FW364 Ecological Problem Solving

Class 17: Spatial Structure

October 30, 2013

Shifting focus: No longer discussing a single population……Instead, a “population of populations”

Adding spatial structure to models

Objectives for Today:Introduce spatial structure / metapopulation analysisIn-class demo of why spatial structure is good

Objectives for Next Class:Cover more on metapopulation theoryManagement of metapopulations

Text (optional reading):Chapter 6

Outline for Today

Metapopulations

Metapopulation:

Group of sub-populationsconnected by dispersal

(emigration, immigration)

“population of populations”

Synonyms for sub-populations:populations, local populations

Metapopulation structure is very important to population growth and persistence

(for both metapopulation and local populations)

Metapopulations

Examples

Birds in a fragmented forest

Fish in lakes in a landscape Deer in an island chain

Sheep living on mountains

Mosquitoes in pitcher plants

What assumption are we dropping from earlier classes? Closed populations (i.e., no immigration or emigration)

Immigration/emigration (dispersal) is KEY to metapopulations

Essence of the metapopulation idea:

while local populations may go extinct at a relatively high frequency,

a set of local populations connected by limited dispersal(i.e., the metapopulation) may persist with a relatively high probability

Metapopulations

Metapopulation Mantra:

“local extinction, global persistence”

Metapopulations

Metapopulation dynamics is a relatively new field of study

especially populations that have become fragmented by human development

Strong utility for threatened / endangered species management

Probabilities

Some math background: Probability of multiple events

What’s the probability of a coin turning up heads (or tails)? 0.5 = 50%

If I flip the coin again, what’s the probability of it turning up heads (or tails)? 0.5 = 50%

Considering both coin flips, what’s the probability of getting heads twice? 0.5 * 0.5 = 0.25 = 25%

Flip 1 then 2:

Flip 1:

Flip 2:

Heads or Tails

Heads or Tails

50% of each

50% of each

Heads-HeadsHeads-TailsTails-HeadsTails-Tails

25% of each

Probabilities

Some math background: Probability of multiple events

What’s the probability of a coin turning up heads (or tails)? 0.5 = 50%

If I flip the coin again, what’s the probability of it turning up heads (or tails)? 0.5 = 50%

Considering both coin flips, what’s the probability of getting heads twice? 0.5 * 0.5 = 0.25 = 25%

What’s the probability of getting three heads in a row? 0.5 * 0.5 * 0.5 = 0.53 = 0.125 = 12.5%

Consecutive coin flips represent multiple independent events

Probability of multiple independent events all occurring

product of probabilities of each event =

Let’s apply probability theory to metapopulations

by considering probability of metapopulation persistence / extinction:

Probability metapopulation extinction = f (probability local extinction)

Parameters we need to define:

pe = probability of a local population going extinct in one time step

pp = probability of local population persisting (not going extinct)

Pe = probability of metapopulation going extinct (i.e., ALL local populations go extinct at same time)

Pp = probability of metapopulation persisting (i.e., at least one sub-population persists)

Probability of Metapopulation Persistence

Let’s apply probability theory to metapopulations

by considering probability of metapopulation persistence / extinction:

Probability metapopulation extinction = f (probability local extinction)

Parameters we need to define:

pe = probability of a local population going extinct in one time step

pp = probability of local population persisting (not going extinct)

Pe = probability of metapopulation going extinct (i.e., ALL local populations go extinct at same time)

Pp = probability of metapopulation persisting (i.e., at least one local population persists)

What is the range of pe , pp , Pe , and Pp values?

How do pp and pe relate mathematically?

How do Pp and Pe relate mathematically?

0 to 1pp = 1 – pe Pp = 1 – Pe

Probability of Metapopulation Persistence

Probability of Metapopulation Persistence

Example:

If pe = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, Pe,

with 5 local populations?

Need to make crucial assumption:

assume populations operate completely independently

i.e., the extinction of any local population is completely independent of extinction in all the other populations

like multiple flips of a coin

Concept check:

Is this a reasonable assumption? Not usually; typically some correlation of extinction risks (more later)

Probability of Metapopulation Persistence

Example:

If pe = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, Pe,

with 5 local populations?

Can use law of probability for independent events:

Pe = pe * pe * pe * pe * pe = pe5

Pe = 0.8 * 0.8 * 0.8 * 0.8 * 0.8 = 0.85 = 0.33

So, probability that all local populations will go extinct is 33%!

(i.e., probability the metapopulation will go extinct is 33%)

Much smaller extinction probability for metapopulation than local populations!

Probability of Metapopulation Persistence

Example:

If pe = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, Pe,

with 5 local populations?

How about probability of metapopulation persistence?

