Metapopulations: Part I

class: left, top, inverse, title-slide

.title[
# <strong>Metapopulations: Part I</strong>
]
.subtitle[
## <a href="https://eligurarie.github.io/EFB370/">EFB 370: Population Ecology</a>
]
.author[
### .white[**Dr. Elie Gurarie]**
]
.date[
### .white[March 20, 2023]
]

---

<!--
## Goals

.pull-left[
Define ***metapopulation***

How does **metapopulation structure** influence **extinction**

Some theoretical models: 
- **External** and **Internal** colonization
- **Independent mortality** vs. **Rescue Effect**
]

.pull-right[Gotelli Chapter 4]
-->

## Blowing up `$N_t$` in space

**Simple population:**  `$$\large N(t)$$`

**Age/stage-structured:**  `$$\large N_i(t) = \{N_1(t), N_2(t), ..., N_k(t)\}$$`

- where `$i$` represents structure, with `$k$` age/stage classes

**Spatial structure:** `$$\large N_i(t) = \{N_1(t), N_2(t), ..., N_k(t)\}$$`

- where `$i$` is location, with `$k$` locations

---
## .darkblue[A metapopulation is a **population of populations**]
.pull-left[

![](images/fritillary_apbio.png)]

.pull-right.large[

1. The local populations MUST be somehow connected via **dispersal**.

2. There must be areas of (near) zero density in between.  The "in-between" is referred to as the **matrix**.

### Canonical examples

- **Fragmented habitats**

- Island populations
]

---

## **Population** vs. **Metapopulation**

.pull-left-40.large[
**WA sea otters**

- Closed population

- only **Birth** and **Death**

- Questions: 
    - growth | dynamics | age structures
    
- **Extinction** of interest mainly due to stochasticity, low numbers
]

.pull-right-50.large[

**ALL sea otters**

- Open population
    - **Immigration!**  **Emigration!**

- Questions:

- given that a local population might go **extinct**, will the metapopulation go **extinct**?
    
    - what is the proportion of occupied patches?
]

---

## What makes it a metapopulation?  **Dispersal distance**

.pull-left[

![](images/metapops_and_dispersal.png)

Sale, Hanski, Kritzer 2006
]

.pull-right.large[

As long as there is *some* local connectivity among populations.

![](images/sea_otter_metapop.png)

]

---

.pull-left-60[
## By that metric ...

### Polar bear (*Ursus maritimus*)

![](images/polar_bear_metapop.png)
]

.pull-right-40[

### and caribou / reindeer (*Rangifer tarandus*)

![](images/rangifer_metapop.jpg)

.center[Are also **metapopulations**]
]

---

## Population persistence of a single population?

.pull-left[

`$e$` = *local* probability of extinction

time steps | Prob. persistence
---|--- 
1: | `$1-e$`
2: | `$(1-e)(1-e)$`
3: | `$(1-e)(1-e)(1-e)$`
4: |  `$(1-e)(1-e)(1-e)(1-e)$`
... | 
`$t$`: | `$(1-e)^t$`
]

.pull-right[
<img src="Metapopulations_PartI_files/figure-html/localpersistence-1.png" width="100%" />

**Take away:** Even with very LOW probability of extinction, you WILL go extinct.  
]

---

## **M0**: Population persistence of a **metapopulation**

.pull-left-40[

`$k$` populations, `$t$` time steps

Pops: | 1 time step|  *t* steps 
---|---|----
1: | `$1-e$` | `$(1-e)^t$`
2: |  `$1-e \times e$` | `$(1-e^2)^t$`
3: | `$1-e \times e \times e$` | `$(1-e^3)^t$`
... | ... | ... 
k: | `$1-e^k$` | `$(1-e^k)^t$`

]

.pull-right-60[

]

---

## Metapopulations are resistant to extinction!

.pull-left-40[

`$$\large P(k,t) = (1-e^k)^t$$`

.darkblue[
Metapopulations **dramatically** spread out / buffer the risk of extinction!]

.darkred[
But still ... if the ONLY process is extinction, you **will go extinct** (sorry!)
]
]

.pull-right-60[

<img src="Metapopulations_PartI_files/figure-html/unnamed-chunk-4-1.png" width="100%" />
]

---

## **M1:** Let's add **colonization**

.pull-left.large[

**Island-Mainland model**

- Every (local) population has a probability of going extinct: `$p_e$`

- But every empty location has a probability of getting colonized: `$p_c$`

> Note - there is an important (implicit) assumption that population very quickly hits **carrying capacity**, so essentially *instant* saturation.

]

.pull-right.large[

![](images/mainland_island.png)

The mainland is a constant, independent source of potential colonizers. Also known as **propagule rain**.

