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)→∞).
GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 9 / 20
ResultsDynamics
Figure: Populational dynamics for different combinations of fear and toxicity.
GROUP 6 (ICTP-SAIFR) Weaker predation in well-protected plants SSMB 2017 10 / 20
ResultsPhase space diagrams
Figure: Phase space for h = 0, µ = 0 (left) and for h = 0.4, µ = 0.6 (right).
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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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