Lecture 4

Lecture 4Stochasticity

WILD3810 (Spring 2021)1 / 36

Readings

Mills 84-92

2 / 36

Assumptions of the B-D modelsRemember from the lecture 3 that our simple models of population growth were based on the following assumptions:3 / 36

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

1) Population closed to immigration and emigration

3 / 36

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

1) Population closed to immigration and emigration

2) Model pertains to only the limiting sex, usually females

3 / 36

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

1) Population closed to immigration and emigration

2) Model pertains to only the limiting sex, usually females

3) Birth and death rates are independent of an individual’s age or biological stage

3 / 36

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

1) Population closed to immigration and emigration

2) Model pertains to only the limiting sex, usually females

3) Birth and death rates are independent of an individual’s age or biological stage

4) Birth and death rates are constant $^1$

3 / 36

$^1$ Do not vary over time or space, across individuals

Assumptions of the B-D modelsIn reality, we know that assumption 4 is almost never true4 / 36

Assumptions of the B-D modelsIn reality, we know that assumption 4 is almost never trueBirth and death rates are determined by many external forces4 / 36

Assumptions of the B-D models

In reality, we know that assumption 4 is almost never true

Birth and death rates are determined by many external forces

These forces often vary across space, time, and individuals:

body condition
temperature
drought
fire

4 / 36

Assumptions of the B-D models

In reality, we know that assumption 4 is almost never true

Birth and death rates are determined by many external forces

These forces often vary across space, time, and individuals:

body condition
temperature
drought
fire

The B-D population model ignored all of these sources of variation $^2$

4 / 36

$^2$ As we'll see, this is not only unrealistic, ignoring variation can change our conclusions about population growth.

Stochasticity5 / 36

StochasticityIn many cases, we do not know exactly why populations go up or down from year-to-year6 / 36

StochasticityIn many cases, we do not know exactly why populations go up or down from year-to-yearOften don't know what these causes are so from our perspective, the variation appears to be random6 / 36

StochasticityIn many cases, we do not know exactly why populations go up or down from year-to-yearOften don't know what these causes are so from our perspective, the variation appears to be randomProcesses that are governed by some element of chance (randomness) are referred to as stochastic processes6 / 36

Stochasticity

In many cases, we do not know exactly why populations go up or down from year-to-year

Often don't know what these causes are so from our perspective, the variation appears to be random

Processes that are governed by some element of chance (randomness) are referred to as stochastic processes

Just because something is stochastic doesn't mean it's completely unpredictable $^3$

6 / 36

$^3$ For example, we can't predict with 100% certainty whether a coin flip will be heads or tails or what side a 6-sided dice will land on (there is some stochasticity in these processes) but that doesn't mean we have no idea what the outcomes will be

StochasticityWe can predict how often we expect different random outcomes using probability7 / 36

Stochasticity

We can predict how often we expect different random outcomes using probability

Probability distributions characterize the frequency of different outcomes $^4$

7 / 36

Stochasticity

We can predict how often we expect different random outcomes using probability

Probability distributions characterize the frequency of different outcomes $^4$

In population models, we can use probability to characterize and account for stochastic processes that effect population growth

7 / 36

$^4$ For example, many of you are probably familiar with the normal (or bell-shaped) distribution. This is a probability distribution with a single "most probable" outcome and decreasing probability of more "extreme" values.

Two types of stochastic processesWith regards to population models, we generally distinguish between two types of stochasticity:8 / 36

Two types of stochastic processes

With regards to population models, we generally distinguish between two types of stochasticity:

1) Environmental stochasticity

variation in the mean demographic parameters and population growth that occurs due to random changes in environmental conditions

8 / 36

Two types of stochastic processes

With regards to population models, we generally distinguish between two types of stochasticity:

1) Environmental stochasticity

variation in the mean demographic parameters and population growth that occurs due to random changes in environmental conditions

2) Demographic stochasticity

variability in demographic parameters and population growth that arises from random outcomes among individual survival and reproductive fates due to random chance alone

8 / 36

Environmental stochasticity9 / 36

Environmental stochasticity

The expected value of demographic parameters can fluctuate over time in response to:

rainfall
temperature
fires and disturbance
competitors
predators
pathogens

10 / 36

Environmental stochasticity

If environmental attributes change stochastically over time, so will demographic vital rates and population growth rate

