# Lecture 6 - Susan Schwinning

```EXPLOITATION
+
Species 2
(predator P)
Species 1
(victim V)
-
Classic predation theory is built upon the idea of time
constraint (foraging theory):
A 24 hour day is divided into time spent unrelated to eating:
social interactions
mating rituals
grooming
sleeping
And eating-related activities:
searching for prey
pursuing prey
subduing the prey
eating the prey
digesting (may not always exclude other activities)
The time constraints on foraging
other
essential
activities
foraging
Foraging time
The time constraints on foraging
Foraging time
other
essential
activities
Handling
time
search
handling
Search
time
Search time: all activities up to the point of spotting the prey
searching
Handling time: all activities from spotting to digesting the prey
pursuing
subduing, killing
eating
(transporting, burying, regurgitating, etc)
digesting
Caveat: not all activities may be mutually exclusive
ex. Digesting and non-eating related activities
The time constraints on foraging
Foraging time
other
essential
activities
Handling
time
eating
pursuing
&
subduing
Search
time
search
eating
pursuit &
time subdue time
Different species will allocate foraging time differently:
Filter feeder:
Sit & wait predator (spider)
eating
waiting
eating
digesting
subduing
Time allocation also depends on victim density and predator status:
Well-fed mammalian
predator:
Starving mammalian
Predator (victims at low dnsity):
eating
eating
pursuing
&
subduing
searching
pursuing searching
&
subduing
The math of predation:
(After C.S. Holling)
ts
th
Total search time per day
Total handing time per day
tmax  ts  th
C.S. (Buzz) Holling
Total foraging time is fixed (or
cannot exceed a certain limit).
1) Define the per-predator capture rate as the number of
victims captured (n) per time spent searching (ts):
n
ts
2) Capture rate is a function of victim density (V). Define a
as capture efficiency.
n
 aV
ts
3) Every captured victim requires a certain time for
“processing”.
th  hn
t  ts  th
n
ts 
aV
th  hn
n
t
 hn
aV
n
aV

t 1  ahV
n/t =
capture rate
n
aV

t 1  ahV
0.1
Capture rate
0.09
0.08
0.07
0.06
0.05
0.04
Capture rate
limited by prey
density and
capture efficiency
0.03
0.02
0.01
0
0
50
Capture rate
limited by
predator’s
handling time.
100
Prey density (V)
150
Damselfly
nymph
(Thompson 1975)
Asymptote: 1/h
Decreasing prey size
The larger the prey, the greater the handling time.
(Thompson 1975)
Three Functional Responses
(of predators with respect to prey abundance):
Holling Type I:
Consumption per predator depends only on
capture efficiency: no handling time constraint.
Holling Type II:
Predator is constrained by handling time.
Holling Type III: Predator is constrained by handling time but
also changes foraging behavior when victim
density is low.
Per predator consumption rate
Type I (filter feeders)
Type II (predator with
significant handling
time limitations)
Type I:
n
 aV
t
Type II
n
aV

t 1  ahV
Type III (predator who
pays less attention to
victims at low density)
victim density
Type III
n
(aV )

2
t 1  (ahV )
2
Holling Type I functional response:
Type I functional response
Daphnia path
Daphnia
(Filter feeder on microscopic
freshwater organism)
Thin algae
suspension
culture
Thick algae
suspension
culture
Holling Type II functional response:
Slug eating grass
Cattle grazing in sagebrush grassland
Holling Type III functional response:
Paper wasp, a generalist predator, eating shield beetle larvae:
The wasp learns to hunt for other prey, when the beetle larvae becomes scarce.
The dynamics of predator prey
systems are often quite complex
and dependent on foraging
mechanics and constraints.
Gause’s Predation Experiments:
Didinium nasutum eats
Paramecium caudatum:
Gause’s Predation Experiments:
1) Paramecium in oat medium:
logistic growth.
2) Paramecium with Didinium in
oat medium: extinction of both.
3) Paramecium with Didinium in
oat medium with sediment:
extinction of Didinium.
