Interactions Part II: Predation

class: left, top, title-slide

.title[
# Interactions Part II: <strong>Predation</strong>
]
.subtitle[
## <a href="https://eligurarie.github.io/EFB370/">EFB 370: Population Ecology</a>
]
.author[
### <strong>Dr. Elie Gurarie</strong>
]
.date[
### April 3, 2023
]

---

## Competition ....

.darkorange[ 
`$${dN_1 \over dt} = r_1 N \left(1-{N_1 \over K_1} - \alpha {N_2 \over K_1}\right)$$`
]

.darkblue[ 
`$${dN_2 \over dt} = r_2 N \left(1-{N_2 \over K_2} - \beta {N_1 \over K_2}\right)$$`
]

is the outcome of 2 density dependent phenomena:

1. Intra-specific competition  ($K_1$ and `$K_2$`)
2. Inter-specific competition  ($\alpha$ and `$\beta$`)

Patterns depend on the *scale* and *particulars* of the competition intensity.  
]

---

## Focus on the the isoclines of zero growth

.pull-left[
.darkorange[$$N_2  = {K_1 \over \alpha} - {1\over \alpha} N_1$$]
.darkblue[$$N_2 = K_2 - \beta N_1$$]
]

.pull-right[

![](Interactions_PartII_CompetitionAndPredation_files/figure-html/unnamed-chunk-2-1.png)

]

---

## Extinction vs. Equilibria

.pull-left[

If:

- `${1\over\beta} < {K_1 \over K_2} > \alpha$`; .darkorange[ `$N_1$` ] wins
- `${1\over\beta} > {K_1 \over K_2} < \alpha$`; .darkblue[ `$N_2$` ] wins
- `${1\over\beta} < {K_1 \over K_2} < \alpha$`; .darkorange[ `$N_1$` ] or .darkblue[ `$N_2$` ] wins
- `${1\over\beta} > {K_1 \over K_2} > \alpha$`; .darkorange[ `$N_1$` ] and .darkblue[ `$N_2$` ] coexist

Equilibria (stable or not) at:

- `$N^*_1 = {K_1 - \alpha K_2 \over 1-\alpha\beta}$`
- `$N^*_2 = {K_2 - \beta K_1 \over 1-\alpha\beta}$`

]

.pull-right[

![](Interactions_PartII_CompetitionAndPredation_files/figure-html/unnamed-chunk-3-1.png)
]

---

##  Co-existence essentially occurs when ....

.pull-left[

**Intra**-specific > **Inter**-specific competition

`$${1\over\beta} > {K_1 \over K_2} > \alpha$$`

`$\beta$` and `$\alpha$` small, in a very particular way relative to the ratio of carrying capacities. 
]

.pull-right[
<center>
<img src='images/CrissCrossStable.png' width='100%'/>
</center>
]

---
class: inverse

# **Predation**

.center[<img src='images/fawn.jpg' width='70%'/>]

---
class: inverse

## Predation

.pull-left[
![](images/fawn.jpg)
]

.pull-right.large[

an ecological process where one organism (the predator) consumes another (the prey).

- Provides most of the principle route of energy flow through ecosystems

- Strong selective pressure

- **Chief source of density dependent effects** in regulation of many animal (and plant) populations
]

---

## Predation is *any* transfer of energy up a trophic level!

.pull-left.large[
- **Herbivory:** animals eat plants

- **granivores:** eat grain

- **frugivores:** eat fruit

- **Parasitism:** animals eat other animals without (immediately) killing them

- **Carnivory:** animals eat other animals, killing them

- **Cannibalism:** animals eat their conspecifics
]

.pull-right[

### Basic principles (to model)

.large[
- Growth in **predator** population depends on the number of **prey**

- Growth in **prey** population depends on the number of **predators**

- As an extension of competition

- An extreme form of competition
]
]

---

.pull-left-40[
![](images/asiaticcheetah.png)
![](images/persianleopard.png)
]

.pull-right[

## Intra-guild predation

![](images/skull.png)
![](images/cheetahleopardtitle.png)
]

---

## Start with competition ...

.pull-left-40[

.darkorange[$${dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right)$$]