11 / 36

Modeling environmental stochasticity

If we want to take into account changes in demographic parameters as environmental conditions change, we need time-specific demographic parameters:

$\huge N_{t+1} = N_t \times (1 + b_t - d_t)$

12 / 36

Modeling environmental stochasticity

If we want to take into account changes in demographic parameters as environmental conditions change, we need time-specific demographic parameters:

$\huge N_{t+1} = N_t \times (1 + b_t - d_t)$ $\huge N_{t+1} = N_t \times \lambda_t$

12 / 36

Modeling environmental stochasticity

If we want to take into account changes in demographic parameters as environmental conditions change, we need time-specific demographic parameters:

$\huge N_{t+1} = N_t \times (1 + b_t - d_t)$ $\huge N_{t+1} = N_t \times \lambda_t$

Over $T$ years, abundance will be:

$\huge N_{T} = N_0 \times \big(\lambda_0 \times \lambda_1 \times \lambda_2 \times...\lambda_{T-1}\big)$

12 / 36

Modeling environmental stochasticity

For example, if $N_0 = 50$ and:

$\large \lambda_0 = 1.1$

$\large \lambda_1 = 0.9$

$\large \lambda_2 = 0.7$

$\large \lambda_3 = 1.2$

$\large \lambda_4 = 1.1$

then:

13 / 36

Modeling environmental stochasticity

For example, if $N_0 = 50$ and:

$\large \lambda_0 = 1.1$

$\large \lambda_1 = 0.9$

$\large \lambda_2 = 0.7$

$\large \lambda_3 = 1.2$

$\large \lambda_4 = 1.1$

then:

$\large N_{5} = 50 \times 1.1 \times 0.9 \times 0.7 \times 1.2 \times 1.1 = 46$

13 / 36

Modeling environmental stochasticityIt's clear that this population is decliningthe average value of λ must be <1  
14 / 36

Modeling environmental stochasticityIt's clear that this population is decliningthe average value of λ must be <1  
What is the average λ (which we'll call ˉλ)?14 / 36

Modeling environmental stochasticity

An obvious way to calculate $\large \bar{\lambda}$ is the arithmetic mean of the annual $\large \lambda$ 's:

$\Large \bar{\lambda} = \frac{\lambda_0+ \lambda_1 + \lambda_2 +...\lambda_{T-1}}{T}$

15 / 36

Modeling environmental stochasticity

An obvious way to calculate $\large \bar{\lambda}$ is the arithmetic mean of the annual $\large \lambda$ 's:

$\Large \bar{\lambda} = \frac{\lambda_0+ \lambda_1 + \lambda_2 +...\lambda_{T-1}}{T}$

In our example, this equals:

$\Large \bar{\lambda} = \frac{1.1 + 0.9 + 0.7 + 1.2 + 1.1}{5} = \frac{5}{5} = 1$

15 / 36

Modeling environmental stochasticity

But that can't be right!

$\large \bar{\lambda} = 1$ means the population should have, on average, remained at 50 individuals

16 / 36

Modeling environmental stochasticity

But that can't be right!

$\large \bar{\lambda} = 1$ means the population should have, on average, remained at 50 individuals

The issue that the population growth is a multiplicative process

This means that shrinking by 30% $(\large \lambda = 0.7)$ and then growing by 30% $(\large \lambda = 1.3)$ does not get you back to where you started

For example, $100 \times 0.7 = 70$ but $70 \times 1.3 = 91$ .

16 / 36

Modeling environmental stochasticity

To estimate the average of a multiplicative process, we need to take the geometric mean rather than the arithmetic mean:

$\Large \bar{\lambda} = \big(\lambda_0 \times \lambda_1 \times \lambda_2 \times...\lambda_{T-1}\big)^\frac{1}{T}$

17 / 36

Modeling environmental stochasticity

To estimate the average of a multiplicative process, we need to take the geometric mean rather than the arithmetic mean:

$\Large \bar{\lambda} = \big(\lambda_0 \times \lambda_1 \times \lambda_2 \times...\lambda_{T-1}\big)^\frac{1}{T}$

For our population, that means:

$\Large \bar{\lambda} = \big(1.1 \times 0.9 \times 0.7 \times 1.2 \times 1.1\big)^\frac{1}{5} = 0.98$

17 / 36

Modeling environmental stochasticityAs you can see, the geometric mean < arithmetic mean18 / 36

Modeling environmental stochasticityAs you can see, the geometric mean < arithmetic meanThis is a critical point about environmental stochasticity18 / 36

Modeling environmental stochasticity

As you can see, the geometric mean < arithmetic mean

This is a critical point about environmental stochasticity

Populations with variable growth rates will tend to grow more slowly (or decrease faster) than populations in constant environments even if their mean vital rates are the same.