.green[(echoes of *biogeography*).]
]

---

## **M1:** Island-Mainland Model

.pull-left[

### Q: How many occupied patches might we expect?

`$$E(N_{t+1}) = N_t - p_e N_t + (K - N_t)\,p_c$$`

define proportion of populated patches: `$f_t = E(N_t)/K$`, and define *equilibirum*:
`$$f^* := f_{t+1} = f_t$$`

.center[*...then some math happens...*]

.red[$$\large f^* = {p_c \over p_c + p_e}$$]
]

.pull-right[

![](images/mainland_island.png)
> The equilibrium is a balance between colonization and extinction rate.

]

---

## Continuous time formulation

.pull-left-40[
Very general metapopulation model:

.red[

`$$\huge {df \over dt} = c(f) - e(f)$$`
]

Where *c* = colonization rate, *e* = extinction rate. Can be (often are!) functions of `$f$` (occupied proportion).

.green[Note: this is similar to 
`$$\large {dN \over dt} = b(N) - d(N).$$`

which is the foundation of population growth models)
]]

.pull-right-50[

> #### Assumptions: 
> 
> - Deterministic (i.e. `$k \to \infty$`)
> - Continuous-time, unstructured extinction / colonization process
> - "Rates" are like infinitesemal probabilities

But - lots of elegant analyses can be made messing with this model. 
]

---

.pull-left[

## **M1:** Mainland-Island

.large.red[  `$$\large {df \over dt} = c - e$$` ]

Colonization is constant, so proportional to **available** patches: 
`$$c = p_c(1-f)$$`

Extinction is constant, so proportional to **occupied** patches:

`$$e = p_e f$$`

so: 
$$ {df \over dt} = p_c(1-f) - p_ef$$

]

.pull-right[

> The rate of change of the occupied patches GROWS in proportion to unoccupied patches and FALLS in proportion with occupied patches.

]

---

## **M2**: Internal Colonization

.red[$$\Large {df \over dt} = p_c f(1-f) - p_ef$$]

.pull-left-30[
![](images/colonization.png)
]

.pull-right-60.large[

Extinction *rate* is constant, as before: 
`$$e = p_e f$$`

Colonization can only come from **occupied** patches: 
`$$c = p_c \, f \, (1-f)$$`

If no patch is colonized ( `$f=0$` ), nothing can colonize.

If the population is 100% occupied ( `$f = 1$` ), there is nothing to colonize. 
]

---

.pull-left[

## **M2**: Internal Colonization - with Schematic

.red[
$$ \large {df \over dt} = p_c f(1-f) - p_ef$$
]

Extinction is constant, as before: 
`$$e = p_e f$$`

Colonization can only come from **occupied** patches: 
`$$c = p_c \, f \, (1-f)$$`

.blue[The maximum rate of colonization occurs when `$f = 1/2$`.

]

.pull-right[

**Equilibrium** occurs when: 
.darkred[

`$$f^* = \begin{cases} 1-p_e/p_c & \text{when} & p_e < p_c  \\ 0 & \text{when} & p_e \geq p_c\end{cases}$$`

]]

---

## **M3:** Rescue Effect

.red[
$$ \Large {df \over dt} = p_c (1-f) - p_ef(1-f)$$
]

.pull-left-40[

Assumes that if you have a lot of neighbors some loose "propagules" will buffer you from extinction.

**Equilibrium states:**

.darkred[

`$$f^* = \begin{cases} p_c/p_e & \text{when} & p_e > p_c  \\ 1 & \text{when} & p_e \leq p_c\end{cases}$$`

]

Even with higher extinction rate than colonization rate, there will always be some occupied patches!
]

.pull-right[

<img src="Metapopulations_PartI_files/figure-html/RescueEffect-1.png" width="100%" />
]

---

## **M4:** Rescue Effect with Internal Colonization

<font color = "red">
$$ \large {df \over dt} = p_c f(1-f) - p_ef(1-f)$$
</font>

.pull-left[

Only equilibria: 0, if `$p_e > p_c$` or 1, if `$p_e \leq p_c$`.

Fundamental conclusions:

> **metapopulation** under equilibrium MUST be rare! 
> Either everything colonizes or nothing colonizes.**
]

.pull-right[

<img src="Metapopulations_PartI_files/figure-html/FinalSchematic-1.png" width="100%" />
]

---

## Four models

With rather different predictions!  (Nice synthesis -  mainly due to Gotelli.)

---

.pull-left-70[
## Some characters

### Richard Levins (1930-2016)

![](images/Levins.jpg)

- "Scholarship that is indifferent to human suffering is immoral."
- "Our truth is the intersection of independent lies."
]

.pull-right-30[

### Ilkka Hanski (1953-2016)

![](images/ilkka.jpg)
]