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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.