18 / 36

Modeling environmental stochasticityLet's see what that looks like:19 / 36

Modeling environmental stochasticity

Let's see what that looks like:

Initially, assume a large population $\large (N_0 = 500)$ with:

$\bar{b_t} = 0.55$
$\bar{d_t} = 0.50$
and neither parameter varies over time

Thus:

$\Large \bar{\lambda} = (1 + \bar{b_t} - \bar{d_t}) = (1 + 0.55 - 0.5) = 1.05$

19 / 36

Modeling environmental stochasticity

Over 100 years, the population growth will be:

20 / 36

Modeling environmental stochasticity

Now let's look at the dynamics of a second population with the same starting population size, the same mean demographic rates but with annual variation of 20%

21 / 36

Modeling environmental stochasticity

Now let's look at the dynamics of a second population with the same starting population size, the same mean demographic rates but with annual variation of 20%
Notice that the mean rates are the same so intuitively,

$\Large \bar{\lambda} = (1 + \bar{b_t} - \bar{d_t}) = (1 + 0.55 - 0.5) = 1.05$

21 / 36

Modeling environmental stochasticity

$\Large \bar{\lambda} = (1 + \bar{b_t} - \bar{d_t}) = (1 + 0.55 - 0.5) = 1.05$

21 / 36

Modeling environmental stochasticity

Finally, let's add a third population with the same starting population size, the same mean demographic rates but now with annual variation of 40%.

22 / 36

Modeling environmental stochasticityBecause environmental stochasticity tends to reduce ˉλ regardless of population size, it has important consequences for the extinction risk of both small and large populationsWe'll explore this idea more in lab23 / 36

Demographic stochasticity24 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variables25 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variablesIf the mean survival probability in a given year is 80%, then exactly 80% of the population survived and 20% died  
25 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variablesIf the mean survival probability in a given year is 80%, then exactly 80% of the population survived and 20% died  
This is not a realistic assumptionImagine flipping a coin 10 times:25 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variablesIf the mean survival probability in a given year is 80%, then exactly 80% of the population survived and 20% died  
This is not a realistic assumptionImagine flipping a coin 10 times:we know that the probability of getting a heads is 50% 
25 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variablesIf the mean survival probability in a given year is 80%, then exactly 80% of the population survived and 20% died  
This is not a realistic assumptionImagine flipping a coin 10 times:we know that the probability of getting a heads is 50% 
but we would not be surprised to get 4 heads. Or 7  
25 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variablesIf the mean survival probability in a given year is 80%, then exactly 80% of the population survived and 20% died  
This is not a realistic assumptionImagine flipping a coin 10 times:we know that the probability of getting a heads is 50% 
but we would not be surprised to get 4 heads. Or 7  
we wouldn't even be that surprised to get 2 or 9  
25 / 36

Demographic  stochasticityIn the preceding examples, we treated the demographic outcomes (individual survival and reproductive success) at each time step as non-random variablesIf the mean survival probability in a given year is 80%, then exactly 80% of the population survived and 20% died  
This is not a realistic assumptionImagine flipping a coin 10 times:we know that the probability of getting a heads is 50% 
but we would not be surprised to get 4 heads. Or 7  
we wouldn't even be that surprised to get 2 or 9  
This is because the outcome of the coin flip is a random variable - the outcome is governed by chance25 / 36

Demographic  stochasticityThe same is true to demographic outcomes26 / 36

Demographic  stochasticityThe same is true to demographic outcomesEven if the expected survival rate is 80%, the realized survival rate could be higher or lowerLikewise, we might expect individuals to produce 3 offspring on average, but some individuals will have more and some fewer based on chance 526 / 36

Demographic  stochasticityThe same is true to demographic outcomesEven if the expected survival rate is 80%, the realized survival rate could be higher or lowerLikewise, we might expect individuals to produce 3 offspring on average, but some individuals will have more and some fewer based on chance 5Demographic stochasticity is essentially the difference between the expected survival/reproductive rate and the realized rates in our population.26 / 36

