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 analysis In-class demo of why spatial structure is good Objectives for Next Class: Cover more on metapopulation theory Management of metapopulations Text (optional reading) : Chapter 6 Outline for Today

Metapopulations Metapopulation: Group of sub-populations connected 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 50% of each Heads-Heads Heads-Tails Tails-Heads Tails-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 = = = 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: p e = probability of a local population going extinct in one time step p p = probability of local population persisting (not going extinct) P e = probability of metapopulation going extinct (i.e., ALL local populations go extinct at same time) P p = 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: p e = probability of a local population going extinct in one time step p p = probability of local population persisting (not going extinct) P e = probability of metapopulation going extinct (i.e., ALL local populations go extinct at same time) P p = probability of metapopulation persisting (i.e., at least one local population persists) What is the range of p e, p p, P e, and P p values? How do p p and p e relate mathematically? How do P p and P e relate mathematically? 0 to 1 p p = 1 – p e P p = 1 – P e Probability of Metapopulation Persistence

Example: If p e = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, P e, 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 p e = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, P e, with 5 local populations?  Can use law of probability for independent events: P e = p e * p e * p e * p e * p e = p e 5 P e = 0.8 * 0.8 * 0.8 * 0.8 * 0.8 = = 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 p e = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, P e, with 5 local populations? How about probability of metapopulation persistence? If P e = 0.33, then P p is: P p = 1 – P e P p = 1 – 0.33 P p = 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 p e = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, P e, with 5 local populations? What’s the probability ALL local populations will persist? If p e = 0.8, then p p = 0.2 Probability ALL persist = p p * p p * p p * p p * p p = p p 5 Probability ALL persist = 0.2 * 0.2 * 0.2 * 0.2 * 0.2 = = 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 calculating probability of metapopulation persistence ( P p ): P p = 1 – (p e ) x Where x = number of local populations For example above, P p = 1 – (0.8) 5 P p = 67%  The more local populations ( x ), the smaller (p e ) x becomes so P p gets larger with a greater number of local populations Example: If p e = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, P e, with 5 local populations?

Probability of Metapopulation Persistence General equation for calculating probability of metapopulation persistence ( P p ): P p = 1 – (p e ) x Where x = number of local populations For example above, P p = 1 – (0.8) 5  P p = 67%  The more local populations ( x ), the smaller (p e ) x becomes so P p gets larger with a greater number of local populations Example: If p e = 0.8 per year for local population extinction (high extinction prob), what is probability (per year) of metapopulation going extinct, P e, with 5 local populations? Let’s look at some figures

Probability of Metapopulation Persistence pepe pepe pepe pepe pepe pepe pepe

Challenge: Why is this line straight?

Probability of Metapopulation Persistence Challenge: Why is this line straight? When x = 1, P p = 1 – p e x Reduces to P p = 1 – p e

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 corridor that allows for migration between local populations

Correlated Fluctuations We can make an opposite extreme assumption: All populations are have completed correlated fluctuations i.e., all local populations fluctuate (go extinct) together Earlier, we made the crucial assumption that all 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 extinction is equal to probability of local extinction P e = p e

Correlated Fluctuations We can make an opposite extreme assumption: All populations are have completed correlated fluctuations i.e., all local populations fluctuate (go extinct) together Earlier, we made the crucial assumption that all 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 extinction is equal to probability of local extinction P e = p e In reality, metapopulations fall somewhere in between these extremes The degree to which fluctuations are correlated among habitat patches is 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 p e = 0.5 Demonstration rules: Coin flips determine local population extinction Heads: Local population persists Tails: Local population goes extinct P p = 1 – (p e ) x x = 4 local populations P p = 1 – (0.5) 4 P p = 0.94 Parameters: Two groups of four flipping at same time I’ll record results in Excel

Looking Ahead Next Class : Metapopulation theory Metapopulation management