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
Top Related