Hawkmoths: pollinators that are herbivores

Hawkmoths: pollinators that are herbivores

Julia Oshima, Laila Kazimierski, Lucas S. Simões, Lucas Nascimento, Virgilio Vazquez Hipólito, Yazmín H.

Zurita-Gutiérrez

January 11th, 2015

T

ABLE OF

C

ONTENTS BACKGROUND

The System Biological Facts Objectives

MATHEMATICALMODEL

System of Equations Analytical Analysis RESULTS

Analytical Solutions Simulations

Graphics CONCLUSIONS

BIBLIOGRAPHY

R

EFERENCE

P

APER

T

HE

S

YSTEM

Figure: Hawkmoth Lifecycle

Photo: http://insected.arizona.edu/manduca

T

HE

S

YSTEM

Figure: Model Diagram

B

IOLOGICAL

F

ACTS

I Datura wrightii(P) is highly self-compatible

I Plant leaves are the only food ofManduca sexta larvae(M_L)

I ButM. sextaadults(MA) don’t depend exclusively onP

I Pollinated flowers byM_Aproduce more fruit and seeds

I The floral visitation component of the moth-plant interaction is mutualistic

O

BJECTIVES

Identify the key biological aspects of the system.

Create a minimal model that describes the population dynamics of the systemP-M_L-M_A.

Can this model predict coexistence of a plant and a pollinator that is also its herbivore?

S

YSTEM OF

E

QUATIONS

dP dt =

✓

n+ M_A 1+µM_A

◆ P

✓

1 P

K

◆ aM_LP 1+ahP dML

dt = bMA mLML MLP 1+uP dM_A

dt = M_LP

1+uP m_AM_A

S

YSTEM OF

E

QUATIONS

Setting some parameters:

dP dt =

✓

1+ 3MA

1+M_A

◆

P(1 P) MLP 1+0.3P dM_L

dt = bM_A 0.2M_L M_LP 1+uP dMA

dt = MLP

1+uP 0.2MA

A

NALYTICAL

A

NALYSIS

Now we search for equilibria:

dP

dt = dM_A

dt = dM_L dt =0

P^⇤ = 1

(b mA) mLmA u M^⇤_L =

✓

n+ 3M^⇤_A 1+M^⇤_A

◆

(1+hP^⇤)(1 P^⇤)

M^⇤_A = P^⇤M^⇤_L m_A(1+uP^⇤)

A

NALYTICAL

A

NALYSIS

Psolution:

Gives a condition in order to get a biologically relevant equilibrium:

P^⇤ = 1

(b mA)

mLmA u ) u

< b m_A mLmA

A

NALYTICAL

A

NALYSIS

Figure:MLSolution

A

NALYTICAL

A

NALYSIS

Figure:MASolution

A

NALYTICAL

A

NALYSIS

J = 0 BB

@

P^⇤(1+ _1+M^3M⇤A_⇤

A + _(1+hP^0.3M_⇤^⇤L₎₂) _1+hP^P⇤_⇤ ^3P_(1+M^⇤(1 _⇤^P⇤)

A)² M^⇤_L

(1+uP^⇤)² 0.2 _1+uP^P⇤_⇤ b

M^⇤_L (1+uP^⇤)²

P^⇤

1+uP^⇤ 0.2

1 CC A

G

RAPHICS

Settingb=0.5,u=10 (real part of eigenvalues negative):

0.00 0.25 0.50 0.75 1.00

0 100 200 300

time

value

variable pMLpMA pP

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0.8 0.9 1.0

0.0 0.2 0.4

pML

pP

0 100 200 time300

G

RAPHICS

Settingb=1.4,u=10 (real part of eigenvalues negative):

0.0 0.5 1.0 1.5

0 100 200 300

time

value

variable pMLpMA pP

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0.0 0.2 0.4 0.6 0.8

0.5 1.0 1.5

pML

pP

0 100 200 time300