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

Potential Functional ResponsesPotential Functional ResponsesNet Effect = Gross Benefit - Cost

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

Some simulations…Some simulations…