Species 1(victim V)

Species 2(predator P)

+

-

EXPLOITATION

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 interactionsmating ritualsgroomingsleeping

And eating-related activities:

searching for preypursuing preysubduing the preyeating the preydigesting (may not always exclude other activities)

foraging

otheressentialactivities

Foraging time

The time constraints on foraging

otheressentialactivities

search

handling

Foraging time

Handlingtime

Searchtime

The time constraints on foraging

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, killingeating (transporting, burying, regurgitating, etc)digesting

Caveat: not all activities may be mutually exclusive

ex. Digesting and non-eating related activities

otheressentialactivities

search

eating

pursuing &subduing

Foraging time

Handlingtime

Searchtime

eatingtime

pursuit &subdue time

The time constraints on foraging

Different species will allocate foraging time differently:

Filter feeder:

eatingdigesting

Sit & wait predator (spider)

subduing

eating

waiting

Time allocation also depends on victim density and predator status:

Well-fed mammalian predator:

eating

pursuing&

subduing

searching

Starving mammalian Predator (victims at low dnsity):

searching

eating

pursuing&

subduing

The math of predation:(After C.S. Holling)

C.S. (Buzz) Holling

Total search time per dayst

Total handing time per dayht

Total foraging time is fixed (or cannot exceed a certain limit).hs ttt max

1) Define the per-predator capture rate as the number of victims captured (n) per time spent searching (ts):

st

n

2) Capture rate is a function of victim density (V). Define as capture efficiency.

Vt

n

s

3) Every captured victim requires a certain time for “processing”.

hnth

 

hs ttt

hnth

V

nts

hnV

nt

hV

V

t

n

1

n/t = capture rate

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 50 100 150

hV

V

t

n

1

Ca

ptu

re r

ate

Prey density (V)

Capture rate limited by prey density and capture efficiency

Capture rate limited by predator’s handling time.

Damselfly

nymph

(Thompson 1975)

The larger the prey, the greater the handling time.

Decreasing prey size

Asymptote: 1/h

(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

pre

dato

r co

nsum

ptio

n ra

te

victim density

Type I (filter feeders)

Type II (predator with significant handling time limitations)

Type III (predator who pays less attention to victims at low density)

Vtn

Type I:

hV

V

t

n

1

Type II

2

2

)(1

)(

hV

V

t

n

Type III

Daphnia(Filter feeder on microscopic

freshwater organism)

Type I functional response

Thin algae suspensionculture

Daphnia path

Thick algae suspensionculture

Holling Type I functional response:

Slug eating grass Cattle grazing in sagebrush grassland

Holling Type II 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.

Holling Type III functional response:

The dynamics of predator prey systems are often quite complex

and dependent on foraging mechanics and constraints.

Didinium nasutum eats Paramecium caudatum:

Gause’s Predation Experiments:

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

Parasitoid wasp(Burnett 1959)

Laboratory experiment

Spider mites

Predatory mite

spider mite on its own with predator in simple habitat

with predator in complex habitat

(Laboratory experiment)

(Huffaker 1958)

(Laboratory experiment)

Azuki bean weevil and parasitoid wasp

(Utida 1957)

collared lemming stoat

lemmingstoat

(Greenland)

(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 bust predator boom

prey bust

4. Predator and prey coexist in stable ratios.

Putting together the population dynamics:

Predators (P):

Victims (V):

dt

dPVictim consumption rate * Victim Predator conversion efficiency

- Predator death rate

dt

dVVictim renewal rate – Victim consumption rate

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)

Choices, choices….

The simplest predator-prey model(Lotka-Volterra predation model)

VPrVdtdV

qPVPdtdP

Exponential victim growth in the absence of predators.Capture rate proportional to victim density (Holling Type I).

Isocline analysis:

r

Pdt

dV :0

q

Vdt

dP :0

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

P P

reda

tor

isoc

line

:

q

V

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

P P

reda

tor

isoc

line

:

q

V

dV/dt < 0dP/dt > 0

dV/dt > 0dP/dt < 0

dV/dt > 0dP/dt > 0

dV/dt < 0dP/dt < 0

Show me dynamics

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

P P

reda

tor

isoc

line

:

q

V

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

P P

reat

or

iso

clin

e:

q

V

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

P P

reat

or

iso

clin

e:

q

V Neutrally stable cycles!Every new starting point has its own cycle, except the equilibrium point.

The equilibrium is also neutrally stable.

Logistic victim growth in the absence of predators.Capture rate proportional to victim density (Holling Type I).

VPK

VrV

dt

dV

1

qPVPdtdP

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

r

rc

P

V

Stable Point !Predator and Prey cycle move towards the equilibrium with damping oscillations.

Exponential growth in the absence of predators.Capture rate Holling Type II (victim saturation).

DV

VPrV

dt

dV

qPDV

VP

dt

dP

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim

isocli

ne:

rkD

P

V

Unstable Equilibrium Point!Predator and prey move away from equilibrium with growing oscillations.

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.):

qPDV

VP

dt

dP

DV

VP

K

VrV

dt

dV

1

Victim population growth rate (with type II funct. resp.):

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

Rosenzweig-MacArthur Model

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

Rosenzweig-MacArthur Model

If the predator needs high victim density to survive, competition between victims is strong, stabilizing the equilibrium!

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

Rosenzweig-MacArthur Model

If the predator drives the victim population to very low density, the equilibrium is unstable because of strong mutualistic victim interactions.

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

Rosenzweig-MacArthur Model

However, there is a stable PP cycle. Predator and prey still coexist!

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.