Lecture 12

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# Lecture 12
## Reproductive value, sensitivity, and elasticity
### WILD3810 (Spring 2021)

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## Readings

> Mills 103-109

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## 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?

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## Common teasel example

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## How does initial stage-distribution effect growth?

--
#### All populations reach the same stable stage distribution and have the same `$\Large \lambda_{SSD}$`

--
- But they do not have the same `$\Large N_T$`

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# Reproductive value

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## Reproductive value

> the number of offspring that an individual is expected to contribute to a population over its remaining life span (after adjusting for the growth rate of the population)

#### Intuitive, but complex to compute

--
#### Factors that influence reproductive value:

--
- Expected future reproductive output

--
- Survival probability

--
- Age at maturity

--
- Population growth rate

+ if a population is growing, future offspring will be smaller contribution to `$N$`
    + if a population is shrinking, future offspringwill be larger contribution to `$N$`

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## Reproductive value

#### Teasel example

- Dormant seeds, year 1: `$\Large 1.00$`

- Dormant seeds, year 2: `$\Large 0.04$`

- Small rosette: `$\Large 9.19$`

- Medium rosette: `$\Large 152.4$`

- Large rosette: `$\Large 972.1$`

- Flowering plant: `$\Large 2,633$`

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## Reproductive value

.left-column[
<img src="figs/lobiopa.jpg" width="100%" style="display: block; margin: auto;" />
]

.right-column[
<img src="figs/lobiopa_rv.PNG" width="100%" style="display: block; margin: auto;" />
]

???

Image courtesy of Scott Justis

Figure from Greco et al. (2017) *Life history traits and life table analysis of Lobiopa insularis (Coleoptera: Nitidulidae) fed on strawberry* PlosOne

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## Reproductive value

.left-column[
<img src="https://upload.wikimedia.org/wikipedia/commons/e/ea/Forest_elephant.jpg" width="100%" style="display: block; margin: auto;" />
]

.right-column[
<img src="figs/elephant_rv.PNG" width="100%" style="display: block; margin: auto;" />
]

???

Image courtesy of dsg-photo.com, via Wikicommons

Figure from Turkalo et al. (2018) *Demography of a forest elephant population* PlosOne
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# Population inertia

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## Population inertia

> difference between the long-term population size of a population that experiences transient dynamics and the long-term population size of a population that grows at the SSD

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## Population inertia

#### Grey wolf population matrix

`$$\mathbf A = \begin{bmatrix}
    0 & 0.44 & 0.62 & 0.62 & 0.62 & 0.62 & 0.62 & 0.62 & 0.22\\
    0.69 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
    0 & 0.77 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\
    0 & 0 & 0.77 & 0 & 0 & 0 & 0 & 0 & 0\\
    0 & 0 & 0 & 0.77 & 0 & 0 & 0 & 0 & 0\\
    0 & 0 & 0 & 0 & 0.77 & 0 & 0 & 0 & 0\\
    0 & 0 & 0 & 0 & 0 & 0.77 & 0 & 0 & 0\\
    0 & 0 & 0 & 0 & 0 & 0 & 0.77 & 0 & 0\\
    0 & 0 & 0 & 0 & 0 & 0 & 0 & 0.55 & 0.37
\end{bmatrix}$$`

--
`$$\large \lambda_{SAD} = 1.097$$`

--
#### Reproductive values: 
`$$\large [1.00, 1.59, 1.69, 1.61, 1.49, 1.31, 1.07, 0.72, 0.30]$$`

???

Adapted from Carroll et al. (2003) *Cons Biol*

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## Population inertia

#### What is the projected growth if reintroduced population starts at SAD?

.left-column[
#### SAD:
`$$\large \begin{bmatrix}
    0.33\\
    0.21\\
    0.15\\
    0.10\\
    0.07\\
    0.05\\
    0.04\\
    0.03\\
    0.02
\end{bmatrix}$$`
]

.right-column[
 
 
<img src="Lecture12_files/figure-html/unnamed-chunk-11-1.png" width="468" style="display: block; margin: auto auto auto 0;" />
]

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## Population inertia

#### What is the projected growth if reintroduced population starts with all 9 year olds?

.left-column[
`$$\large \begin{bmatrix}
    0\\
    0\\
    0\\
    0\\
    0\\
    0\\
    0\\
    0\\
    1
\end{bmatrix}$$`
]

.right-column[
 
 
<img src="Lecture12_files/figure-html/unnamed-chunk-12-1.png" width="468" style="display: block; margin: auto auto auto 0;" />
]

