
(frequency; y-axis)
Summing down previous column
Previous column divided by 10,000 trials

10. Although mean N1937 = 35 muskox population size will not grow
Risk Table
What is the chance the population size will be 31 muskox or smaller in 1937?
Although mean N1937 = 35 muskox
there is 10% chance that
population size will not grow
(N1937 ≤ 31 muskox)
Nc = 31 muskox
Nc=31 indicates concern that population will not grow
Population sizes (x-axis)
Height of bars
(frequency; y-axis)
Summing down previous column
Previous column divided by 10,000 trials

11. Risk Table
What is the chance the population size will be 31 muskox or smaller in 1937?
Last column is a “cumulative probability distribution”
Can be plotted as a “risk curve”
Plot: Probability (y-axis) vs. Population size (x-axis)

12. Risk Curve - Extinction
“Risk curve” (a.k.a. “Quasi-extinction curve”)
Probability of decline to Nc
(Nc)
Useful for answering the question:
What is the probability the population will fall below some threshold (Nc)?

13. Risk Curve - Extinction
“Risk curve” (a.k.a. “Quasi-extinction curve”)
To use:
Pick a threshold size (x-axis)
Go up until contact curve
y-axis gives probability that the population will reach threshold size or less
Demographic stochasticity is more severe for small population sizes!
Probability of decline to Nc
(Nc)
Useful for endangered species management:
Often interested in predicting the risk that the population size will fall below a certain threshold that corresponds to minimum viable population size
Useful for answering the question:
What is the probability the population will fall below some threshold (Nc)?

14. Risk Curve - Extinction
Two Important Notes on Risk Curves:
Our risk curve example involved demographic stochasticity
Risk curves can include environmental stochasticity as well
Our risk curve example was for extinction risk (e.g., endangered species)
 Can also build explosion risk curves (e.g., pest species)

15. Stochastic Variation
Two types of stochastic variation that we can include in models:
Demographic stochasticity
Environmental stochasticity
Demographic stochasticity:
Variation that arises as a consequence of working with individuals
Strongest in small populations
Environmental stochasticity:
Unpredictable change in the environment in time and space
Often changes due to weather

16. Environmental Stochasticity
Key distinction between demographic and environmental stochasticity:
Environmental stochasticity affects all populations regardless of size
The vital rates (population parameters: b’, d’, λ) vary randomly
(population parameters did not vary for demographic stochasticity – remember harbor seal game)
Example from Nature:
Snowfall changes from year-to-year (environmental variability),
which somehow affects b’ and/or d’ each year

17. Environmental Stochasticity
Key distinction between demographic and environmental stochasticity:
Environmental stochasticity affects all populations regardless of size
The vital rates (population parameters: b’, d’, λ) vary randomly
(population parameters did not vary for demographic stochasticity – remember harbor seal game)
How to incorporate environmental stochasticity:
Rather than just use the average λ for every time period in our models,
we include variation in λ by time period

18. Environmental Stochasticity
Deterministic model
(λ does not vary):
Nt+1 = Nt λ
Stochastic model
(λ varies each time step: λt):
Nt+1 = Nt λt
λt indicates that λ varies each time step (e.g., year-to-year)
i.e., a different value of λ is used each time step
By using different λs,
we build in variability in birth and death rates each time step
How do we know which λ values to use (i.e., how much should λ vary)?
 Use range of values seen in population during previous years
Let’s look at an example!

19. Environmental Stochasticity
Sufficient data on muskox to build a distribution of λ
Can use the distribution to constrain our selection of λt each year
Fig 1.4 λ
Fig 2.5 λ

20. Environmental Stochasticity
Sufficient data on muskox to build a distribution of λ
Can use the distribution to constrain our selection of λt each year
These “historic” data are our guide for muskox λ in the future
Fig 2.5
In a model with environmental stochasticity, there are two parameters that determine λ for each time step (λt):
average λ
standard deviation of λ
Nt+1 = Nt ( λ ± errort )
Can revise model from:
Nt+1 = Nt λt
to:
The error term varies randomly (but within bounds) each time step

21. Let’s illustrate with an activity!
Environmental Stochasticity Activity
Let’s illustrate with an activity!
You are all Alaskan snowstorms
Chance of muskox starvation is dependent on winter conditions:
During mild winters, survival is high (low d’)
During harsh winters, survival is low (high d’)
Mild winter:
greater λ
Average winter:
close to average λ
Harsh winter:
low λ

22. Environmental Stochasticity Activity
Rules:
We want to forecast muskox population growth between 1968 and 1978 taking potential environmental variation into consideration
Bowl contains a distribution of λs for muskox: λ ± errort
based on 1947 to 1964 data (λ = 1.15 ± SD)
Pass around bowl of λs
Randomly pick one λ from the bowl (λt)
Read the λ out loud
Replace λ in bowl (a key part of simulation)
Pass bowl to next person
We will do three trials, each with 10 time steps
Each time step will have a random λ, i.e., λt
I will record all data in Excel

