Limits on Population Growth

class: center, top, title-slide

.title[
# <strong>Limits on Population Growth</strong>
]
.subtitle[
## <a href="https://eligurarie.github.io/EFB370/">.white[EFB 370: Population Ecology]</a>
]
.author[
### <strong>Dr. Elie Gurarie</strong>
]
.date[
### February 26, 2024
]

---

class:large

.pull-left-70[
## As populations grow ...

### they always hit **Limits to growth**

.box-blue.large[
- Space limits

- Resource limitations

- Competition

- Predation

- Disease 
]

.large[**All of these can and do interact**]
]

.pull-right-30[
![](images/limits.jpg)
]

---

## Fundamental population equation

`$$\huge \Delta N = B - D + I - E$$`
.large[

Exponential growth assumes these (especially .red[**Birth**] & .red[**Death**]) are proportional to .red[**N**].

But at high N ... B can fall, or D can rise, or I can decrease or E can increase.

]

### Density dependence

Means that the *rate* of a parameter, e.g. `$b = {B/N}$` or `$d = {D/N}$` is 
- (a) NOT constant
- (b) dependent on total population (or density) `$N$`

---
class:inverse

.pull-left-30[

![](images/purple-squirrel-deliverance-1.gif)

![](images/oakieisland.png)
]

.pull-right-70.large[
# Meet the Squirlicorn  (*Sciurus monocerus*)
### **Natural History Facts**

**Limited range:** Only found on .lightred[Oakie island] 
{{content}}

]

**Extremely territorial:** Strictly ONE squirlicorn per oak tree. 
{{content}}

**One colonist** ( `$\color{yellow}{N_0 = 1}$` ) appears on Oakie Island, which contains  `$\color{yellow} K$` oak trees 
{{content}}

Reproduces **asexually** with some fixed probability `$\color{yellow} {p_b}$` per year.
{{content}}

Offspring disperses to nearest tree. 
{{content}}

If tree is occupied, existing squirlicorn **assassinates** newcomer with its horn.
{{content}}

Is ***immortal*** (*of course*).

---

.pull-left-70[
## Simulating limits on growth

<html>
<head>
	<meta charset="utf-8" />
	<meta name="generator" content="R package animation 2.7">
	<title>Logistic growth ...</title>
	<link rel="stylesheet" href="css/reset.css" />
	<link rel="stylesheet" href="css/styles.css" />
	<link rel="stylesheet" href="css/scianimator.css" />
  <link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/highlight.js/8.3/styles/github.min.css">

</head>
<body>

</body>
</html>

( note - if you can't see animation try clicking [here](logisticgrowth.html) )
]

.pull-right-30[

### On Oakie Island there are:
- `$K = 64$` trees 
- `$b = 0.2$` offspring / year
]

---

.pull-left[

## Some math

`$$\large E(N_{t+1}) = N_t + b \times N_t \times p_{c}$$`
`$b$` - is birth rate, `$p_c$` - probability of successful colonization

> Remember `$E()$` is *expectation*  (just a way to accept that we're talking about random variables and are interested in their MEAN behavior ... but we'll drop it from here on out).

Key calculation ... the **probability of colonization** is just the **proportion of available trees**:

`$$p_c = (K - N_t) /K$$`
`$K$` is the *number of available trees*. Plugging back in:

`$$N_{t+1} = N_t + b N_t \left(K - N_t \over K \right)$$`
Rewrite in terms of `$\Delta N = N_{t+1} - N_t$`

$$\Delta N = b N_t \left(1 - {N_t \over K} \right) $$
]

.pull-right[

Slightly more generally with a death rate: `$\lambda = b - d$`

`$$\Delta N_t = \lambda N_t \left(1 - {N_t \over K} \right)$$`

Let's just divide both sides by `$\Delta t$`, squeeze time, and replace `$\lambda / \Delta t$` with `$r_0$` (much like in exponential growth) to make this differential equation:

`$${\Delta N_t \over \Delta t} = r_0 N_t \left((1 - {N_t \over K} \right)$$`
`$${dN \over dt} = r_0 N_t \left(1 - {N_t \over K}\right)$$`

This is the:

.center.large.green[**logistic population growth model**]. 
]

---

## Properties and Assumptions

`$$\Large {dN \over dt} = r N_t \left(1 - {N_t \over K}\right)$$`

.pull-left[

The key assumption is that *growth rate* decreases *linearly* with *population*.