.darkblue[$${dV \over dt} = r_v V \left(1-{V \over K_v} - \beta {P \over K_v}\right)$$]
]

.pull-right-60[

![](Interactions_PartII_CompetitionAndPredation_files/figure-html/unnamed-chunk-4-1.png)

]

---

## And ADD some extra terms

**Species 1** (competitor AND predator `$P$`) benefits:

.darkorange[ 
`$${dP \over dt} = r_p P \left(1-{P \over K_p} - \alpha {V \over K_p}\right) + \gamma VP$$`
]

**Species 2** (competitor AND prey `$V$`) suffers:

.darkblue[
`$${dV \over dt} = r_v V \left(1-{V \over K_v} - \beta { P\over K_v}\right) - \sigma V P$$`
]

- `$\gamma$` (gamma) = **conversion factor** - how much does predators population benefit from eating prey?

- `$\sigma$` (sigma) = **capture efficiency** - how much does prey population suffer from getting eaten?

---

## Inspect the isoclines!

.pull-left[

`$$P^* = K_p - \left(\alpha - {\gamma K_p \over r_p}\right) V$$`

`$$V^* = K_v - \left(\beta + {\sigma K_v \over r_v}\right) P$$`

Qualitatively

- `$P$` isocline slope becomes STEEPER (and y - intercept increases).

- `$V$` isocline also becomes STEEPER (and x-intercept decreases)

]

.pull-right-40[
![](images/rotatingisoclines.png)
]

---

## Rotating isocline

**Equilibrium shifts UP toward the predator `$K_1$` and DOWN from prey `$K_2$`.**

See: https://egurarie.shinyapps.io/isoclines/

<!--
## Let's toss out competition: {.columns-2}

![](Interactions_PartII_CompetitionAndPredation_files/figure-html/unnamed-chunk-5-1.png)

-->

---

## The Lotka-Volterra Predator-Prey Model

That was complicated!   Let's simplify RADICALLY and eliminate ALL competition and density dependence:

`$$\Large \alpha = \beta = 0; K_v = K_p = \infty$$`

.pull-left-40[

.darkorange[$$\large {dP \over dt}  = -q P + \gamma VP$$]

.darkblue[$$\large {dV \over dt}  = r V - \sigma VP$$]
]

.pull-right[
**Predators** are always dying at rate `$q$`, BUT grow in propotion to **prey**.

**Prey** grow exponentially at rate `$r$`, BUT die off in proportion to **predator**.
]

---

## Assumptions

.large[

Lots!  And mainly unrealistic!

- Standard continuous modeling stuff:

- No age structure
  - Closed population 
  - No time lag

- Instantaneous **Birth** responses of **P** to **V** and **Death** responses to **V** to **P**

- **Mass action**:  perfect population mixing in proportion to number of individuals

**BUT** it's still a fun model to play with - and a good starting point.  
]

---

.center[
 `$P$` - predators |  `$V$` - prey
 
 `$q$` - mortality rate | `$\gamma$` - "conversion" efficiency
]

.pull-left[
### .darkorange[Prey dynamics]

.darkorange[ `$$\Large {dV \over dt}  = r V - \sigma VP$$`]

No density dependence `$K$`, No competition  `$\alpha P$`, only intrinsic growth rate `$r$` and removal by `$P$` at rate `$\sigma$`.

- Large `$\sigma$` - strong effect: each **moose** killed by each **wolf** has big impact

- Small `$\sigma$` - weak effect: **bats** and  **mosquitoes** ... hard for bats to make an impact. 
]

.pull-right[

### .darkblue[Predator dynamics]

.darkblue[ $$\Large {dP \over dt}  = - qP + \gamma VP $$]

- Constant mortality; Growth depends entirely on Prey

- High `$\gamma$`: high dependency on each discrete prey item, e.g. 1 .green[**anaconda**] eats 1 .darkred[**capybara**]

- Low `$\gamma$`: low dependency on each discrete prey item, e.g. 1 .blue[**blue whale**] eats 1 .red[**krill**]. 
]

---

## Obtain equilibria

.pull-left[

### .darkorange[Prey] equilibrium occurs when

.darkorange[$$\Large P^* = {r\over \sigma}$$]