$^5$ or at least what appears to use to be chance

Demographic  stochasticityDemographic  stochasticity occurs because our population is a finite sample27 / 36

Demographic  stochasticityDemographic  stochasticity occurs because our population is a finite sampleGoing back to the coin flip, if we flip the coin 1000 times, we would expect to get pretty close to 50% headsBut as the number of flips gets smaller, the realized success rate could differ more from the expected value27 / 36

Effects of demographic stochasticityDemographic stochasticity is mainly an issue at small population sizes because each individual is a larger proportion of the total sample size28 / 36

Effects of demographic stochasticity

Demographic stochasticity is mainly an issue at small population sizes because each individual is a larger proportion of the total sample size

To see this in action, we can use `R` to simulate the abundance of populations that experience demographic stochasticity

28 / 36

Effects of demographic stochasticityStart with 100 populations with a relatively large population size (N=500) 6Further assume that the mean survival and reproductive rates remain constant so all stochastisticity is demographic  
29 / 36

Effects of demographic stochasticity

Start with 100 populations with a relatively large population size $(N = 500)$ $^6$

Further assume that the mean survival and reproductive rates remain constant so all stochastisticity is demographic

29 / 36

Effects of demographic stochasticity

Start with 100 populations with a relatively large population size $(N = 500)$ $^6$

Further assume that the mean survival and reproductive rates remain constant so all stochastisticity is demographic

With $\large N_0=500$ no populations went extinct

29 / 36

$^6$ Since each population starts with the same initial population size, any differences at the end of the simulation are due to stochasticity

Effects of demographic stochasticity

What happens if we start with 10 individuals instead?

30 / 36

Effects of demographic stochasticity

What happens if we start with 10 individuals instead?

Now 41% went extinct $^7$

30 / 36

$^7$ Remember that if at any point during the time series $N=0$ , the population is extinct. Because our model assumes no movement, a population that goes extinct stays extinct

Effects of demographic stochasticity

Demographic stochasticity increase extinction risk of small populations because there's an increased chance that, purely due to randomness, more individuals die than are born

At large abundances, this is much less likely

31 / 36

Stochasticity and extinction risk over time

Another important consequences of stochasticity is that, over long-enough time periods, populations that experience stochasticity (both demographic and environmental) will eventually go extinct

32 / 36

Stochasticity and extinction risk over time

Another important consequences of stochasticity is that, over long-enough time periods, populations that experience stochasticity (both demographic and environmental) will eventually go extinct

Given long enough, each population will eventually experience a string of years with high mortality and low reproductive success

32 / 36

Stochasticity and extinction risk over time

Another important consequences of stochasticity is that, over long-enough time periods, populations that experience stochasticity (both demographic and environmental) will eventually go extinct

Given long enough, each population will eventually experience a string of years with high mortality and low reproductive success

The time it takes for this to occur will be longer for large populations but even still, it will eventually happen

32 / 36

Stochasticity and extinction risk over time

Again, we can use data simulations to show this:

33 / 36

Stochasticity and extinction risk over time

Again, we can use data simulations to show this: If we start with populations of 100 individuals and simulate 50 years of population change, 1% of populations go extinct

33 / 36

Stochasticity and extinction risk over time

If we extend our simulation out to 200 years, 23% of populations go extinct

34 / 36

Stochasticity and extinction risk over time

500 years? 57% of populations go extinct

35 / 36

Stochasticity and extinction risk over time

10,000 years? 99% of populations go extinct $^8$

36 / 36

$^8$ We could go further if we wanted and that last population would eventually go extinct as well

So why isn't extinction more common in nature?

↑, ←, Pg Up, k	Go to previous slide
↓, →, Pg Dn, Space, j	Go to next slide
Home	Go to first slide
End	Go to last slide
Number + Return	Go to specific slide
b / m / f	Toggle blackout / mirrored / fullscreen mode
c	Clone slideshow
p	Toggle presenter mode
t	Restart the presentation timer
?, h	Toggle this help

Lecture 4

Stochasticity

WILD3810 (Spring 2021)

Readings

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

Assumptions of the B-D models

Remember from the lecture 3 that our simple models of population growth were based on the following assumptions:

