Intraguild mutualism

Carlos H. MantillaFabio Daura-Jorge

Maria Florencia AssaneoMarina Magalhaes da CunhaMurilo Dantas de Miranda

Yangchen Lin

January 22, 2012

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 1 / 19

Background

Guild: species with common resource

Focus on competition/predation

Intraguild mutualism(Crowley & Cox 2011 Trends Ecol. Evol. 26:627–633)

Consequences for biodiversity and stabilitye.g. Gross 2008 Ecol. Lett. 11:929–936

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19

Background

Guild: species with common resource

Focus on competition/predation

Intraguild mutualism(Crowley & Cox 2011 Trends Ecol. Evol. 26:627–633)

Consequences for biodiversity and stabilitye.g. Gross 2008 Ecol. Lett. 11:929–936

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19

Background

Guild: species with common resource

Focus on competition/predation

Intraguild mutualism(Crowley & Cox 2011 Trends Ecol. Evol. 26:627–633)

Consequences for biodiversity and stabilitye.g. Gross 2008 Ecol. Lett. 11:929–936

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19

Background

Guild: species with common resource

Focus on competition/predation

Intraguild mutualism(Crowley & Cox 2011 Trends Ecol. Evol. 26:627–633)

Consequences for biodiversity and stabilitye.g. Gross 2008 Ecol. Lett. 11:929–936

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19

Intraguild mutualism

Crowley & Cox 2011

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 3 / 19

Groupers and moray eelsBshary et al. 2006 PLoS Biol. 4:e431

deardivebuddy.com

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Objectives

Does intraguild mutualism promote consumer coexistence?

General dynamical model (any mutualistic functional form)

Stability analysis and simulation

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 5 / 19

Objectives

Does intraguild mutualism promote consumer coexistence?

General dynamical model (any mutualistic functional form)

Stability analysis and simulation

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 5 / 19

Objectives

Does intraguild mutualism promote consumer coexistence?

General dynamical model (any mutualistic functional form)

Stability analysis and simulation

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 5 / 19

Particular model 1

X = X(a21Y Z + a1Z − d1)

Y = Y (a12XZ + a2Z − d2)

Z = r(S − Z)− (e1Y Z + e2XZ + e12Y ZX)

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Simulation

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 7 / 19

Particular model 2

X = X(a12Y Z + a1Z − d1)

Y = Y (a21XZ + a2Z − d2)

Z = Z(r − e1X − e2Y − e12Y X)

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General model

X = X [Zf(Y, ~α1)− d1]

Y = Y [Zf(X, ~α2)− d2]

Z = Z [r − e1Xf(Y, ~α1)− e2Y f(X, ~α2)]

Conditions for mutualism

∂f

∂X= g(X, ~α2) > 0

∂f

∂Y= g(Y, ~α1) > 0

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 9 / 19

General model

X = X [Zf(Y, ~α1)− d1]

Y = Y [Zf(X, ~α2)− d2]

Z = Z [r − e1Xf(Y, ~α1)− e2Y f(X, ~α2)]

Conditions for mutualism

∂f

∂X= g(X, ~α2) > 0

∂f

∂Y= g(Y, ~α1) > 0

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 9 / 19

Fixed points

Z∗f(Y ∗, ~α1)− d1 = 0

Z∗f(X∗, ~α2)− d2 = 0

e1X∗f(Y ∗, ~α1) + e2Y

∗f(X∗, ~α2) = r

(0, 0, 0) (0, Y ∗, Z∗) (X∗, 0, Z∗) (X∗, Y ∗, Z∗)

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 10 / 19

Fixed points

Z∗f(Y ∗, ~α1)− d1 = 0

Z∗f(X∗, ~α2)− d2 = 0

e1X∗f(Y ∗, ~α1) + e2Y

∗f(X∗, ~α2) = r

(0, 0, 0) (0, Y ∗, Z∗) (X∗, 0, Z∗) (X∗, Y ∗, Z∗)

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 10 / 19

Fixed point (X∗, Y ∗, Z∗)

0 X∗Z∗g(Y ∗, ~α1) X∗f(Y ∗, ~α1)

Y ∗Z∗g(X∗, ~α2) 0 Y ∗f(X∗, ~α2)

