Lecture 10

class: center, middle, inverse, title-slide

# Lecture 10
## Matrix population models
### <br/><br/><br/>WILD3810 (Spring 2021)

---

## Readings

> Mills 98-103

---
## Assumptions of the B-D models

#### Over the coming weeks, we will learn about why and how to relax assumption 3:

1) Population closed to immigration and emigration  
<br/>

2) Model pertains to only the limiting sex, usually females  
<br/>

**3) Birth and death rates are independent of an individual’s age or biological stage**  
<br/>

4) Birth and death rates are constant

--
### **How can we model population dynamics of populations with complex age structure?**

---
## Age-structured demography

### Consider the following life table:
<br/>

???

What do `$m_x$` and `$P_x$` represent?

What can you say about the reproductive and survival schedules of this population?

---
## Age-structred demography

#### List population abundance in discrete 1-year age classes `$\Large (n_i)$`

- e.g., `$\large n_1$`, `$\large n_2$`, `$\large n_3$`

- `$\large n_i$` is the number of individuals about to experience their `$\large i^{th}$` birthday  
<br/>

---
## Age-structured demography

<br/>

---
## Age-structured demography

<br/>

---
## Age-structured demography

---
## Age-structured demography

---
## Age-structured demography

#### How many individuals will be in the population next year?

`$\LARGE n_{1,t+1} = m_1P_0n_{1,t}+m_2P_0n_{2,t} + m_3P_0n_{3,t}$`

---
## Age-structured demography

#### How many individuals will be in the population next year?

`$\LARGE n_{1,t+1} = m_1P_0n_{1,t}+m_2P_0n_{2,t} + m_3P_0n_{3,t}$`

`$\LARGE n_{2,t+1} =P_1n_{1,t}$`

---
## Age-structured demography

#### How many individuals will be in the population next year?

`$\LARGE n_{1,t+1} = m_1P_0n_{1,t}+m_2P_0n_{2,t} + m_3P_0n_{3,t}$`

`$\LARGE n_{2,t+1} =P_1n_{1,t}$`

`$\LARGE n_{3,t+1} =P_2n_{2,t}$`

---
## Leslie matrix model

#### Rather than modeling the dynamics using the previous equations, we can use **matrix projection models**

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

--
- one column for each age class

--
- developed by Sir Patrick H. Leslie for application to population biology

--
- a matrix with age-specific birth and survival rates is called a **Leslie matrix**  
<br/>

`$$\LARGE \mathbf A = \begin{bmatrix}
    m_1P_0 & m_2P_0 & m_3P_0\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix}$$`

---
class: inverse, middle, center

# Review of matrix algebra

---
## Review of matrix algebra

#### Matrix addition

- Matrices must be of the same dimension

- Add corresponding elements of the matrices

`$$\LARGE \mathbf A = \begin{bmatrix}
    2 & 0 \\
    4 & 6
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    1 & 5 \\
    2 & 7
\end{bmatrix}$$`
<br/>

`$$\LARGE \mathbf A + \mathbf B = \begin{bmatrix}
    3 & 5 \\
    6 & 13
\end{bmatrix}$$`

---
## Review of matrix algebra

#### Matrix subtraction

- Matrices must be of the same dimension

- Subtract corresponding elements of the matrices

`$$\LARGE \mathbf A = \begin{bmatrix}
    2 & 0 \\
    4 & 6
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    1 & 5 \\
    2 & 7
\end{bmatrix}$$`

`$$\LARGE \mathbf A - \mathbf B = \begin{bmatrix}
    1 & -5 \\
    2 & -1
\end{bmatrix}$$`

---
## Review of matrix algebra

#### Matrix multiplication

- Dimensions are specified as **rows** by **columns**

- Matrix multiplication does not require matrices to have the same dimensions

- But they must have the same **inner** dimension

+ for example, a 3x3 matrix can be multiplied by a 3x1 matrix

`$$\LARGE \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;3\times3\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;3\times1$$`
`$$\LARGE \mathbf A = \begin{bmatrix}
    a & b & c \\
    d & e & f \\
    g & h & i 
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    x \\
    y \\
    z
\end{bmatrix}$$`

---
## Review of matrix algebra

#### Matrix multiplication

- Dimensions are specified as **rows** by **columns**

- Matrix multiplication does not require matrices to have the same dimensions

- But they must have the same **inner** dimension

+ but a 3x3 matrix **cannnot** be multiplied by a 1x3 matrix

`$$\LARGE \;\;\;\;\;\;\;\;\;\;\;\;3\times3\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;1\times3$$`
`$$\LARGE \mathbf A = \begin{bmatrix}
    a & b & c \\
    d & e & f \\
    g & h & i 
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    x & y & z\\
\end{bmatrix}$$`

---
## Review of matrix algebra

#### How does matrix multiplication work?

`$$\LARGE \mathbf A = \begin{bmatrix}
    a & b & c \\
    d & e & f \\
    g & h & i 
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    x \\
    y \\
    z
\end{bmatrix}$$`
<br/>

`$$\LARGE \mathbf A = \begin{bmatrix}
    a\times x + b\times y + c\times z\\
    d\times x + e\times y + f\times z\\
    g\times x + h\times y + i\times z
\end{bmatrix}$$`