- Number of predators associated with 0 growth in prey population

- ratio of **prey growth** to predator **capture efficiency**

- i.e. **inputs**/**outputs**

]

.pull-right[

### .darkblue[Predator] equilibrium occurs when

.darkblue[$$\Large V^* = {q\over \gamma}$$]

- Number of prey associated with 0 growth in predator population

- ratio of predator **death rate** to **conversion**

- i.e. **outputs**/**inputs**
]

---

.pull-left[
## Prey Isoclines

]

.pull-right[
## Predator Isoclines

<img src="Interactions_PartII_CompetitionAndPredation_files/figure-html/unnamed-chunk-7-1.png" width="100%" />
]

---

.pull-left[
## Graphical analysis

![](images/LVtrajectories.png)
]

.pull-right[
## Numerical analysis

.center[https://egurarie.shinyapps.io/predatorprey/]
]

---

.pull-left[
## How does this look?

- Persistent oscillations with no convergence

- 1/4 cycle out of phase
  - `$P_{max}$` and `$P_{min}$` occur when `$V$` is at median
  - `$V_{max}$` and `$V_{min}$` occur when `$P$` is at median
  
2 exceptions:

- equilibrium point
  - extreme starting point

]

.pull-right[

![](images/LVexample2.png)

]

---

## Very famous Snowshoe Hare and Lynx dataset

Based (mainly) on fur sales from the Hudson Bay Company in Canada over 100 years. Roughly a 9 to 11 year, fairly synchronous, cycle.

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

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

.center[**But are these really predator prey oscillations!?**]

---

## Are predator prey oscillations real?

A theoretical wrinkle ... add *any* carrying capacity to the model:

- https://egurarie.shinyapps.io/predatorprey/

- https://egurarie.shinyapps.io/isoclines/

---

## Are predator prey oscillations real?

... add *any* carrying capacity to the model:

![](images/LVexampleWithCC.png)

And we lose all our oscillations!

---

.pull-left-70[
## Are predator prey oscillations real?

Back to the very controlled laboratory environment ...

![](images/UnicellularPredation.jpg)

*Didinium nausutum* vs. *Paramecium caudatum*.
]

.pull-right-30[

![](images/Gause.jpg)

Georgiy Frantsevich Gause
</center>
]

---

## First attempt

Just put them together and see what happens

... **Maybe one oscillation?**

---

## Second attempt

Add a little prey refuge which *Didinium* can't reach.

**Paramecia hide until all predators die off, then rebound.**

---

## Third attempt

Dribble in 1 *Paramecium* and 1 *Didinium* into the dish every 3 days.

**Conclusion:** Oscillations are not intrinsic to predator prey dynamics, but require some local back-and-forth, immigration - emigration, local dynamics.

---

.pull-left-60[

## Are Hare - Lynx  Lotka-Volterra?

![](images/hare_lynx.png)
]

.pull-right-40[

### Some quibbles

- Lynx peaks *follow* Hare peaks!

- Hare also oscillate where there are no Lynx

- Peaks are synchronized **across** North America

![](images/harelynx_capture.jpg)

]

---
background-image: url("images/StensethBackground.png")
background-size: cover

## Maybe it's food-web structure?

**Hare** cycle with vegetation (bottom up control) and multi-species predation (top-down control), whereas **Lynx** depend strongly just on hare - which solely drive the cycle.

---
background-image: url("images/YanBackground.png")
background-size: cover

## Maybe it's climate?

.pull-left[

Maybe large scale climate oscillations? (sunspots, NAO)

]

---
background-image: url("images/DengBackground.png")
background-size: cover

.pull-left[

## Is it the trappers?

Why do Lynx lag behind Hare?  Maybe the **Hare** eat the **Lynx**? (HEL hypothesis).

Or ... if you add trappers themselves to the equation ... you can reshift the cycle.

]

---

## On balance ....

.pull-left[

- Lotka-Volterra is a nice, but highly theoretical framework ... just too many unrealistic assumptions.

- Like any **null model**, thinking specifically about WHY it breaks down (which is **always**) can lead to real insights.

- Predator-prey phase-spaces is a very useful graphical way to understand dynamic systems.

- More versatile approaches to predation modeling coming next week... 
]

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