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

.pull-right.large[
- At `$N = 0$` - growth = 0

- At `$N$` slightly above 0 - growth `$\approx r_0$`

- At `$N = K$` - growth = `$0$`

- At `$N > K$` - growth < `$0$`
]

---

## Logistic Growth Curves

.pull-left[

Here's how the process looks, at values: `$r_0 = 1$`, `$K = 36$` and a selection of values for `$N_0$`.

![](LogisticGrowth_PartI_files/figure-html/logisticcurves-1.png)
]

.pull-right[

Starts out looking ***Exponential*** (at low `$N_0$`, low values of time)

Typical "S" shape coming from below ... Slows down at `$K$`

Decays to `$K$` if `$N_0 > K$`

> ***K*** is famously known as ....
>
>
> .darkblue.center.large[***Carrying Capacity***]
]

---

## Different models of density dependence

.pull-left-30[

**What is it that depends on density?**

Is it birth?  Is it death?  Is it linear?  Is it curvy?

> What changes for the squirlicorn? Is it linear?

]

.pull-right-70[
![](images/DensityDependence.png)

.center[Fryxell chapter 8.]
]

---

## Density Dependence
.pull-left[

![](images/DifferentFormsOfDD.png)
]

.pull-right.large[
***Any*** relationship where `$r$` or `$\lambda$` depends on `$N$` is called **Density Dependence**.

It does not *have* to be linear. (That's just easiest to model)

These different curves are different **forms** of density dependence. 
]

---

## Typically for large mammals ...

.pull-left-30[

Calf / pup / juvenile mortality is highest when densities are highest.

**Fecundity** (# of offspring per female) falls at high densities...

... but this effect mainly kicks in at very high numbers (not linear).  
]

.pull-right-70.center[

**Fecundity** (offspring / mother)

![](images/FowlerExample.png)

.center[Fowler (1981)]
]

---

###  Most density dependence is curvy

![](images/SiblyExample.png)

.footnote[Sibly, et al. Science 309.5734 (2005): 607-610.]

### .pull-right-70[in fact, most density dependence might be concave]

---

## Continuous time logistic growth curves

.pull-left[
![](LogisticGrowth_PartI_files/figure-html/unnamed-chunk-2-1.png)
]

.pull-right[

- Exponential growth at low `$N_0$`

- Typical "S" shape (**sigmoidal**) coming from below ... Slows down at `$K$`

- Decays to `$K$` if `$N_0 > K$`
]

---

## Exponential vs. Logistic Growth in Three Graphs

.pull-left[
  ***Exponential*** | ***Growth***
  ---:|:---
  differential equation: | `${dN \over dt} = r N$`
  solution: | `$N(t) = N_0e^{rt}$`
]

.pull-right[
  ***Logistic*** | ***Growth***
  ---:|:---
  differential equation: | `${dN \over dt} = r_0 N\left(1-{N\over K}\right)$`
  solution: | `$N(t) = {K \over 1 + \left({K-N_0 \over N_0}\right)e^{-r_0 t}}$`
]

---

## Properties

Inflection point - where rate of growth is maximum - occurs at  `$N(t^*) = K/2$`. 
`$$t^* = {1\over r} \log\left({K- N_0 \over N_0}\right)$$`
<img src="LogisticGrowth_PartI_files/figure-html/ThreeFigures2-1.png" width="100%" />

---

.pull-left-30[
## Pierre François Verhulst

]

