This document analyzes how intraguild mutualism, where species within a guild provide benefits to each other, can promote coexistence of consumer species. A general dynamical model is developed and analyzed to show that under certain parameter conditions, mutualism can allow for the stable coexistence of both consumer species. Simulations demonstrate oscillatory coexistence for a specific functional form of mutualism. The findings suggest intraguild mutualism may enhance stability within ecological communities.
1. Intraguild mutualism
Carlos H. Mantilla
Fabio Daura-Jorge
Maria Florencia Assaneo
Marina Magalh˜es da Cunha
a
Murilo Dantas de Miranda
Yangchen Lin
January 22, 2012
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 1 / 19
2. Background
Guild: species with common resource
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19
3. Background
Guild: species with common resource
Focus on competition/predation
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19
4. Background
Guild: species with common resource
Focus on competition/predation
Intraguild mutualism
(Crowley & Cox 2011 Trends Ecol. Evol. 26:627–633)
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19
5. 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 stability
e.g. Gross 2008 Ecol. Lett. 11:929–936
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 2 / 19
6. Intraguild mutualism
Crowley & Cox 2011
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 3 / 19
7. Groupers and moray eels
Bshary et al. 2006 PLoS Biol. 4:e431
deardivebuddy.com
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8. Objectives
Does intraguild mutualism promote consumer coexistence?
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 5 / 19
9. Objectives
Does intraguild mutualism promote consumer coexistence?
General dynamical model (any mutualistic functional form)
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 5 / 19
10. Objectives
Does intraguild mutualism promote consumer coexistence?
General dynamical model (any mutualistic functional form)
Stability analysis and simulation
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11. Particular model 1
˙
X = X(a21 Y Z + a1 Z − d1 )
˙
Y = Y (a12 XZ + a2 Z − d2 )
˙
Z = r(S − Z) − (e1 Y Z + e2 XZ + e12 Y ZX)
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12. Simulation
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13. Particular model 2
˙
X = X(a12 Y Z + a1 Z − d1 )
˙
Y = Y (a21 XZ + a2 Z − d2 )
˙
Z = Z(r − e1 X − e2 Y − e12 Y X)
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14. General model
˙
X = X [Zf (Y, α1 ) − d1 ]
˙
Y = Y [Zf (X, α2 ) − d2 ]
˙
Z = Z [r − e1 Xf (Y, α1 ) − e2 Y f (X, α2 )]
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15. General model
˙
X = X [Zf (Y, α1 ) − d1 ]
˙
Y = Y [Zf (X, α2 ) − d2 ]
˙
Z = Z [r − e1 Xf (Y, α1 ) − e2 Y f (X, α2 )]
Conditions for mutualism
∂f
= g(X, α2 ) > 0
∂X
∂f
= g(Y, α1 ) > 0
∂Y
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16. Fixed points
Z ∗ f (Y ∗ , α1 ) − d1 = 0
Z ∗ f (X ∗ , α2 ) − d2 = 0
e1 X ∗ f (Y ∗ , α1 ) + e2 Y ∗ f (X ∗ , α2 ) = r
Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 10 / 19
17. Fixed points
Z ∗ f (Y ∗ , α1 ) − d1 = 0
Z ∗ f (X ∗ , α2 ) − d2 = 0
e1 X ∗ f (Y ∗ , α1 ) + e2 Y ∗ 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
18. Fixed point (X ∗ , Y ∗ , Z ∗ )
X ∗ Z ∗ g(Y ∗ , α1 ) X ∗ f (Y ∗ , α1 )
0
Y ∗ Z ∗ g(X ∗ , α ) 0 Y ∗ f (X ∗ , α2 )
2
∗
Z [−e1 f (Y ∗ , α1 )− Z ∗ [−e1 X ∗ g(Y ∗ , α1 )− 0
e2 Y ∗ g(X ∗ , α2 )] e2 f (X ∗ , α2 )]
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19. Fixed point (X ∗ , Y ∗ , Z ∗ )
X ∗ Z ∗ g(Y ∗ , α1 ) X ∗ f (Y ∗ , α1 )
0
Y ∗ Z ∗ g(X ∗ , α ) 0 Y ∗ f (X ∗ , α2 )
2
∗
Z [−e1 f (Y ∗ , α1 )− Z ∗ [−e1 X ∗ g(Y ∗ , α1 )− 0
e2 Y ∗ g(X ∗ , α2 )] e2 f (X ∗ , α2 )]
−λ3 = mλ − b
λ0 < 0 ⇒ R(λ1 ) = R(λ2 ) > 0
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20. Fixed point (X ∗ , 0, Z ∗ )
X ∗ d1
0 X ∗ Z ∗ g(0, α1 )
Z
0 Z ∗ f (X ∗ , α2 ) − d2 0
−Z ∗ e1 f (0, α1 ) −e1 Z ∗ X ∗ g(0, α1 )− 0
e2 Z ∗ f (X ∗ , α2 )
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21. Fixed point (X ∗ , 0, Z ∗ )
X ∗ d1
0 X ∗ Z ∗ g(0, α1 )
Z
0 Z ∗ f (X ∗ , α2 ) − d2 0
−Z ∗ e1 f (0, α1 ) −e1 Z ∗ X ∗ g(0, α1 )− 0
e2 Z ∗ f (X ∗ , α2 )
λ = Z ∗ f (X ∗ , α2 ) − d2
f (X ∗ , α2 ) f (0, α1 )
>
d2 d1
f (Y ∗ , α1 ) f (0, α2 )
Fixed point (0, Y ∗ , Z ∗ ) >
d1 d2
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23. Simulations
f (Y ∗ , α1 ) = α12 Y + α1
f (X ∗ , α2 ) = α21 X + α2
Parameter Value
r 0.5
d1 0.1
d2 0.08
e1 1.002
e2 1.001
α1 0.01
α2 0.02
α12 0.0009
α21 0.00026
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24. Simulations
Resource
Consumer 1
Consumer 2
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25. Behaviour
Mutualism Strenght
0.10
0.08
0.06
0.04
0.02
r
0.0 0.2 0.4 0.6 0.8 1.0
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26. Conclusion
Parameters exist for mutualistic coexistence
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27. Conclusion
Parameters exist for mutualistic coexistence
Resultant dynamics are oscillatory
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28. Conclusion
Parameters exist for mutualistic coexistence
Resultant dynamics are oscillatory
Works for any mutualistic functional form
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29. 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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30. Directions
Empirical data
More species
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31. Directions
Empirical data
More species
Realistic network structure
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32. Directions
Empirical data
More species
Realistic network structure
Stochasticity
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33. Acknowledgements
Roberto Andr´ Kraenkel
e
Paulo In´cio Prado
a
Ayana de Brito Martins
Gabriel Andreguetto Maciel
Renato Mendes Coutinho
Funded by FAPESP and ICTP-SAIFR
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