Z∗[−e1f(Y ∗, ~α1)− Z∗[−e1X∗g(Y ∗, ~α1)− 0

e2Y∗g(X∗, ~α2)] e2f(X

∗, ~α2)]

−λ3 = mλ− b

λ0 < 0⇒ R(λ1) = R(λ2) > 0

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 11 / 19

Fixed point (X∗, Y ∗, Z∗)

0 X∗Z∗g(Y ∗, ~α1) X∗f(Y ∗, ~α1)

Y ∗Z∗g(X∗, ~α2) 0 Y ∗f(X∗, ~α2)

Z∗[−e1f(Y ∗, ~α1)− Z∗[−e1X∗g(Y ∗, ~α1)− 0

e2Y∗g(X∗, ~α2)] e2f(X

∗, ~α2)]

−λ3 = mλ− b

λ0 < 0⇒ R(λ1) = R(λ2) > 0

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 11 / 19

Fixed point (X∗, 0, Z∗)

0 X∗Z∗g(0, ~α1)X∗d1Z

0 Z∗f(X∗, ~α2)− d2 0

−Z∗e1f(0, ~α1) −e1Z∗X∗g(0, ~α1)− 0

e2Z∗f(X∗, ~α2)

λ = Z∗f(X∗, ~α2)− d2

f(X∗, ~α2)

d2>f(0, ~α1)

d1

Fixed point (0, Y ∗, Z∗)f(Y ∗, ~α1)

d1>f(0, ~α2)

d2

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Fixed point (X∗, 0, Z∗)

0 X∗Z∗g(0, ~α1)X∗d1Z

0 Z∗f(X∗, ~α2)− d2 0

−Z∗e1f(0, ~α1) −e1Z∗X∗g(0, ~α1)− 0

e2Z∗f(X∗, ~α2)

λ = Z∗f(X∗, ~α2)− d2

f(X∗, ~α2)

d2>f(0, ~α1)

d1

Fixed point (0, Y ∗, Z∗)f(Y ∗, ~α1)

d1>f(0, ~α2)

d2

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 12 / 19

Stability characteristics

λ1 > 0 ∧ λ2 > 0⇒ coexistence

λ1 < 0 ∧ λ2 > 0⇒ competitive exclusion

λ1 < 0 ∧ λ2 < 0⇒ bistability

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Simulations

f(Y ∗, ~α1) = α12Y + α1

f(X∗, ~α2) = α21X + α2

Parameter Value

r 0.5d1 0.1d2 0.08e1 1.002e2 1.001α1 0.01α2 0.02α12 0.0009α21 0.00026

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Simulations

ResourceConsumer 1Consumer 2

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Behaviour

0.0 0.2 0.4 0.6 0.8 1.0r

0.02

0.04

0.06

0.08

0.10

Mutualism Strenght

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Conclusion

Parameters exist for mutualistic coexistence

Resultant dynamics are oscillatory

Works for any mutualistic functional form

Intraguild mutualism may enhance stability

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Conclusion

Parameters exist for mutualistic coexistence

Resultant dynamics are oscillatory

Works for any mutualistic functional form

Intraguild mutualism may enhance stability

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 17 / 19

Conclusion

Parameters exist for mutualistic coexistence

Resultant dynamics are oscillatory

Works for any mutualistic functional form

Intraguild mutualism may enhance stability

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 17 / 19

Conclusion

Parameters exist for mutualistic coexistence

Resultant dynamics are oscillatory

Works for any mutualistic functional form

Intraguild mutualism may enhance stability

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 17 / 19

Directions

Empirical data

More species

Realistic network structure

Stochasticity

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 18 / 19

Directions

Empirical data

More species

Realistic network structure

Stochasticity

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 18 / 19

Directions

Empirical data

More species

Realistic network structure

Stochasticity

Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 18 / 19

Acknowledgements

Roberto Andre KraenkelPaulo Inacio Prado

Ayana de Brito MartinsGabriel Andreguetto MacielRenato Mendes Coutinho

Funded by FAPESP and ICTP-SAIFR

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