Lecture 11

class: center, middle, inverse, title-slide

# Lecture 11
## Stage-structured populations
### <br/><br/><br/>WILD3810 (Spring 2021)

---

## Leslie matrix

#### Leslie matrices are defined as :

- square matrix that summarizes the demography of **age**-specific life cycles

- one column for each **age** class

- matrix elements contain **age**-specific birth and survival rates

- individuals cannot stay in the same **age**-class for more than a single time step `$^1$`
<br/> 
<br/> 
`$$\LARGE \mathbf A = \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix}$$`

???

`$^1$` In some cases, for example when we define identify individuals juveniles (age 1), pre-adults (age 2), and adults (age 3+), the last element in the Leslie matrix might not be zero. Individuals can remain in the "adult" class for more than 1 year

---
## Stage-structured populations

#### In some cases, age is not a relevant predictor of survival and birth rates

--
#### Instead, survival and birth rates vary with **stage**

--
- life cycle stage

.pull-left[
<img src="figs/lifecycle_diagram.jpg" width="75%" style="display: block; margin: auto;" />
]

.pull-right[
<img src="figs/lifecycle.gif" width="75%" style="display: block; margin: auto;" />
]

---
## Stage-structured populations

#### In some cases, age is not a relevant predictor of survival and birth rates

#### Instead, survival and birth rates vary with **stage**

- life cycle stage

- size

.left-column[
<img src="https://upload.wikimedia.org/wikipedia/commons/a/af/Gasterosteus_aculeatus_t%C3%BCsk%C3%A9s_pik%C3%B3.jpg" width="100%" style="display: block; margin: auto;" />
]

.right-column[
<img src="figs/stickleback_fecundity.png" width="60%" style="display: block; margin: auto;" />
]

???

Image courtesy of Kókay Szabolcs, via Wikimedia Commons
Figure from Moser et al. (2012)

---
## Stage-structured populations

#### L. Lefkovitch relaxed an assumption of the age-structured matrix model developed by Leslie

--
#### **Lefkovitch matrices ** allow individuals to remain in life-stages (or size classes) longer than one time step

--
#### Useful for plants and animals with stage-dependent demography

---
## Stage-structured matrix

`$$\LARGE \mathbf A = \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix} \rightarrow
\mathbf A = \begin{bmatrix}
    P_1 & F_2 & F_3\\
    G_1 & P_2 & 0\\
    0 & G_2 & P_3
\end{bmatrix}$$`
<br/>
- `$\large F_x$` is still **recruitment**, the number of offspring recruited into stage class 1 per adult in stage `$\large x$`

- `$\large P_x$` is the probability of **surviving** from year `$\large t$` until year `$\large t+1$` and **remaining** in stage `$\large x$`

- `$\large G_x$` is the probability of **growing and surviving** to stage `$\large x + 1$` during `$\large t$` to  `$\large t+1$`

---
## Stage-structured matrix model

`$$\Large \mathbf N_{t+1} = \mathbf A \times \mathbf N_t$$`
`$$\LARGE = \begin{bmatrix}
    P_1 & F_2 & F_3\\
    G_1 & P_2 & 0\\
    0 & G_2 & P_3
\end{bmatrix} \times 
\begin{bmatrix} 
    n_{1,t}\\
    n_{2,t}\\
    n_{3,t}
\end{bmatrix}$$`

<br/>
#### Matrix multiplication is the  same as in the Leslie matrix model!!

---
## Common teasel example

.pull-left[
<img src="https://upload.wikimedia.org/wikipedia/commons/5/56/Dipsacus_fullonum_002.JPG" width="80%" style="display: block; margin: auto;" />
]

.pull-right[

- *Dypsacus sylvestris*

- native to Europe

- invasive species in United States

- stage-structured dynamics studied intensively by Patricia Werner and Hal Caswell
]

???

Image courtesy of H. Zell, via Wikicommons

---
## Common teasel example

.pull-left[
<img src="figs/teasel_rosette.png" width="80%" style="display: block; margin: auto;" />
]

.pull-right[

#### Complex stage structure

1) Dormant 1st year seeds

2) Dormant 2nd year seeds

3) Small rosettes `$(< 2.5cm)$`

4) Medium rosettes `$2.5-18.9cm$`

5) Large rosettes `$\geq 19cm$`

6) Flowering plants  
]

---
## Common teasel example

.pull-left[
<img src="figs/teasel_states.png" width="80%" style="display: block; margin: auto;" />
]

.pull-right[

#### Complex stage structure

1) Dormant 1st year seeds

2) Dormant 2nd year seeds

3) Small rosettes `$(< 2.5cm)$`

4) Medium rosettes `$2.5-18.9cm$`

5) Large rosettes `$\geq 19cm$`

6) Flowering plants  
]

---
## Common teasel example

<br/>
 
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 15px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 322.280 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.966 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.013 </td>
   <td style="text-align:center;"> 0.01 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 3.448 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.238 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 30.170 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.245 </td>
   <td style="text-align:center;"> 0.167 </td>
   <td style="text-align:center;"> 0.862 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.023 </td>
   <td style="text-align:center;"> 0.750 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
</tbody>
</table>

