Predator-prey interactions are prevalent

The building block

Predator-prey interactions are the building blocks of a food web and affect community stability

• Links are species specific (e.g., specialist or generalist)
• Differ in strength (strong and weak interactions)

Predator

• Carnivore
• Herbivore
• Omnivore

They take a variety of predation tactics (active, ambush hunting, etc.)

Prey

• Behavioral avoidance
• Predator satiation (e.g., synchronized emergence)
• Mechanical
• Chemical
• Mimicry

Prey adopt a variety of anti-predator tactics

Prey adaptation

The front door

https://youtu.be/l5Py02wqloM

The back door

https://youtu.be/qbefo_vUzog

The model

Lotka-Volterra predator-prey model

Let \(x\) and \(y\) be prey and predator density

\[ \frac{dx}{dt} = rx - cxy\\ \frac{dy}{dt} = b(cxy) - dy \]

Prey dynamics

Lotka-Volterra predator-prey model

\[ \frac{dx}{dt} = rx - cxy\\ \]

The term \(rx\) represents the population growth of the prey

• \(r\) is the intrinsic population growth rate

Prey dynamics

Lotka-Volterra predator-prey model

\[ \frac{dx}{dt} = rx - cxy\\ \]

The term \(cxy\) represents the predator-prey interaction

• \(xy\) is proportional to the frequency of prey and predator meet
• \(c\) is the rate of predation if they happen to meet

Predator dynamics

Lotka-Volterra predator-prey model

\[ \frac{dy}{dt} = b(cxy) - dy \]

The term \(b(cxy)\) represents the population growth of the predator

• predator population growth depends on prey density \(x\)
• \(b\) is the birth rate or efficiency that predators convert prey to reproduction

Predator dynamics

Lotka-Volterra predator-prey model

\[ \frac{dy}{dt} = b(cxy) - dy \]

The term \(dy\) represents the death rate of the predator

• \(d\) is the death rate

Prediction

Unfortunately, the discrete equivalent does not exist (some closely-related models exist, such as the Nicholson-Bailey model) - we will use R package deSolve to solve the equations

library(deSolve)

# parameters
param <- c(r = 1,
           b = 0.1,
           c = 0.1,
           d = 0.1)

# initial values for state variables
state <- c(x = 5,
           y = 5)

Prediction

# function
lotka_voltera <- function(t, state, param) {
  with(as.list(c(state, param)), {
    
    # prey
    dx <- r * x - c * x * y
    
    # predator
    dy <- b * (c * x * y) - d * y
    
    list(c(dx, dy))
  })
}

Prediction

times <- seq(0, 100, by = 0.01)

pred <- ode(y = state,
            times = times,
            func = lotka_voltera,
            parms = param)

df_pred <- data.frame(pred)

Prediction

Make figures

1. figure 1 - time on x-axis and prey and predator density on y-axis

2. figure 2 - prey density on x-axis and predator density on y-axis

Food web complexity

Food webs are more complex in nature

• Predators eat multiple prey species
• Interaction strength differs
• and more…

Food web complexity and stability

Does food web complexity incease community stability?

Pimm’s work (1980) looked at if a species deletion causes futher species loss in the system

Pimm 1980, Oikos

Sample food webs

Which of the following is more sensitive to the basal species deletion?

Sample food webs

Which of the following is more sensitive to the top predator deletion?

Complexity destabilizes the system

General conclusion is greater complexity (connectance or number of species) destabilizes the system

Complexity destabilizes the system

Opposing effects of complexity

1. Complexity increases the stability against the basal species removal
   
2. Complexity decreases the stability against the predator removal

1 < 2 - as a result, complexity decreases the overall stability

Food web complexity

There are many studies exploring the stability-complexity relationship

• May’s paradox (May 1974, Stability and Complexity in Model Ecosystems)
  • Develop a random food web (interactions are randomly drawn)
  • Analyze the relationship between community stability* and complexity
  • Food web complexity destabilizes the food web

*the maximum eigenvalue of the community interaction matrix

Food web complexity

The following properties can reverse the stability-complexity relationship

• Dominance of weak interactions (McCann et al. 1998 Nature)
• Adaptive foraging (Kondoh 2003, Science)
• Mutualistic interactions (Mougi and Kondoh 2012, Science)

How do we study predator prey interactions?