Assumptions of the B-D models

In reality, we know that assumption 4 is almost never true

Assumptions of the B-D models

In reality, we know that assumption 4 is almost never true

Birth and death rates are determined by many external forces

Assumptions of the B-D models

In reality, we know that assumption 4 is almost never true

Birth and death rates are determined by many external forces

These forces often vary across space, time, and individuals:

Assumptions of the B-D models

In reality, we know that assumption 4 is almost never true

Birth and death rates are determined by many external forces

These forces often vary across space, time, and individuals:

The B-D population model ignored all of these sources of variation 2^2

Stochasticity

Stochasticity

In many cases, we do not know exactly why populations go up or down from year-to-year

Stochasticity

In many cases, we do not know exactly why populations go up or down from year-to-year

Often don't know what these causes are so from our perspective, the variation appears to be random

Stochasticity

In many cases, we do not know exactly why populations go up or down from year-to-year

Often don't know what these causes are so from our perspective, the variation appears to be random

Processes that are governed by some element of chance (randomness) are referred to as stochastic processes

Stochasticity

In many cases, we do not know exactly why populations go up or down from year-to-year

Often don't know what these causes are so from our perspective, the variation appears to be random

Processes that are governed by some element of chance (randomness) are referred to as stochastic processes

Just because something is stochastic doesn't mean it's completely unpredictable 3^3

Stochasticity

We can predict how often we expect different random outcomes using probability

Stochasticity

We can predict how often we expect different random outcomes using probability

Probability distributions characterize the frequency of different outcomes 4^4

Stochasticity

We can predict how often we expect different random outcomes using probability

Probability distributions characterize the frequency of different outcomes 4^4

In population models, we can use probability to characterize and account for stochastic processes that effect population growth

Two types of stochastic processes

With regards to population models, we generally distinguish between two types of stochasticity:

Two types of stochastic processes

With regards to population models, we generally distinguish between two types of stochasticity:

Two types of stochastic processes

With regards to population models, we generally distinguish between two types of stochasticity:

Environmental stochasticity

Environmental stochasticity

The expected value of demographic parameters can fluctuate over time in response to:

Environmental stochasticity

If environmental attributes change stochastically over time, so will demographic vital rates and population growth rate

Modeling environmental stochasticity

If we want to take into account changes in demographic parameters as environmental conditions change, we need time-specific demographic parameters:

Modeling environmental stochasticity

If we want to take into account changes in demographic parameters as environmental conditions change, we need time-specific demographic parameters:

Modeling environmental stochasticity

If we want to take into account changes in demographic parameters as environmental conditions change, we need time-specific demographic parameters:

Over TT years, abundance will be:

Modeling environmental stochasticity

For example, if N0=50N_0 = 50 and:

Modeling environmental stochasticity

For example, if N0=50N_0 = 50 and:

Modeling environmental stochasticity

It's clear that this population is declining

Modeling environmental stochasticity

It's clear that this population is declining

What is the average λ\large \lambda (which we'll call ˉλ\large \bar{\lambda})?

Modeling environmental stochasticity

The B-D population model ignored all of these sources of variation $^2$

Just because something is stochastic doesn't mean it's completely unpredictable $^3$

Probability distributions characterize the frequency of different outcomes $^4$

Probability distributions characterize the frequency of different outcomes $^4$

Over $T$ years, abundance will be:

For example, if $N_0 = 50$ and:

For example, if $N_0 = 50$ and:

What is the average $\large \lambda$ (which we'll call $\large \bar{\lambda}$ )?

An obvious way to calculate $\large \bar{\lambda}$ is the arithmetic mean of the annual $\large \lambda$ 's:

An obvious way to calculate $\large \bar{\lambda}$ is the arithmetic mean of the annual $\large \lambda$ 's:

$\large \bar{\lambda} = 1$ means the population should have, on average, remained at 50 individuals

$\large \bar{\lambda} = 1$ means the population should have, on average, remained at 50 individuals

This means that shrinking by 30% $(\large \lambda = 0.7)$ and then growing by 30% $(\large \lambda = 1.3)$ does not get you back to where you started

Initially, assume a large population $\large (N_0 = 500)$ with:

Because environmental stochasticity tends to reduce $\large \bar{\lambda}$ regardless of population size, it has important consequences for the extinction risk of both small and large populations

Likewise, we might expect individuals to produce 3 offspring on average, but some individuals will have more and some fewer based on chance $^5$