1. Populations II: population growth and viability
Bio 415/615

2. Questions
1. What are two objectives of population viability analysis (PVA)?
2. In a population projection matrix, what do the row and column values represent?
3. In a stage-structured model, what else can happen to an individual besides growing, dying, and reproducing?
4. What is the parameter lambda (λ), and how does it relate to PVA?
5. How do stochastic and deterministic models differ?

3. Managing rare species
Those tasked with protecting rare species are primarily concerned with whether populations are VIABLE—will they persist long into the future?
We can use models of how populations grow (births and deaths) to determine population viability.

4. Assumption of equal individuals
How have we defined population growth so far?
Births (fecundity) and Deaths (survival)
fast growing populations can have lots of births, or few deaths
What did we assume about the chance of births and deaths among individuals?
Constant.
Can we adequately model populations this way?
Can we manage populations this way?

5. More detail needed: accounting for age
In our logistic and exponential models, we considered b and d for all individuals simultaneously.
Now we assign b and d to specific age classes. Instead of per capita birth and death rates, we term these
FECUNDITY: average number of offspring per individual of certain age in one time interval
SURVIVAL: probability of moving from one age to the next
These are akin to births per age class and the inverse of deaths per age class.

6. Age-structured model Standard population trajectory: N(1) = 10
Age-structured trajectory:
N0(1) = 5
N1(1) = 2
N2(1) = 2
N3(1) = 1
N0(2) = 10
N1(2) = 4
N2(2) = 4
N3(2) = 2
N0(3) = 20
N1(3) = 8
N2(3) = 8
N3(3) = 4
N0(4) = 40
N1(4) = 16
N2(4) = 16
N3(4) = 8
Column of values is a vector.

7. Still have assumptions
Individuals of same age are identical
Closed population (no dispersal or emigration)
Equal sex ratio (esp. for fecundity)
Some assumptions that can be relaxed:
Density dependence (how do survival and fecundity parameters depend on density?) Often ignored with rare populations.
Stochasticity: demographic and environmental
Often added for PVA

8. Example: helmeted honeyeater
Endangered bird endemic to Victoria, Australia eucalypt swamps.

9. Example: helmeted honeyeater
Data:
4 years of population censuses
Ages determined for all individuals
Individuals start breeding at age 1

10. Example: helmeted honeyeater
1. Calculate survival
Sx is the proportion of individuals in one year that make it to the next
Can use weighted mean for the overall Sx , and variance

11. Example: helmeted honeyeater
2. Calculate fecundities
Can calculate Fx as the offspring produced in the next year, divided by all reproductive individuals
Can we get other information to calculate age-specific fecundities?

12. Example: helmeted honeyeater
3. Create Leslie matrix
What are rows and columns?
Leslie matrix (L) is a method of matrix multiplication
Once S and F for all age classes are known, we can predict age distribution of next year
Look familiar?

13. Example: helmeted honeyeaterFx Sx
Can now use this matrix as a projection matrix to predict age structure given any starting conditions.

14. Example: helmeted honeyeaterFx Sx

15. Stable age distribution
If the matrix elements stay the same, then regardless of the initial age structure, the proportion of species of each age will equilibrate.

16. Stable age distribution
But the population overall can be growing, declining, or stable!
A matrix property called the dominant eigenvalue is equal to lambda.

17. Is age always the most critical difference between individuals?
Age versus size:
Modular organisms (plants)
Age is sometimes difficult to measure, but size is relatively easy
If reproduction or survival is based more on environmental circumstances, then age may be a poor correlate of vital rates (eg, trees in an understory)
Can we construct similar models using stages, rather than ages?

18. Stage (size)-structured models
Demographic (vital) rates best described by size or life stage
Other assumptions still apply (density independence, stochasticity, within-class equality, etc)

19. Age vs. stage
Differences?

20. Age vs. stage
Now individuals can grow, die, reproduce, do nothing (stasis), or get smaller (retrogression)!

21. Silene regia
G3 (rare): populations

22. Modeling transitions Determine life stages seedling vegetative
small flwr
med flwr
large flwr

23. Modeling transitions stasis seedling 0% vegetative small flwr 11%
med flwr
51%
stasis
61%
large flwr
67%

24. Modeling transitions growth seedling 0% vegetative small flwr 11%
med flwr
51%
30%
57%
61%
large flwr
growth
21%
67% 4%

