Limits on Population Growth: Part II

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.title[
# <strong>Limits on Population Growth: Part II</strong>
]
.subtitle[
## <a href="https://eligurarie.github.io/EFB370/">.white[EFB 370: Population Ecology]</a>
]
.author[
### <strong>Dr. Elie Gurarie</strong>
]
.date[
### February 28, 2025
]

---

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# Topics

.Large[

- Wolf carrying capacity example & experiment

- Delayed Density Dependence

- Discrete Logistic Growth

- Delayed Discrete Density Dependence

]

---

.pull-left-50[
## Basics of Wolf population dynamics

-	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)
]

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##  Human-wolf experiment

.pull-left-60[

### basics of model
- 8 possible territories
- 1 initial dispersing wolf (female)

### each season ...
- One female / pack gives birth to 2 offspring
- Offspring can choose whether to disperse or not 
- about 1/4 of all wolves die each year
]

.pull-right-40[
![](images/wolfmother.jpg)
]

---

.pull-left-60[

###  Summary Concepts

Growth of natural populations is always *eventually* limited

When population rates (*b*, *d*, also *i*, *e*) depend on the **total population** or **density** (***N***), this is called: .darkred[***Density Dependence***].

The maximum growth rate ( `$r_0  = max (b-d)$`) is called the .darkred[**intrinsic growth rate**].

In density dependent growth, the .darkred[actual growth rate (***r***)] falls with higher density until it is 0.  The point where that happens is the .darkred[***Carrying Capacity*** (**K**)].  If ***N*** ever exceeds ***K***, the growth becomes negative.

.darkred[***Logistic growth***] is a specific kind of .darkred[Density Dependent] growth where the relationship between ***r*** and ***N*** is .darkred[**linear**]. The formula is:
`$$r = r_0 (1 - N/K)$$`
which leads to the following differential equation:
`$${dN \over dt} = r_0 N \left( 1 - {N \over K}\right)$$`

]
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## *Delayed* Density Dependence

*Sometimes* (or often?) the *density dependent response* kicks in with a *time lag*.

.pull-left[

**Logistic Growth**

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

]

.pull-right[

**Logistic Growth** .red[**with delay**]

`$$\Large {dN \over dt} = r N_t \left(1 - {N_{t - \color{red}{\tau}} \over K}\right)$$`

]
  
> Can you think of examples?

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## simulation (sadly - not mine)

.center[https://demonstrations.wolfram.com/DelayLogisticEquation/]

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## Delayed Logistic Growth

.pull-left-30[

The *greater* the delay (`$\tau$`), the *greater* the **amplitude** of oscillation.

Oscillations are defined by:

- **amplitude**
- **period**

.green[**mathemagically:** the period is always `$= 4\tau$`  
]

If `$$r \times \tau > 1.57$$`   
the oscillation attains a **stable limit cycle**

]

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

"*stable limit cycle: Like a faulty heating system that always overheats & overcools*"
]

---

## 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:

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

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## 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/

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## 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

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## Very nice video on "logistic mapping"

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

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# .Huge[More "Chaotic" dynamics]

.center[![](images/bifurcation.gif)]

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.pull-left-60[
## Single population dynamics: Small Rodents

Many, relatively easy long-term studies of small-mammal populations.

Observed that populations fluctuate with higher amplitude and different periods / randomness across latitudinal gradients.

Norway lemming *(Lemmus lemmus)* Tundra vole *(Microtus oeconomus)*
]

.pull-right-30[

Turchin and Hanski (1997)
]

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## Delayed Density Dependence

`$$\log(N_t) = a_0 + a_1 \log(N_{t-1}) + a_2 \log(N_{t-2})$$`

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## Lots of data and lots of debate

.pull-left-30[

Are we observing deterministic **chaos** or are we observing **stochasticity**?

What *IS* chaos in a stochastic world?

> But ... methodologically ... it is ALL done by fitting **simple linear models**! 
]

.pull-right-70.center[
<img src='images/Stenseth_megafigure.png' width = "100%">
Stenseth (1999) *Oikos*
]

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## Top 10 subsection heading of all time

.pull-left-60[
![](images/StensethQuote.png)
]

.pull-right-40.large[

> **My conclusion:**
>
> Argumentative or not, vole/lemming population ecologists certainly have a high **self-regard**!

]

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## Next week (and beyond) ....

We blow up: `$$\huge N_t$$`

.pull-left[
**into:**

>- sex / age classes
>- multiple sub-populations
>- multiple species (competitors / predator-prey)
>- infected, susceptible, recovered
]

.pull-right[

**relying on:**

>- difference and differential equations
>- probability and stochasticity
>- visualization and simulation
>- statistics (esp. linear modeling)
>- ***natural history and biological intuition!***

]