Direct survey
Dissect the stomach and see what’s in there

• High resolution
• Reflect short-term diet
• Time consuming

How do we study predator prey interactions?

Stable isotopes

Alternative approach is the use of stable isotopes

The same atomic # (proton), but different neutron #

Carbon \(^{12}C\), \(^{13}C\) and nitrogen \(^{14}N\), \(^{15}N\)

• \(^{12}C\) 98.89%, \(^{13}C\) 1.11%
• \(^{14}N\) 99.63%, \(^{15}N\) 0.37%

Delta expression

Stable istopes are expressed in \(\delta\) values

carbon example:

\[ \delta ^{13}C (\unicode{x2030}) = 1000(\frac{R_{sample}}{R_{standard}} - 1) \]

where \(R = \frac{^{13}C}{^{12}C}\)

Fractionation

Predator cannot convert all the prey into their body tissues - predators lose some through metabolic processes

• respiration
• urine, feces

Excrete lighter isotopes first, then heavier

The ratio in predators differs from that in prey

Fractionation

The change in \(\delta\) values through predator-prey interactions is referred to as trophic enrichment factor* (TEF)

Carbon: \(0.4 \pm 1.3 \unicode{x2030}\)

Nitrogen: \(3.4 \pm 1.0 \unicode{x2030}\)

Carbon may reflect the basal resources, while nitrogen may reflect the vertical position in a food web

Post 2002, Ecology 83: 703-718

*there are other names…

Fractionation

Two sources

Again, predators eat multiple prey items…

How stable isotopes help ditinguish their contributions to the diet?

Two sources

Think caborn isotopes only

\[ \begin{align} \delta ^{13}C (\unicode{x2030}) &= 3.0 = Y &&\text{Predator}\\ \delta ^{13}C (\unicode{x2030}) &= 1.0 = X_1 &&\text{Prey 1}\\ \delta ^{13}C (\unicode{x2030}) &= 3.0 = X_2 &&\text{Prey 2}\\ \end{align} \]

Two sources

Make up \(Y\) with \(X_1\) and \(X_2\)

\[ \begin{align} Y &= \alpha (X_1 + 0.4) + (1-\alpha) (X_2 + 0.4)\\ &\text{where}\\ Y &= 3.0, X_1 = 1.0, and ~ X_2 = 3.0 \end{align} \]

Calculate \(\alpha\)

\(\alpha = ...\)

Mixing model

\(Y\) is the weighted mean of \(X_1\) and \(X_2\)
(\(\alpha\) is the proportional contribution of \(X_1\))

This is the basis of mixing model

Mixing model

General formula

\[ \begin{align} \delta_{j,predator} &= \sum^n_{i=1} \alpha_{i}(\delta_{j,i} + \Delta_j)\\ 1 &= \sum^n_{i=1} \alpha_{i} \end{align} \]

• \(i\) is the prey indicator (1, 2, …, n)
• \(j\) is the element indicator (carbon, nitrogen,…)

Mixing model

Different types of mixing models

• IsoSource\(^1\) (the basic model)
• MixSIR\(^2\) (prior, uncertainties in TEF)
• SIAR\(^3\) (prior, uncertainties in TEF)
• IsoWeb\(^4\) (whole food web analysis)

\(^1\)Phillips and Gregg 2003, Oecologia
\(^2\)Moore and Semmens 2008, Ecology Letters
\(^3\)Parnell et al. 2010, Plos one
\(^4\)Kadoya et al. 2012, Plos one
bold: Bayesian implmentation

Caution

• Low resolution (think if \(X_1\) and \(X_2\) values are very close)
• Reflect long-term diet (turnover time vaies by taxa)

Comparison

Alternative

One drawback of stable isotopes is the low resolution

However, stable isotope ratios provide the integrated measures of a food web

• Niche width
• Food chain length

Niche width

Niche can be defined in a variety of ways

• habitat
• physiology
• food
• …

Stable isotopes can be used as a composite measure of variation in resource use among individuals