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## Population inertia

#### What is the projected growth if reintroduced population starts with all 3 year olds?

.left-column[
`$$\large \begin{bmatrix}
    0\\
    0\\
    1\\
    0\\
    0\\
    0\\
    0\\
    0\\
    0
\end{bmatrix}$$`
]

.right-column[
 
 
<img src="Lecture12_files/figure-html/unnamed-chunk-13-1.png" width="468" style="display: block; margin: auto auto auto 0;" />
]

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## Population inertia and reproductive value in practice

#### What age/stage distribution will lead to largest `$\large N$` for introduced population?

#### What age/stage should be the target for reducing invasive species?

#### What age/stage distribution can be harvested to maintain stable population of game species?

???
Image courtesy of USFWS Mountain-Prairie via WikiCommons

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# Sensitivity and elasticity

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## Sensitivity

#### Remember that `$\Large \lambda$` is determined by the birth and death rates of a population

`$$\Large \lambda = 1 + b - d$$`

--
#### As managers, we might want to increase or decrease `$\Large \lambda$` of certain species by manipulating age-specific `$\large b$` and `$\large d$` rates

#### But `$\lambda$` does not respond equally to all vital rates

- in some cases, a small change in adult survival may result in a large change in `$\large \lambda$`

- in other cases, a small change in fecundity or juvenile survival may result in a large change in `$\large \lambda$`

#### Which vital rates should we focus our management efforts on?

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## Sensitivity

#### Sensitivity

> the change in `$\large \lambda$` caused by a small change in a vital rate

`$$\Large s_{i,j} = \frac{\delta \lambda}{\delta a_{i,j}} = \frac{v_iw_j}{\sum_{k=1}v_kw_k}$$`

where `$v_i$` and `$w_j$` are the reproductive value and stable stage distribution of stage `$i$`

--
- large reproductive values or large stable stable distribution lead to large sensitivity

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## Sensitivity

.pull-left[

**Common frog (*Rana temporaria*)**

3 stages:

- pre-juvenile (egg - tadpole)

- juvenile (tadpole - 2 years)

- adult (> 2 years)
]

.pull-right[
 
 
 
`$$\Large \mathbf A = \begin{bmatrix}
 0 & 52 & 279.5\\
 0.019 & 0.25 & 0\\
 0 & 0.08 & 0.43\\
\end{bmatrix}$$`

`$$\LARGE \lambda = 1.338$$`
]

???

Image courtesy of H. Krisp, via Wikicommons

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## Estimating sensitivity

#### By hand

`$$\Large a_{2,1} = 0.019, \lambda = 1.338$$`

--
`$$\Large a_{2,1}' = 0.029, \lambda = 1.571$$`

`$$\Large s_{2,1} = \frac{1.571 - 1.338}{0.01} = 23.3$$`
--

#### Analytically

`popbio::sensitivity(A)[2,1] = 26.05`

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## Estimating sensitivity

#### By hand

`$$\Large a_{1,2} = 52, \lambda = 1.338$$`

--
`$$\Large a_{1,2}' = 52.01, \lambda = 1.338$$`

`$$\Large s_{1,2} = \frac{1.338 - 1.338}{0.01} = 0.00$$`
--

#### Analytically

`popbio::sensitivity(A)[1,2] = 0.006`

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## Elasticity

#### Does it make sense to compare a change of 0.01 in a survival value to a change of 0.01 in fecundity?

- 0.01 is about 52% of 0.019

- 0.01 is about 0.02% of 52

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#### Elasticity

> the change in `$\large \lambda$` caused by a small *proportional* change in a vital rate

`$$\Large e_{i,j} = s_{i,j} \bigg[\frac{a_{i,j}}{\lambda}\bigg]$$`

- Elasticity scales sensitivity to account for the magnitude of the vital rate

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## Elasticity

#### Common frog example

##### By hand
`$$\Large e_{2,1} = 26.05 \bigg[\frac{0.019}{1.338}\bigg] = 0.3699$$`

`$$\Large e_{1,2} = 0.006 \bigg[\frac{52}{1.338}\bigg] = 0.2332$$`

##### Analytically

`popbio::elasticity(A)[2,1] = 0.3699`

`popbio::elasticity(A)[1,2] = 0.251`

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## Life history variation

#### Organisms have limited resources to investment between growth, reproduction, and survivorship

#### Evolution selects for different combinations of *life history traits*

> Demographic traits that influence fitness (i.e., `$\lambda$`)

- size at birth  
- growth pattern  
- age at maturity  
- fecundity schedule  
- mortality schedule  
- length of life

#### Trade-offs between traits directly related to sensitivity/elasticity

#### We'll explore variation in life history traits during the next lecture