Populations II: population growth and viability Bio 415/615

Populations II: population growth

and viabilityBio 415/615

Questions1. 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?

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.

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?

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.

Age-structured model

t = 1 t = 2 t = 3 t = 4

N(1) = 10

N(2) = 20

N(3) = 40

N(4) = 80

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

Age-structured trajectory:

Standard population trajectory:

Column of values is a vector.

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

Example: helmeted honeyeater

Endangered bird endemic to Victoria, Australia eucalypt swamps.


Data:

4 years of population censuses

Ages determined for all individuals

Individuals start breeding at age 1


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


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?


3. Create Leslie matrix

Once S and F for all age classes are known, we can predict age distribution of next year

What are rows and columns?• Leslie matrix (L) is a method of matrix multiplication

Look familiar?


Can now use this matrix as a projection matrix to predict age structure given any starting conditions.

Fx

Sx


Fx

Sx

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.

Stable age distribution

But the population overall can be growing, declining, or stable!

A matrix property called the dominant eigenvalue is equal to lambda.

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?

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)

Age vs. stage

Differences?

Age vs. stage

Now individuals can grow, die, reproduce, do nothing

(stasis), or get smaller (retrogression)!

Silene regia

G3 (rare): 20-100 populations

Modeling transitions

seedling

vegetative

small flwr

med flwr

large flwr

Determine life stages


seedling

vegetative

small flwr

med flwr

large flwr

0%

11%

51%

61%

67%

stasis


seedling

vegetative

small flwr

med flwr

large flwr

30%

57%

4%

21%

0%

11%

51%

61%

67%

growth


seedling

vegetative

small flwr

med flwr

large flwr

30%

57%

4%

21%

0%

11%

51%

61%

67%

growth

11%

1%


seedling

vegetative

small flwr

med flwr

large flwr

30%

57%

4%

21%

0%

11%

51%

61%

67%

14%

17%

17%

growth

11%

1%

retrogression


seedling

vegetative

small flwr

med flwr

large flwr

30%

57%

4%

21%

0%

11%

51%

61%

67%

14%

17%

17%retrogression

growth

11%

1%

fecundity

5.3 12.7

30.9


= Projection matrix


= Projection matrix

Simulation: can be deterministic or stochastic


= Projection matrix

Simulation: can be deterministic or stochastic

Deterministic: result does not depend on initial conditions


= 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


seedling

vegetative

small flwr

med flwr

large flwr

30%

57%

4%

21%

0%

11%

51%

61%

67%

14%

17%

17%

growth

11%

1%

fecundity

5.3 12.7

30.9

retrogression

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

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

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


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.


Is demographic stochasticity more important at high or low population sizes? Why?

P(extinction) = (d/b)^No


= Projection matrix

How could we make this projection model stochastic?

- Choose years randomly

- Choose parameters from a sampling distribution

Is a population sustainable, and how is it to be sustained?

Several objectives:

1.Targeting research: which factors are most relevant to extinction probability?

2. Assessing vulnerability: which populations or species are of immediate conservation concern?

3. How to manage rarity: what are the relative effects of reintroduction, translocation, weed control, burning, captive breeding, etc?

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%

Population viability analysis (PVA)

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

“How large must a grizzly bear population be to remain viable, unthreatened, and naturally regulated?”

i.e. persist

i.e. in the face of environmental and demography stochasticity

i.e. exceed the official threatened standard of <100 bears

Survival by age

Fecundity by ageAge of first and last birth

maternity

Data: demographic parameters for different bear populations

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?

Mean survival by age

Resulting Leslie matrix

Mean fecundity by age

PVA results from matrix

Threatened!

Viable

Examples of Minimum Viable Populations –

Bighorn Sheep in the southwestern U.S.

Examples of Minimum Viable Populations –

Channel Island Birds (California)

Four threats to small populations:

1.Loss of genetic variability

2.Demographic variation

3.Environmental variation

4.Natural catastrophes

Documents

Populations II: population growth and viability Bio 415/615