Variation in stable isotope signatures

Recall: carbon may reflect the basal resources, while nitrogen may reflect the vertical position in a food web

*Points represent isotope signatures of individuals

Variation in stable isotope signatures

Recall: carbon may reflect the basal resources, while nitrogen may reflect the vertical position in a food web

Layman et al. 2007 Ecology Letters

Variation in stable isotope signatures

Recall: carbon may reflect the basal resources, while nitrogen may reflect the vertical position in a food web

Layman et al. 2007 Ecology Letters

Tottabetsu river

Tottabetsu River, Japan

Ground beetles

Ground beetle - generalist predator

R exercise: install SIAR

Install siar & tidyverse packages

install.packages("siar")
install.packages("tidyverse")

Call packages in the current R session
This will add additional functions to R

library(siar)
library(tidyverse)

functions in packages must be called through library() everytime you open a new R session; otherwise you’ll get error messages

R exercise: read data

Download sample_data.csv and read it into R

dat <- read.csv("data/sample_data.csv")
head(dat)

##         site     date     species   d13C  d15N
## 1 tottabetsu 6/9/2014 grasshopper -28.03 -0.04
## 2 tottabetsu 6/9/2014 grasshopper -27.86  2.53
## 3 tottabetsu 6/9/2014 grasshopper -28.18  0.99
## 4 tottabetsu 6/9/2014 grasshopper -28.24  0.81
## 5 tottabetsu 8/9/2014 grasshopper -26.94  0.23
## 6 tottabetsu 8/9/2014 grasshopper -27.92  1.65

R exercise: data formatting

Check species column

unique(dat$species)

## [1] "grasshopper"  "L_noguchii"   "B_stenoderus"

R exercise: data formatting

Filter rows

# L. nogutii
dat_ln <- filter(dat, species == "L_noguchii")
unique(dat_ln$species)

## [1] "L_noguchii"

# B. stenoderus
dat_bs <- filter(dat, species == "B_stenoderus")
unique(dat_bs$species)

## [1] "B_stenoderus"

R exercise: convex hull

Calculate convex hull with convexhull

x carbon data
y nitrogen data

estln <- convexhull(x = dat_ln$d13C, y = dat_ln$d15N)
estbs <- convexhull(x = dat_bs$d13C, y = dat_bs$d15N)

R exercise: covex hull

Inspect L. noguchii

estln

## $TA
##          [,1]
## [1,] 12.95555
## 
## $xcoords
## [1] -22.10 -25.40 -26.92 -27.46 -25.50 -24.85 -22.23 -22.10
## 
## $ycoords
## [1] 4.32 3.05 4.39 5.13 7.03 6.85 5.57 4.32
## 
## $ind
## [1] 18 29 39 37  5 38  1 18

R exercise: covex hull

Inspect B. stenoderus

estbs

## $TA
##        [,1]
## [1,] 12.545
## 
## $xcoords
## [1] -23.97 -22.94 -27.38 -27.80 -27.33 -26.76 -23.97
## 
## $ycoords
## [1] 6.24 4.25 4.52 5.57 8.63 8.47 6.24
## 
## $ind
## [1] 18  4 30 33 25 21 18

R exercise: visualize

Visualize the convex hulls

# set plot region
plot(0, type = "n",
     xlim = range(dat_ln$d13C, dat_bs$d13C),
     ylim = range(dat_ln$d15N, dat_bs$d15N),
     ylab = "delta N", xlab = "delta C")

# for L.noguchii
points(d15N ~ d13C, data = dat_ln, pch = 19) # add points
polygon(estln$xcoords, estln$ycoords) # draw polygon

# for B.stenoderus
points(d15N ~ d13C, data = dat_bs, pch = 21,
       col = NA, bg = grey(0, 0.2)) # add points
polygon(estbs$xcoords, estbs$ycoords,
        col = grey(0, 0.2), border = grey(0, 0.5)) # draw polygon

R exercise: visualize