Lecture 14

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# Lecture 14
## Metapopulation dynamics
### WILD3810 (Spring 2021)

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## Readings

> Mills 175-188

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### Population ecology is the study of the distribution of individuals in a population over time and space

`$$\Huge N = B + I + D +E$$`

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## Assumptions of the exponential growth model

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

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## Assumptions of the exponential growth model

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~~  
- **Stochasticity, density-dependence**

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## Assumptions of the exponential growth model

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~~  
- **Stage-structured dynamics**

4) ~~Birth and death rates are constant~~  
- **Stochasticity, density-dependence**

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## Assumptions of the exponential growth model

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~~  
- **Stage-structured dynamics**

4) ~~Birth and death rates are constant~~  
- **Stochasticity, density-dependence**

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# Dispersal

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## Dispersal

> Movement of an individual from one location to another for the purposes of breeding

- also defined as movements that *can* result in gene flow

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#### Types of dispersal

1) Natal dispersal

> Movement of an individual from its birth location to its first breeding location

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2) Breeding dispersal

> Movement of an individual between breeding attempts

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## Dispersal

#### Three stages

1) Emigration

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## Dispersal

#### Three stages

1) Emigration

2) Search

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## Dispersal

#### Three stages

1) Emigration

2) Search

3) Immigration

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## Why disperse?

#### Find mates

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## Why disperse?

#### Find mates

#### Avoid inbreeding

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## Why disperse?

#### Find mates

#### Avoid inbreeding

#### Avoid kin competition

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## Why not disperse?

#### In some cases, remaining in your current patch (**philopatry**) is beneficial because:

- Search phase can be risky and energetically costly

- "Home field" advantage

- Loss of kin cooperation

- Competition in new patch

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Trade-offs between the costs and benefits of dispersal may differ between sexes, populations, and species

- In birds, females often disperse more than males (males benefit from defending territory)

- In mammals, males often disperse more than females (males benefit from defending mates)

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## How does dispersal influence population dynamics?

#### Movement of individuals can:

- reduce local population size `$(E)$`

- increase local population size `$(I)$`

- increase genetic diversity

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#### Dispersal can link the dynamics of populations that would otherwise be independent

- metapopulation: a population of populations

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## When are populations a metapopulation?

#### Fragmentation creates patches of suitable habitat surrounded by unsuitable habitat

- if *connectivity* between patches is low, dynamics of local populations will be independent
 + patches that go extinct have little chance of being colonized
 
<img src="figs/isolated.png" width="30%" style="display: block; margin: auto;" />

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## When are populations a metapopulation?

#### Fragmentation creates patches of suitable habitat surrounded by unsuitable habitat

- if connectivity between patches is very high, populations will act as a single, large population
 + very low probability that patches will go extinct
 + "patchy population"
 
<img src="figs/patchy.png" width="30%" style="display: block; margin: auto;" />

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## When are populations a metapopulation?

#### Fragmentation creates patches of suitable habitat surrounded by unsuitable habitat

- if connectivity between patches is moderate, populations can go extinct and be re-colonized
 + at any given time, some sites will be occupied and some will not
 + patches will "blink" on and off
 + metapopulation
 
<img src="figs/metapopulation.png" width="30%" style="display: block; margin: auto;" />

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## When are populations a metapopulation?

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## Levins' metapopulation model

#### Developed originally by Richard Levins

- simple model describing the occupancy of local patches

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- each patch can be occupied (1) or unoccupied (0)
    + ignores local population dynamics

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- at each time step, occupied patches go extinct with probability `$\Large \epsilon$`

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- unoccupied patches can be colonized with probability `$\Large \gamma$`

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- objective is to estimate the fraction of patches occupied, `$\Large \psi$`

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## Levins' metapopulation model

#### Spatial configuration of patches is ignored
- all patches have the same probability of being colonized regardless of their location
 
<img src="figs/levins1.png" width="50%" style="display: block; margin: auto;" />

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## Levins' metapopulation model

#### At any point in time, what proportion of sites will go extinct?

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`$$\Large E = \psi \epsilon$$`

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## Levins' metapopulation model

#### At any point in time, what proportion of sites will be colonized?

`$$\Large I = \gamma \psi (1 - \psi)$$`

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## Levins' metapopulation model

#### At any point in time, what proportion of sites will be occupied?

`$$\Large \psi = 1 - \frac{\epsilon}{\gamma}$$`

- Although `$\large \psi$`% of patches are expected to be occupied at any point in time, *which* sites are occupied will change
 + leads to patches "blinking" on and off
 
<img src="Lecture14_files/figure-html/unnamed-chunk-13-1.png" width="396" style="display: block; margin: auto;" />
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## Metapopulation persistance

#### If `$\Large \epsilon$` is the probability that 1 patch goes extinct:

- probability that it doesn't go extinct (persistence) is `$\large 1 - \epsilon$`

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#### What is the probability that a metapopulation composed of two patches does **not** go extinct?

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`$$\LARGE 1 - \epsilon \times \epsilon$$`

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#### What is the probability that a metapopulation composed of `$\large n$` patches does not go extinct?

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`$$\LARGE 1 - \epsilon^n$$`

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## Metapopulation persistance

#### More patches = lower metapopulation extinction risk 
- unoccupied sites are "rescued" by colonization from occupied sites
- critical to maintain patches, even if they are currently unoccupied
- connectivity of patches important for increasing colonization probability

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## Assumptions of the Levins model

- `$\large \gamma$` and `$\large \epsilon$` are the same for every patch 
 
- `$\large \gamma$` and `$\large \epsilon$` are constant over time 
 
- `$\large \gamma$` and `$\large \epsilon$` are independent of patch size 
 
- `$\large \gamma$` and `$\large \epsilon$` are independent of distance of patch to other patches 
 
- `$\large \gamma$` and `$\large \epsilon$` are independent of population density 
 
- Local birth-death dynamics are ignored