Midterm 2 Results

Midterm 2 Results

Highest grade: 43.5Lowest grade: 12Average: 30.9

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)

Tawny owl

Wood mouse

(field observation: England)

(Southern 1970)

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):

Victim consumption rate -> predator birth rate

Constant predator death rate

Victims (V):

Victim consumption rate -> victim death rate

Logistic growth in the absence of predators

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

PP

reda

tor

isoc

line

:

q

V

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

PP

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

PP

reda

tor

isoc

line

:

q

V

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

PP

reat

or

iso

clin

e:

q

V

Victim density

Pre

dato

r de

nsity

Victim isocline:

r

PP

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

rcShow me dynamics

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

Show me dynamics

P

V

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

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

At high density, victim competition stabilizes: stable equilibrium!

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

Rosenzweig-MacArthur Model

At low density, victim mutualism destabilizes: unstable equilibrium!

Victim density

Pre

dato

r de

nsity

Pre

dato

r is

oclin

e:

Victim isocline:

Rosenzweig-MacArthur Model

At low density, victim mutualism destabilizes: unstable equilibrium!

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.