population dynamics and mutualism: functional responses of benefits and costs j. nathaniel holland,...
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Population Dynamics and Population Dynamics and Mutualism:Mutualism:
Functional Responses of Benefits and CostsFunctional Responses of Benefits and Costs
J. Nathaniel Holland,
Donald L. DeAngelis,
Judith L. Bronstein
Presented by
Kate Buenau
Generalizations for MutualismGeneralizations for Mutualism
Benefits—can’t be obtained in absence of partner: nutrients, transport, protection
Costs—investments in attracting mutualists, substances to reward them, energy and time to obtain rewards
Both affect reproduction and survival Benefits and costs tend to be density dependent Positive feedback between mutualists Negative feedback needed for stability
The GoalThe Goal
Develop functional responses in terms of benefits and costs as a function of the population of the mutualist partner
General modelCase study of senita cactus
Some Basic ModelsSome Basic Models
General model:Functional responses:
linear:
saturating:
unimodal:
2222
2 gNdNNBdt
dNn
11
11
1 N
NBn
12
12
11
11
11 N
N
N
NBn
amNBn 1
Senita Cacti and MothsSenita Cacti and Moths
Flowers must be pollinated by mothsMoths lay eggs in flowersNot all flowers have fruit, and moth larvae
eats some immature fruitFruit with larvae cannot produce seedsFruit may abort, and eggs and larvae die
(negative feedback)
Benefits and costs to cacti depends upon abundance of moths compared to flower production (M/FP) …unimodal net effect
Benefit: Flowers are pollinated with a saturating function, using a Poisson searching process
21
2
1
exp11
*exp1))(1(
gPPdFP
M
FP
MFPa
dt
dP
The The
ModelModel
FP
M1
FP
MFP 1exp1
Building the equation II…Building the equation II…
Costs (larval infestation) also saturate
(1-a) represents fraction of flowers potentially setting fruit
= fraction of fruit cacti
21
2
1
exp11
*exp1))(1(
gPPdFP
M
FP
MFPa
dt
dP
FP
M2exp11
The Other The Other Side…Side…
Moth benefits: number of flowers effectively oviposited
Costs: fruit abortion (-a)
MdFP
M
FP
MFPa
dt
dM
22
1
exp1*
exp1))(1(
Variation Variation Correlates pollination and
oviposition
Less likelihood of extinction
21
1
2
1
exp1
exp11
*exp1))(1(
gPPd
FPM
FPM
FP
MFPa
dt
dP