Models & Density-independent growth

class: center, middle, inverse, title-slide

.title[
# Models & Density-independent growth
]
.author[
### Christopher Moore
]
.date[
### Week 02
]

---

### Discrete
* Variables in the model change at predefined, constant increments
* Generations can be *non-overlapping*
* There is a delay between "events"
* Commonly used for periodic (e.g., seasonal) systems
---

## Parts of dynamical models
### State variables: change over time
* Examples in non-ecological models include: concentration, position, temperature, current, allele frequency, GDP
* Examples in ecological models include: abundance, density, position, yield, resource concentration
--

***
### Parameters: constants that generally don't change over time
* Examples in non-ecological models: slection coefficients, distance, viscosity, mutation, decay
* Examples in ecological models: births, deaths, strength of density dependence, sign (e.g., predators always have a - effect on prey)
---

## General anatomy of a dynamical model
### State variable(s)
* For this course, we are focused on abundance or density, commonly abbreviated as `$N$`
* Time will be abbreviated as `$t$`

### Generally
$$ \frac{\Delta N}{\Delta t} = f(N) $$
--

#### Some conventions
* For discrete-time models we use `$N_t$`
* For continuous-time models we use `$N(t)$`. Although for this class, and generally throughout the ecological literature (including in the book), we simply write `$N$`

---
## Anatomy of a dynamical model: discrete time
* General dyanmic model
$$f(N) = \frac{\Delta N}{\Delta t} = $$
--

* Geometric growth: constant growth, discrete time 
`$$N_{t+1} = \lambda N_t$$`

* Writing a function in `R`

```r
geom_growth <- function(Nt, lambda) {
*Nt1 <- lambda*Nt
 return(Nt1)
}
```

* An example with `$N_t = 1$` and `$\lambda = 1.1$`:

```r
geom_growth(Nt = 1, lambda = 1.1)
```

```
## [1] 1.1
```

---
## Anatomy of a dynamical model: continuous time
* General dyanmic model
$$f(N) = \frac{\Delta N}{\Delta t} = $$
--

* Exponential growth: constant growth, continuous time 
`$$\frac{\mathrm{d}N}{\mathrm{d}t} = rN$$`

--
* Writing a function in `R`

```r
exp_growth <- function(Nft, r) {
* dNdt <- r*Nft
 return(dNdt)
}
```

--
* An example with `$N = 1$` and `$r = 0.1$`:

```r
exp_growth(Nft = 1, r = 0.1)
```

```
## [1] 0.1
```

---
## Continuous and discrete time (with equations)
### Continous
* Variables in the model change all of the time
* Generations are *overlapping*
* There is no inherent delay between "events"
* No built-in seasonality
`$$\frac{\mathrm{d}N}{\mathrm{d}t} = f(N)$$`

---
# The simplest models: constant change
.pull-left[
Geometric growth for `$\lambda = 1.1$` and `$N_1 = 2$`:
$$ \vphantom{\frac{dN}{dt}} \lambda N_t $$
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-5-1.png" width="300" height="300" />
]

.pull-right[
Exponential growth for `$r = 0.1$` and `$N(1) = 2$`:
$$ \frac{\mathrm{d}N}{\mathrm{d}t} = rN $$

<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-6-1.png" width="300" height="300" />
]

---
# Note that continous time is always greater than discrete time
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-7-1.png" width="400" height="400" style="display: block; margin: auto;" />

---
# The simplest models: constant change
.pull-left[
Geometric growth for `$\lambda = 0.75$` and `$N_1 = 10$`:
$$ \vphantom{\frac{dN}{dt}} N_{t+1} = \lambda N_t $$
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-8-1.png" width="300" height="300" />
]

.pull-right[
Exponential growth for `$r = -0.25$` and `$N(1) = 10$`:
$$ \frac{\mathrm{d}N}{\mathrm{d}t} = rN $$

<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-9-1.png" width="300" height="300" />
]
---
# Note that continous time is always greater than discrete time
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-10-1.png" width="400" height="400" style="display: block; margin: auto;" />
---
class: inverse, center, middle
# So I have a model, what do I do with it?