23. Environmental Stochasticity Activity
Post-game explanation:
All trials used the same distribution of λs (same mean and SD)
However, actual λ for each year (λt) varied due to random selection
 Yielded trials that had different trajectories
Average λ for each trial was similar to λ ( = 1.15)
Different trials yielded variation in final population size (N1978)
Could conduct many trials ( > 1,000) and calculate the average final population size (N1978) and variance (σ2) of final population size
 Would give an indication of likely final population size in future,
given possible environmental variation
 Average final population size should be similar to deterministic outcome

24. Environmental Stochasticity Activity  Variance would increase
Post-game explanation:
Could conduct many trials (often <1,000) and calculate the average final population size (N1978) and variance (σ2) of final population size
 Would give an indication of likely final population size in future,
given some expected environmental variation
 Average final population size should be similar to deterministic outcome
Challenge:
What would happen to variance (σ2) as we predict further and further into the future?
 Variance would increase

25. Sources of Stochastic Variation - Comparison
Demographic stochasticity
Variation due to differences among individuals
The population parameters (b’, d’, λ) do not vary each time step,
Variation arises from random chance involved in the reproduction or
death of individuals
Simulated by random chance of reproduction or death of individuals
Has strongest effect on small population sizes
Environmental stochasticity
Variation due to environmental influence that affects entire population
The population parameters (b’, d’, λ) vary each time step
Variation arises from the environment causing unpredictable increases
or decreases in average growth rates for population
Simulated by random choice of population parameters for each time step
Affects all populations, regardless of size

26. RAMAS Example
Let’s do another example using Ramas
Software allows us to do many trials very quickly
Easy to tweak population parameters (b’, d’, λ)
Can include both demographic and environmental stochasticity
Ramas Example:
Forecast muskox population size between 1936 and 1948 (12 year duration)
We will include demographic and environmental stochasticity
Will run 1,000 replicate trials
Use average λ ± SD from (regular census during that time)
Givens: λ = ± SD (Ramas assumes normal distribution)
d’ = s’ = 0.921
Start Ramas

27. RAMAS Example
Dashed line:
Average trajectory of model trials
Vertical lines:
1 SD above and below the mean trajectory
Triangles:
Max and min of all trials
Black dots:
Observed data – Used to compare model output
Fig. 2.6
Actual population sizes in 1947 and 1948 were pretty unusual
 Population growth rate was lower in than in
from which parameters were derived
 Suggested to be due to greater loss of animals on ice floes or hunting

28. Could just be due to demographic and/or environmental stochasticity:
RAMAS Example
Dashed line:
Average trajectory of model trials
Vertical lines:
1 SD above and below the mean trajectory
Triangles:
Max and min of all trials
Black dots:
Observed data – Used to compare model output
Fig. 2.6
Actual population sizes in 1947 and 1948 were pretty unusual
Could just be due to demographic and/or environmental stochasticity:
Actual data still within range of possibilities based on model predictions

29. RAMAS Example Like before, we can construct risk curves
Previously calculated using consecutive time step model as:
cumulative number of
trials reaching Nc
total number of trials
Instead, we will now use min and max of trials
Dashed line:
Average trajectory of model trials
Vertical lines:
1 SD above and below the mean trajectory
Triangles:
Max and min of all trials
Black dots:
Observed data – Used to compare model output
Fig. 2.6
Actual population sizes in 1947 and 1948 were pretty unusual
Could just be random:
Actual data still within range of possibilities based on model predictions

30. Risk Curves Extinction risk curve
Defines chance of falling below a threshold value at any time during the next 12 years (the range of our forecast)
Calculated based on the minimum population size for each trial
Used for endangered species management
Fig. 2.7
Based on the frequency each minimum appeared

31. Risk Curves Extinction risk curve
Defines chance of falling below a threshold value at any time during the next 12 years (the range of our forecast)
Calculated based on the minimum population size for each trial
Used for endangered species management
Explosion risk curve
Defines chance of exploding above a threshold value at any time during the next
12 years (the range of our forecast)
Calculated based on the maximum population size for each trial
Used for pest management (invasive species)
Gypsy moth larvae
Fig. 2.7

32. Deterministic vs. Stochastic Models
Deterministic models:
Nt+1 = Nt λ
Stochastic models:
Nt+1 = Nt λt
Nt+1 = Nt ( λ ± errort )
λ < 1 population will go extinct
λ > 1 population will not go extinct
λ < 1 population will go extinct (eventually)
λ > 1 population might go extinct
Easy to conceptualize & use,
but less realistic
More difficult to conceptualize & use,
but more realistic
We will use stochastic models (most of) the remainder of course

33. Looking Ahead This Week: Lab Today (Section 2):
Wednesday: Introduce population regulation
Lab Today (Section 2):