---
## Review of matrix algebra

#### How does matrix multiplication work?

`$$\LARGE \mathbf A = \begin{bmatrix}
    1 & 0 & 5 \\
    0 & 4 & 3 \\
    2 & 6 & 0 
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    3 \\
    2 \\
    1
\end{bmatrix}$$`
<br/>
<br/>

`$$\LARGE \mathbf A = \begin{bmatrix}
    1\times 3 + 0\times 2 + 5\times 1\\
    0\times 3 + 4\times 2 + 3\times 1\\
    2\times 3 + 6\times 2 + 1\times 1
\end{bmatrix}=\begin{bmatrix}
    8 \\
    11 \\
    18
\end{bmatrix}$$`

---
## Review of matrix algebra

#### Transpose of a matrix

`$$\LARGE \mathbf B = \begin{bmatrix}
    3 \\
    2 \\
    1
\end{bmatrix},\;\; \mathbf B^T = \begin{bmatrix}
    3 & 2 & 1
\end{bmatrix}$$`
<br/>
<br/>

`$$\LARGE \mathbf C = \begin{bmatrix}
    3 & 2 & 1
\end{bmatrix},\;\; \mathbf C^T = \begin{bmatrix}
    3\\
    2\\
    1
\end{bmatrix}$$`

---
## Review of matrix algebra

#### Transpose of a matrix

- is the following allowed? 
`$$\LARGE \mathbf A = \begin{bmatrix}
    1 & 0 & 5 \\
    0 & 4 & 3 \\
    2 & 6 & 0 
\end{bmatrix},\;\;
\mathbf B = \begin{bmatrix}
    3 & 2 & 1
\end{bmatrix}$$`
<br/>
<br/>
`$$\Huge A \times B$$`

---
## Review of matrix algebra

#### Transpose of a matrix

---
class: inverse, center, middle

# Age-structured matrix models

---
## Age-structured matrix models

#### How many individuals will be in the population next year?

`$\LARGE n_{1,t+1} = m_1P_0n_{1,t}+m_2P_0n_{2,t} + m_3P_0n_{3,t}$`

`$\LARGE n_{2,t+1} =P_1n_{1,t}$`

`$\LARGE n_{3,t+1} =P_2n_{2,t}$`

---
## Age-structured matrix models

#### How many individuals will be in the population next year?

---
## Age-structured matrix models

#### How many individuals will be in the population next year?

---
## Age-structured matrix models

#### How many individuals will be in the population next year?

---
## Age-structured matrix models

#### How many individuals will be in the population next year?

---
## Age-structured matrix models

#### How many individuals will be in the population next year?

---
## Age-structured matrix models

#### Rather than separately modeling fecundity and juvenile survival, Leslie matrices often include **recruitment**

- `$\Large F_x$`: Recruitment (or sometimes **F**ertility)

- `$\Large F_x = m_xP_0$`

`$$\LARGE \mathbf A = \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix}$$`

---
## Age-structured matrix models

####  Recruitment doesn't have to be estimated from the life table

#### Birds:

- Clutch size

- Nest survival

- Chick survival

- Juvenile survival

`$$\LARGE F = cs \times ns \times chs \times js$$`

`$$\LARGE \mathbf A = \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix}$$`

---
## Age-structured matrix models

####  Recruitment doesn't have to be estimated from the life table

#### Plants:

- Seed production

- Seed survival

- Germination rate

- Seedling survival

`$$\LARGE F = sp \times sds \times gr \times sls$$`

`$$\LARGE \mathbf A = \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix}$$`

---
## Projecting abundance

#### The Leslie matrix can be used to project abundance

`$$\LARGE \mathbf N_t = \begin{bmatrix}
    n_{1,t}\\
    n_{2,t}\\
    n_{3,t}
\end{bmatrix}$$`
<br/>

---
## Projecting abundance

#### The Leslie matrix can be used to project abundance

--
- multiply the Leslie matrix

`$$\Large \mathbf A = \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix}$$`

--
- by the abundance matrix

`$$\Large \mathbf N_t = \begin{bmatrix}
    n_{1,t}\\
    n_{2,t}\\
    n_{3,t}
\end{bmatrix}$$`

using the rules of matrix multiplication

---
## Projecting abundance

#### Population abundance is projected through time using matrix multiplication

`$$\Large \mathbf N_{t+1} = \mathbf A \times \mathbf N_t$$`

`$$\Large \begin{bmatrix}
    n_{1,t+1}\\
    n_{2,t+1}\\
    n_{3,t+1}
\end{bmatrix}\Large =  \begin{bmatrix}
    F_1 & F_2 & F_3\\
    P_1 & 0 & 0\\
    0 &P_2 & 0
\end{bmatrix} \times \begin{bmatrix}
    n_{1,t}\\
    n_{2,t}\\
    n_{3,t}
\end{bmatrix}$$`

<br/>

Notice that the inner dimensions match

---
## Projecting abundance

#### Population abundance is projected through time using matrix multiplication

---
## Projecting abundance

#### Population abundance is projected through time using matrix multiplication

---
## Projecting abundance

#### Population abundance is projected through time using matrix multiplication