---
# What do to with models (including density-independent growth and everything we wil cover for this course)

* Analyze it!
    * Simulate it
    * Solve it
    * Find long-term behavior
    * Determine behavior if perturbed
    * Make predictions
  
--

* Vizualize it!
 * Plot a time series of `$t$` and population size ( `$N$` )
 * Plot population size ( `$N$` ) and population growth `$\frac{\Delta N}{\Delta t}$`
 * Plot population size ( `$N$` ) and growth per individual (per-capita) `$\frac{1}{N}\frac{\Delta N}{\Delta t}$`
 
---
# Analyze it!: simulation
.pull-left[
**Code and output** for discrete-time growth (simulating it):

```r
max_time <- 20
timesteps <- 1:100
N <- vector(mode = "numeric", length = max_time)
N[1] <- 2
lambda <- 1.1
for (i in 2:max_time) {
 N[i] <- lambda*N[i-1]
}
round(N[1:3], 2)
## [1] 2.00 2.20 2.42
round(N[(max_time - 2):max_time], 2)
## [1] 10.11 11.12 12.23
```
]

.pull-right[
**Code and output** for continuous-time growth (not simulating it, but using the solution [simulating differential equations is a bit more involved]):

```r
N0 <- 2
r <- 0.1
timesteps <- seq(0, 20, by = 1)
len_time <- length(timesteps)
N <- N0*exp(r*timesteps)
round(N[1:3], 2)
## [1] 2.00 2.21 2.44
round(N[(len_time - 2):len_time], 2)
## [1] 12.10 13.37 14.78
```
]

---
# Analyze it!: solve it

.pull-left[
**Discrete time**:
`$$\begin{align}
N_{t+1} &= \lambda N_t \\
N_{t+2} &= \lambda N_{t+1} \\
N_{t+2} &= \lambda (\lambda N_t) \\
N_{t+2} &= \lambda ^2 N_t \\
N_{t+3} &= \lambda ^3 N_t \\
... \\
N_{t} &= \lambda ^t N_0
\end{align}$$`
]
--
.pull-right[
**Continuous time**:
`$$\begin{align}
\frac{\mathrm{d}N}{\mathrm{d}t} &= rN \\
\frac{1}{N}\mathrm{d}N &= r \mathrm{d}t \\
\int_{N(0)}^{N(T)} \frac{1}{N}\mathrm{d}N &= \int_{0}^{T} r \mathrm{d}t \\
\log _e(N)\Big|_{N(0)}^{N(T)} &= rt\Big|_0^T \\
\log _e\left(\frac{N(T)}{N(0)}\right) &= rT \\
N(T) &= N(0)e^{rT}
\end{align}$$`
]

---
# Analyze it!: Find long-term behavior, determine behavior if perturbed, make predictions
Not too much happening with this simple, linear model:
* Long-term behavior: continues to grow or shrink at a constant rate
* Determinted behavior if perturbed: continues to grow or shrink at a constant rate
* Make predictions: continues to grow or shrink at a constant rate
---
# Vizualize it!: Plot a time series (already did it)
.pull-left[
Geometric growth for `$\lambda = 1.1$` and `$N_1 = 2$`:
$$ \vphantom{\frac{\mathrm{d}N}{\mathrm{d}t}} N_{t+1} = \lambda N_t $$
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-13-1.png" width="300" height="300" />
]

.pull-right[
Exponential growth for `$r = 0.1$` and `$N(1) = 2$`:
$$ \frac{\mathrm{d}N}{\mathrm{d}t} = rN $$

<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-14-1.png" width="300" height="300" />
]
---
# Vizualize it!: Plot population growth against density
.pull-left[
Geometric growth for `$\lambda = 1.1$` and `$N_1 = 2$`:
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-15-1.png" width="300" height="300" />
]
--
.pull-right[
Exponential growth for `$r = 0.1$` and `$N(1) = 2$`:
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-16-1.png" width="300" height="300" />
]
---
# Vizualize it!: Plot per-capita growth against density
.pull-left[
Geometric growth for `$\lambda = 1.1$` and `$N_1 = 2$`:
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-17-1.png" width="300" height="300" />
]
--
.pull-right[
Exponential growth for `$r = 0.1$` and `$N(1) = 2$`:
<img src="Wk02Models-DensIndGrowth_files/figure-html/unnamed-chunk-18-1.png" width="300" height="300" />
]

---
# A note about an assumption of *r*
### `$\lambda$` and `$r$` are the difference between births and deaths

Specific to our discrete-time density-independent models, this means:
`$$N_{t+1} = \lambda N_t = \left(B - D\right)N_t = BN_t - DN_t$$`

And the solution would be:
`$$N_t = N_0(B - D)^t$$`

Specific to our continous-time density-independent models, this means:
`$$\frac{\mathrm{d}N}{\mathrm{d}t} = rN = (b-d)N$$`

And the solution would be:
`$$N(t) = N(0)e^{t(b - d)}$$`