If Pe = 0.33, then Pp is:

Pp = 1 – Pe Pp = 1 – 0.33 Pp = 0.67

So, even though each local population only has a 20% change of persisting,the metapopulation has a 67% chance!

Local extinction, global persistence!

Probability of Metapopulation Persistence

Example:

If pe = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, Pe,

with 5 local populations?

What’s the probability ALL local populations will persist?

If pe = 0.8, then pp = 0.2

Probability ALL persist = pp * pp * pp * pp * pp = pp5

Probability ALL persist = 0.2 * 0.2 * 0.2 * 0.2 * 0.2 = 0.25 = 0.00032

Why isn’t probability all will persist = 67%?

Probability of ALL local population persisting (0.032%) vs.probability of metapopulation persisting (67%)

Probability of Metapopulation Persistence

General equation for calculatingprobability of metapopulation persistence (Pp):

Pp = 1 – (pe)x

Where x = number of local populations

For example above,Pp = 1 – (0.8)5 Pp = 67%

The more local populations (x), the smaller (pe)x becomesso Pp gets larger with a greater number of local populations

Example:

If pe = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, Pe,

with 5 local populations?

Probability of Metapopulation Persistence

General equation for calculatingprobability of metapopulation persistence (Pp):

Pp = 1 – (pe)x

Where x = number of local populations

For example above,Pp = 1 – (0.8)5 Pp = 67%

The more local populations (x), the smaller (pe)x becomesso Pp gets larger with a greater number of local populations

Example:

If pe = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, Pe,

with 5 local populations?

Let’s look at some figures

Probability of Metapopulation Persistence

0 2 4 6 8 10 12 140.0

0.2

0.4

0.6

0.8

1.0 pe = 0.20

pe = 0.40

pe = 0.60

pe = 0.80

pe = 0.90

pe = 0.95

pe = 0.99

Number of local populations, x

Met

apop

ulati

on p

ersi

sten

ce (P

p) pe

pe

pe

pe

pe

pe

pe

Probability of Metapopulation Persistence

0 0.2 0.4 0.6 0.8 10.0

0.2

0.4

0.6

0.8

1.0x = 1

x = 2

x = 4

x = 8

x = 16

Probability of local extinction (pe)

Met

apop

ulati

on p

ersi

sten

ce (P

p)

Challenge:Why is this line

straight?

Probability of Metapopulation Persistence

0 0.2 0.4 0.6 0.8 10.0

0.2

0.4

0.6

0.8

1.0x = 1

x = 2

x = 4

x = 8

x = 16

Probability of local extinction (pe)

Met

apop

ulati

on p

ersi

sten

ce (P

p)

Challenge:Why is this line

straight?

When x = 1, Pp = 1 – pex

Reduces to Pp = 1 – pe

Blinking Light Bulb Analogy

Each bulb represents a local population:when a bulb is dark, the local population is extinct

Unlikely that ALL the bulbs will be dark at any one time if there are:

Many bulbs (x is high)Bulbs do not blink in unison (independent events)Each bulb does not stay dark for too long (rapid blinking rate)

…even though individuals bulbs are dark a lot of the time

Local extinction, global persistence

Blinking Light Bulb Analogy

Dark bulbs represent extinct local populations

there must be a way for bulbs to blink back on

How do local populations re-establish in nature?

colonization from occupied populations

We can think of wiring as a dispersal corridorthat allows for migration between local populations

Correlated Fluctuations

We can make an opposite extreme assumption:

All populations are have completed correlated fluctuationsi.e., all local populations fluctuate (go extinct) together

Earlier, we made the crucial assumption thatall populations operate completely independently

i.e., the extinction of any local population is completely independent of extinction in all the other populations

probability of metapopulation extinctionis equal to

probability of local extinctionPe = pe

Correlated Fluctuations

We can make an opposite extreme assumption:

All populations are have completed correlated fluctuationsi.e., all local populations fluctuate (go extinct) together

Earlier, we made the crucial assumption thatall populations operate completely independently

i.e., the extinction of any local population is completely independent of extinction in all the other populations

probability of metapopulation extinctionis equal to

probability of local extinctionPe = pe

In reality, metapopulations fall somewhere in between these extremes

The degree to which fluctuations are correlated among habitat patchesis a crucial parameter in metapopulation models

Metapopulations Demo

Let’s see an example of how a metapopulation can persist,even when the probability of local population extinction is high

pe = 0.5

Demonstration rules:

Coin flips determine local population extinctionHeads: Local population persistsTails: Local population goes extinct

Pp = 1 – (pe)x

x = 4 local populations

Pp = 1 – (0.5)4 Pp = 0.94

Parameters:Two groups of four flipping at same timeI’ll record results in Excel

Looking Ahead

Next Class:

Metapopulation theoryMetapopulation management