FW364 Ecological Problem Solving Population Variation October 7, 2013

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

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

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, 1936-1937)

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, 1936-1937)  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

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

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

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

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

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

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)

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)?

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)?

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)

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

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

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

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!

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

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

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

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 ± 0.075 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

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

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

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

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 1947-1964 (regular census during that time) Givens: λ = 1.148 ± 0.075 SD (Ramas assumes normal distribution) d’ = 0.079 s’ = 0.921 Start Ramas

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 1936-1948 than in 1947-1964 from which parameters were derived  Suggested to be due to greater loss of animals on ice floes or hunting

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

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

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

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

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

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