.pull-right-70[

***Sur la loi d'accroissement de la population*** - 1844
<center>
<img src='images/Verhulst_04.png' width='70%'/> <br>
<img src='images/Verhulst_figure.png' width='60%'/>
</center>

At the end of paper makes predictions - based purely on extrapolating from the inflection point:

*The extreme limit of the population of France is 40,000,000 souls, and of Belgium is 6,600,000 souls.*

Actual population (in 2020):  65.5 million, and 11.7 million.   **Not bad at all!!**
]

---

## Different functional forms

---

## Discrete logistic growth

Take discrete population growth equation:

`$$N_{t+1} = \lambda N_t$$`

break up into "stable" and "growth" components ( `$\lambda = 1 + r_d$` ):

`$$N_{t+1} = N_t + r_d N_t$$`
add the density dependence "brake" on the growth:

`$$N_{t+1} = N_t + r_d N_t \left(1 - {N_t \over K}\right)$$`

---

## Discrete Logistic: Two formulations

Compute change within a time step:

`$$\Delta N_t = r_d N_t \left(1 - {N_t \over K}\right)$$`

Compute change of a population:

`$$N_{t+1} = N_t + r_d N_t \left(1 - {N_t \over K}\right)$$`

These are *difference* (as opposed to *differential*) equations.  And ... unlike the differential version ... there is NO "analytical" solution of `$N_t$` for any `$N_t$`.

In fact, this little equation **blew up math as we know it!**

> Visit this: https://egurarie.shinyapps.io/DiscreteLogisticMapping/

---

## Unexpected aspects of chaos theory  ....

- It emerged from a basic (and old) *deterministic* population model. 
- It is extremely sensitive to initial conditions
- The patterns are fractal

---

## Very nice video on "logistic mapping"

.center[
<iframe src="https://www.youtube.com/embed/ovJcsL7vyrk?controls=1" width="70%", height = 400>
</iframe>
]

---

.pull-left-50[
## Next Week: Human-Wolf Experiment

-	Dispersal into new area, mainly wolf mating pairs. 
-	Highly territorial! 
-	Wolves produce up to 4 pups per litter that survive
-	If there are no neighbors, wolves will disperse to found new packs
-	Pack with 8 adults or 2 adults, still produces (about) 4 pups per litter 
-	If there are lots of neighbors, packs become larger (more individuals) in smaller territories.

]

.pull-right-50[

**Expansion of Wisconsin Wolves, 1970's to 2000's**

![](images/WI_WolfTerritories.png)
]

---

## Some references

.small[
- Benton, T. G., A. Grant, and T. H. Clutton-Brock. 1995. Does environmental stochasticity matter? Analysis of red deer life-histories on Rum. Evolutionary Ecology 9:559–574.

- Chapman, E. J., and C. J. Byron. 2018. The flexible application of carrying capacity in ecology. Global Ecology and Conservation 13:e00365.

- Fowler, C. W. 1981. Density Dependence as Related to Life History Strategy. Ecology 62:602–610.

- Laidre, K. L., R. J. Jameson, S. J. Jeffries, R. C. Hobbs, C. E. Bowlby, and G. R. VanBlaricom. 2002. Estimates of carrying capacity for sea otters in Washington state. Wildlife Society Bulletin:1172–1181.

- Nowicki, P., S. Bonelli, F. Barbero, and E. Balletto. 2009. Relative importance of density-dependent regulation and environmental stochasticity for butterfly population dynamics. Oecologia 161:227–239.

- Sibly, R. M., D. Barker, M. C. Denham, J. Hone, and M. Pagel. 2005. On the Regulation of Populations of Mammals, Birds, Fish, and Insects. Science 309:607–610.

- Wydeven, A. P., T. R. Van Deelen, and E. J. Heske, editors. 2009. Recovery of Gray Wolves in the Great Lakes Region of the United States: An Endangered Species Success Story. Springer New York, New York, NY.
]