Weaker predation in well-protected plants

GROUP 6: Angelo, Caroline, Marco, Maria, Rodolfo, Vanessa

International Center for Theoretical PhysicsSouth American Institute for Fundamental Research, Brazil

SSMB 2017

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 1 / 20

Introduction

Plants developed a diverse weaponry against herbivory. Top-downcontrol by predators is also very important to keep the population of manyherbivores in check.

Interestingly, recent works have shown that there could be an indirecteffect of plant’s defense in the behavior of predator-prey dynamics in atri-trophic system.

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Introduction

To adress this question, Kaplan & Thaler (2010) studied a tri-trophicsystem:

Caterpillars of the species Manduca sexta feed on the leaves oftomato plants (Solanum lycopersicum), and are also eaten by stinkbugs (Podisus maculiventris).Jasmonate (hormone that regulates toxicity) levels were varied, andlevels of predation and feeding were evaluated.

Figure: A tobacco hornworm caterpillar and a stink bug predator. [3]

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Paper’s highlights

Concepts:Consumptive effects → actual predation → killingNon-consumptive effects → fear → less movement, less eating

→ death by starvation

Kaplan & Thaler’s observation: The number of caterpillars isregulated both by plant chemical defences and by bugs.→ Predation seemed to be weakened in systems with protectedplants.

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Raised questions

Will increasing levels of toxicity weakens predation?

How is the caterpillar’s ”fear” response affected by different levelsof hormone?

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Assumptions and guidelines

The plant’s toxic defence is a parameter h (fixed plant phenotype)→ regulates a carrying capacity K(1− h)→ herbivores can incorporate the toxic compounds of plants,

making the preys less edible for predators: ↑ h ⇒ ↓ predation

Comsumptive (killing) and non-comsumptive (fear) effects→ killing rate b and fear parameter µ→ ↑ h ⇒↓ plant quality ⇒ malnourished cat. ⇒ cat. less

defensive (attenuated fear effect)

Lotka-Volterra-like predator-prey dynamics

Specific predator and herbivore

Closed system

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Parameters considered:r: growth rateK: carrying capacityh: conc. of hormone expressed / toxicity of the plantµ: fear levelk: killing rate (predation of caterpillars)b: predation rate (bug feeding)d: bug’s death rateα: encounter rate (between prey and predator)

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Suggested Model

Growth Term

dC

dt= rC

(1− C

K(1− h)

)− κBC

(1 + µ)− µαBC

1 + e (1 + h)B

dB

dt= −d(1 + h)B +

bBC

(1 + µ)

Predation Term“Fear” TermDeath term

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 8 / 20

Suggested Model

Growth Term

dC

dt= rC

(1− C

K(1− h)

)− κBC

(1 + µ)− µαBC

1 + e (1 + h)B

dB

dt= −d(1 + h)B +

bBC

(1 + µ)

Predation Term

“Fear” TermDeath term

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 8 / 20

Suggested Model

Growth Term

dC

dt= rC

(1− C

K(1− h)

)− κBC

(1 + µ)− µαBC

1 + e (1 + h)B

dB

dt= −d(1 + h)B +

bBC

(1 + µ)

Predation Term“Fear” Term

Death term

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 8 / 20

Suggested Model

Growth Term

dC

dt= rC

(1− C

K(1− h)

)− κBC

(1 + µ)− µαBC

1 + e (1 + h)B

dB

dt= −d(1 + h)B +

bBC

(1 + µ)

Predation Term“Fear” TermDeath term

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 8 / 20

Suggested Model

Growth Term

dC

dt= rC

(1− C

K(1− h)

)− κBC

(1 + µ)− µαBC

1 + e (1 + h)B

dB

dt= −d(1 + h)B +

bBC

(1 + µ)

Predation Term“Fear” TermDeath term

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 8 / 20

ResultsLotka-Volterra

Figure: Lotka-Volterra dynamics obtained putting aside the effect of fear andplant defenses (µ = 0, h = 0, K(1− h)→∞).

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ResultsDynamics

Figure: Populational dynamics for different combinations of fear and toxicity.

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ResultsPhase space diagrams

Figure: Phase space for h = 0, µ = 0 (left) and for h = 0.4, µ = 0.6 (right).

GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 11 / 20

ResultsFixed Point: P = (B(Bugs), C(Caterpillars))

P ∗1 = (0, 0)

P ∗2 = (0, k(1− h))

P ∗3 =

(A+ C

D,d

b(1 + h)(1 + µ)

)

A =re(1 + h)[k(1− h)b− d(1 + h)(1 + µ)] + kb(1− h)[µα(1 + µ)− κ]

kb(1 + µ)(1− µ)(1)

C =

√A2Kb(1− h)− 4κre(1 + h)[Kb− (1 + h)]

Kb(1− h)(2)

D =2κe(1 + h)

(1 + µ)(3)

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ResultsStability

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

C

EB

U

µ

h

Figure: Stability regions in the parameter space

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ResultsBug and Caterpillar stationary populations

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

µ

0

0.5

1

1.5

2

2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

h

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

µ

0 5 10 15 20 25 30 35 40 45

Figure: Mean populations of bugs (left) and caterpillars (right), after atransient time.

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ResultsHighlights

We observed that predator population decreases with higher levelsof plant defence ⇒ cumulative effect of toxins: ↑↑ h ⇒ extinction.

Though intermediate levels of plant toxicity increase the caterpillarpopulation, highest levels might lead to extinction ⇒ possibleplague control strategy (definitive)

Caterpillars eating protected plants are more prone to take risks,hence the fear effect is attenuated.

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Glories and Miseries of the model

Glories

Simple model, with only two variables

Reproduces the most important mechanisms

Miseries

We are not taking into account the plant dynamics

As it is based on Lotka-Volterra model, predation population levelsaren’t realistic ←− LV model problem

Lack of predictability ?

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Perspectives

Different plant phenotypes could be taken into account

The plant dynamics could also be explored

Futher inquiry: plague control strategies

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Thank You!

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Bibliography

Kaplan, I & Thaler, JS.Plant resistance attenuates the consumptive and non-consumptiveimpacts of predators on preyOikos 119-7: 1105-1113. 2010.

Murray, JD (2003).Mathematical Biology. II. Spatial models and Biomedical Applications.3rd Ed.: 82.

Bug, caterpillar and tobacco plant image:http://www.news.cornell.edu/stories/2012/07/

prey-pay-steep-price-elude-predators

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AppendixIncluding the plant population dynamics

Suppose

dC

dt= C[rPPP + rNP (1− PP )]

(1− C

K

)−α[kPPP + kNP (1− PP )]

BC

1 + µ− µαBC

1 + eαC

dB

dt= −φB + γ (kPPP + kNP (1− PP ))

BC

1 +mudPN

dt= f(PN , C)

dPNP

dt= g(PNP , C)

where

PN + PNP = 1 .

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