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This presentation is the result of a one-week student group work during the Southerm-Summer School on Mathematical-Biology, held in São Paulo, BR, in January 2012, http://www.ictp-saifr.org/mathbio . As a follow-up of the subject pat of the group together with F. Lutscher (Univ. Ottawa) and R.M Coutinho (IFT-UNESP, Brazil) published a paper on ecological complexity on this subject, available at http://www.sciencedirect.com/science/article/pii/S1476945X12000773
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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
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 4 / 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
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)
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 6 / 19
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)
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 8 / 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
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
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 12 / 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
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
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 13 / 19
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
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 14 / 19
Simulations
ResourceConsumer 1Consumer 2
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 15 / 19
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
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 16 / 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
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
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 19 / 19