---
## Common teasel example

.pull-left[
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 322.280 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red !important;">0.966</span> </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: orange !important;">0.013</span> </td>
   <td style="text-align:center;"> 0.01 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 3.448 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow !important;">0.007</span> </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.238 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 30.170 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: green !important;">0.001</span> </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.245 </td>
   <td style="text-align:center;"> 0.167 </td>
   <td style="text-align:center;"> 0.862 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.023 </td>
   <td style="text-align:center;"> 0.750 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<img src="figs/teasel_states_from1.png" width="75%" style="display: block; margin: auto 0 auto auto;" />
]

---
## Common teasel example

.pull-left[
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 322.280 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.966 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.013 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red !important;">0.01</span> </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 3.448 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.238 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 30.170 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.245 </td>
   <td style="text-align:center;"> 0.167 </td>
   <td style="text-align:center;"> 0.862 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.023 </td>
   <td style="text-align:center;"> 0.750 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<img src="figs/teasel_states_from2.png" width="75%" style="display: block; margin: auto 0 auto auto;" />
]

---
## Common teasel example

.pull-left[
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 322.280 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.966 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.013 </td>
   <td style="text-align:center;"> 0.01 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red !important;">0.125</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 3.448 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: orange !important;">0.125</span> </td>
   <td style="text-align:center;"> 0.238 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 30.170 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.245 </td>
   <td style="text-align:center;"> 0.167 </td>
   <td style="text-align:center;"> 0.862 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.023 </td>
   <td style="text-align:center;"> 0.750 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<img src="figs/teasel_states_from3.png" width="75%" style="display: block; margin: auto 0 auto auto;" />
]

---
## Common teasel example

.pull-left[
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 322.280 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.966 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.013 </td>
   <td style="text-align:center;"> 0.01 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 3.448 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red !important;">0.238</span> </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 30.170 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: orange !important;">0.245</span> </td>
   <td style="text-align:center;"> 0.167 </td>
   <td style="text-align:center;"> 0.862 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow !important;">0.023</span> </td>
   <td style="text-align:center;"> 0.750 </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<img src="figs/teasel_states_from4.png" width="75%" style="display: block; margin: auto 0 auto auto;" />
]

---
## Common teasel example

.pull-left[
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 322.280 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.966 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.013 </td>
   <td style="text-align:center;"> 0.01 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 3.448 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.238 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
   <td style="text-align:center;"> 30.170 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.245 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red !important;">0.167</span> </td>
   <td style="text-align:center;"> 0.862 </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.023 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: orange !important;">0.75</span> </td>
   <td style="text-align:center;"> 0.000 </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<img src="figs/teasel_states_from5.png" width="75%" style="display: block; margin: auto 0 auto auto;" />
]

---
## Common teasel example

.pull-left[
<table class="table table-striped table-hover table-condensed table-responsive" style="font-size: 14px; margin-left: auto; margin-right: auto;">
 <thead>
  <tr>
   <th style="text-align:center;"> Seed 1 </th>
   <th style="text-align:center;"> Seed 2 </th>
   <th style="text-align:center;"> Small rosette </th>
   <th style="text-align:center;"> Medium rosette </th>
   <th style="text-align:center;"> Large rosette </th>
   <th style="text-align:center;"> Flowering </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: red !important;">322.28</span> </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.966 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.013 </td>
   <td style="text-align:center;"> 0.01 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: orange !important;">3.448</span> </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.007 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.125 </td>
   <td style="text-align:center;"> 0.238 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: yellow !important;">30.17</span> </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.001 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.245 </td>
   <td style="text-align:center;"> 0.167 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: green !important;">0.862</span> </td>
  </tr>
  <tr>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.00 </td>
   <td style="text-align:center;"> 0.000 </td>
   <td style="text-align:center;"> 0.023 </td>
   <td style="text-align:center;"> 0.750 </td>
   <td style="text-align:center;"> <span style="     border-radius: 4px; padding-right: 4px; padding-left: 4px; background-color: transparent !important;">0</span> </td>
  </tr>
</tbody>
</table>
]

.pull-right[
<img src="figs/teasel_states_from6.png" width="75%" style="display: block; margin: auto 0 auto auto;" />
]

---
## Common teasel example

#### What happens to a newly established population?

- Assume population starts with 100 1st year seeds

<br/>

--
`$$\Large \begin{bmatrix}
    0 & 0 & 0 &0 & 0 & 322.38\\
    0.966 & 0 & 0 & 0 & 0 & 0\\
    0.013 & 0.01 & 0.125 & 0 & 0 & 3.448\\
    0.007 & 0 & 0.125 & 0.238 & 0 & 30.17\\
    0.001 & 0 & 0 & 0.245 & 0.167 & 0.862\\
    0 & 0 & 0 & 0.023 & 0.75 & 0\\
\end{bmatrix} \times \begin{bmatrix}
    100\\
    0\\
    0\\
    0\\
    0\\
    0
\end{bmatrix}$$`

---
## Common teasel example

<br/>

---
## Common teasel example
<br/>

---
## Common teasel example

<br/>

--
**Stable stage distribution**

---
# Common teasel example

**Asymptotic growth rate**

---
## Management questions

#### What is the short-term growth of this population given the current age/stage structure?

--
#### What is the long-term growth of this population given the current vital rates?

--
#### Which age/stage contributes most to future population growth?

--
#### Which vital rates have the biggest effect on future growth?

--
#### How would future  population dynamics change if different vital rates were changed?