25. Modeling transitions growth seedling 0% vegetative small flwr 11%
med flwr
51%
30%
57%
61%
large flwr
growth
21%
67%
11% 4% 1%

26. Modeling transitions retrogression growth seedling 17% 0% vegetative
small flwr
14%
11%
med flwr
51%
30%
17%
57%
61%
large flwr
growth
21%
67%
11% 4% 1%

27. Modeling transitions fecundity retrogression growth seedling 5.3 12.7
17% 0%
vegetative
small flwr
30.9
14%
11%
med flwr
51%
30%
17%
57%
61%
large flwr
growth
21%
67%
11% 4% 1%

28. Modeling transitions
= Projection matrix

29. Modeling transitions = Projection matrix
Simulation: can be deterministic or stochastic

30. Modeling transitions = Projection matrix
Simulation: can be deterministic or stochastic
Deterministic: result does not depend on initial conditions

31. Modeling transitions = Projection matrix 1. Define initial conditions
Seedlings: 500
Vegetative: 100
Small flowering: 100
Med flowering: 100
Large flowering: 100
2. Iterate based on transition probabilities

32. Modeling transitions fecundity retrogression growth seedling 5.3 12.7
17% 0%
vegetative
small flwr
30.9
14%
11%
med flwr
51%
30%
17%
57%
61%
large flwr
growth
21%
67%
11% 4% 1%

33. Population stochasticity
What is stochasticity?
Deterministic processes leave nothing to chance
Stochastic models are, to some extent, unpredictable
Why do we model stochasticity?
Because even though the expectation might not change, outcomes can depend on amount of uncertainty

34. Types of population stochasticity
Environmental stochasticity
Births and deaths depend on the environment in a known way, but the environment is itself unpredictable
Demographic stochasticity
Order of births and deaths may fluctuate, even if the rate is generally constant

35. Stochastic parameters: mean and variance
Mean is the expected value; would be the ‘typical’ outcome if you repeated the process many times
Variance describes how unpredictable the expected outcome is

36. Stochastic parameters: mean and variance
The outcome of stochastic population change depends on both the expected pattern (mean) and the amount of uncertainty involved (variance)!
Eg, if the variance is twice as great as the expected (mean) value of r, extinction is very likely.

37. Stochastic parameters: mean and variance
Is demographic stochasticity more important at high or low population sizes? Why?
P(extinction) = (d/b)^No

38. Modeling transitions = Projection matrix
How could we make this projection model stochastic?
Choose years randomly
Choose parameters from a sampling distribution

39. Population viability analysis (PVA)
Is a population sustainable, and how is it to be sustained?
Several objectives:
Targeting research: which factors are most relevant to extinction probability?
Assessing vulnerability: which populations or species are of immediate conservation concern?
How to manage rarity: what are the relative effects of reintroduction, translocation, weed control, burning, captive breeding, etc?

40. Population viability analysis (PVA)
What is a viable population?
A population that can perpetuate itself
Minimum viable population –
estimate of the # of individuals needed to perpetuate a population:
1) For a given length of time, e.g. for 100 or 1000 years
2) With a specified level of certainty, e.g. 95% or 99%

41. Minimum Viable Population – Schaffer (1981)
Formal Definition:
For any given species in any given habitat the MVP is,
the smallest isolated population having a 99% chance of remaining extant for 1000 years despite the foreseeable effects of demographic, environmental, and genetic stochasticity, and natural catastrophes

42. “How large must a grizzly bear population be to remain viable, unthreatened, and naturally regulated?”
i.e. persist
i.e. exceed the official threatened standard of <100 bears
i.e. in the face of environmental and demography stochasticity

43. Data: demographic parameters for different bear populations
Survival by age
maternity
Fecundity by age
Age of first and last birth
Lambda (λ)
What is interesting about mast and non-mast years?
This is a measure of variability between sites. Why is this important? How is it used?

44. Resulting Leslie matrix
Mean fecundity by age
Mean survival by age
PVA results from matrix
Threatened!
Viable

45. Examples of Minimum Viable Populations –
Bighorn Sheep in the southwestern U.S.

46. Examples of Minimum Viable Populations –
Channel Island Birds (California)

47. Four threats to small populations:
Loss of genetic variability
Demographic variation
Environmental variation
Natural catastrophes