A fly and its wasp predator:
Greenhouse whitefly
Laboratory experiment
Parasitoid wasp
(Burnett 1959)
spider mite on its own
with predator in
simple habitat
Spider mites
with predator in complex habitat
(Laboratory experiment)
Predatory mite
(Huffaker 1958)
Azuki bean weevil and parasitoid wasp
(Laboratory experiment)
(Utida 1957)
stoat (Greenland)
collared lemming
lemming
stoat
(Gilg et al. 2003)
Possible outcomes of predator-prey interactions:
1. The predator goes extinct.
2. Both species go extinct.
3. Predator and prey cycle:
prey boom
predator boom
Predator bust
prey bust
4. Predator and prey coexist in stable ratios.
Putting together the population dynamics:
Predators (P):
dP

dt
Victim consumption rate * Victim  Predator conversion efficiency
- Predator death rate
Victims (V):
dV

dt
Victim renewal rate – Victim consumption rate
Choices, choices….
Victim growth assumption:
• exponential
• logistic
Functional response of the predator:
•always proportional to victim density (Holling Type I)
•Saturating (Holling Type II)
•Saturating with threshold effects (Holling Type III)
The simplest predator-prey model
(Lotka-Volterra predation model)
dV
 rV  aVP
dt
dP
 VP  qP
dt
Exponential victim growth in the absence of predators.
Capture rate proportional to victim density (Holling Type I).
Isocline analysis:
dV
r
 0: P 
dt
a
dP
q
 0 :V 
dt


Victim isocline:
Predator isocline:
Predator density
V
q
Victim density
P
r
a
V
Predator density
dV/dt < 0
dP/dt < 0
q

dV/dt < 0
dP/dt > 0
dV/dt > 0
dP/dt < 0
Predator isocline:
Victim isocline:
dV/dt > 0
dP/dt > 0
Victim density
Show me dynamics
P
r
a

Victim isocline:
Predator isocline:
Predator density
V
q
Victim density
P
r
a

Victim isocline:
Preator isocline:
Predator density
V
q
Victim density
P
r
a

Neutrally stable cycles!
Every new starting point has its
own cycle, except the
equilibrium point.
The equilibrium is also neutrally
stable.
Victim isocline:
Preator isocline:
Predator density
V
q
Victim density
P
r
a
dV
 V
 rV 1    aVP
dt
 K
dP
 VP  qP
dt
Logistic victim growth in the absence of predators.
Capture rate proportional to victim density (Holling Type I).
Predator isocline:
Predator density
r
a
Victim density
r
c
Stable Point !
P
Predator and Prey cycle move
towards the equilibrium with
damping oscillations.
V
dV
aVP
 rV 
dt
V D
dP
 VP

 qP
dt V  D
Exponential growth in the absence of predators.
Capture rate Holling Type II (victim saturation).
Predator density
Predator isocline:
r
kD
Victim density
Unstable Equilibrium Point!
Predator and prey move away from
equilibrium with growing oscillations.
P
V
No density-dependence in either
victim or prey (unrealistic model,
but shows the propensity of PP
systems to cycle):
P
V
Intraspecific competition in prey:
(prey competition stabilizes PP
dynamics)
P
V
Intraspecific mutualism in prey
(through a type II functional
response):
P
V
Predators population growth rate (with type II funct. resp.):
dP
 VP

 qP
dt V  D
Victim population growth rate (with type II funct. resp.):
dV
 V  aVP
 rV 1   
dt
 K V D
Predator isocline:
Predator density
Rosenzweig-MacArthur Model
Victim density
If the predator
needs high victim
density to survive,
competition between
victims is strong,
stabilizing the
equilibrium!
Predator isocline:
Predator density
Rosenzweig-MacArthur Model
Victim density
If the predator drives
the victim population
to very low density,
the equilibrium is
unstable because of
strong mutualistic
victim interactions.
Predator isocline:
Predator density
Rosenzweig-MacArthur Model
Victim density
However, there is a
stable PP cycle.
Predator and prey
still coexist!
Predator isocline:
Predator density
Rosenzweig-MacArthur Model
Victim density
The Rosenzweig-MacArthur Model illustrates how the variety
of outcomes in Predator-Prey systems can come about:
1) Both predator and prey can go extinct if the predator is too efficient
capturing prey (or the prey is too good at getting away).
2) The predator can go extinct while the prey survives, if the predator
is not efficient enough: even with the prey is at carrying capacity,
the predator cannot capture enough prey to persist.
3) With the capture efficiency in balance, predator and prey can
coexist.
a) coexistence without cyclical dynamics, if the predator is
relatively inefficient and prey remains close to carrying
capacity.
b) coexistence with predator-prey cycles, if the predators are
more efficient and regularly bring victim densities down below
the level that predators need to maintain their population size.
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