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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
Background




   Guild: species with common resource




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




   Guild: species with common resource
   Focus on competition/predation




  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)




  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 stability
   e.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 eels
Bshary 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?




   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)




   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(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)




   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(a12 Y Z + a1 Z − d1 )
                          ˙
                          Y = Y (a21 XZ + a2 Z − d2 )
                          ˙
                          Z = Z(r − e1 X − e2 Y − e12 Y 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 − e1 Xf (Y, α1 ) − e2 Y f (X, α2 )]




   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 − e1 Xf (Y, α1 ) − e2 Y f (X, α2 )]



                             Conditions for mutualism

                                ∂f
                                   = g(X, α2 ) > 0
                                ∂X
                                ∂f
                                   = g(Y, α1 ) > 0
                                ∂Y

   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
                          e1 X ∗ f (Y ∗ , α1 ) + e2 Y ∗ f (X ∗ , α2 ) = r




   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
                          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
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 )]




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




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




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

   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 ) = 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

   Group 3 (ICTP-SAIFR)         Intraguild mutualism    January 22, 2012   14 / 19
Simulations




                           Resource
                          Consumer 1
                          Consumer 2


   Group 3 (ICTP-SAIFR)   Intraguild mutualism   January 22, 2012   15 / 19
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




   Group 3 (ICTP-SAIFR)           Intraguild mutualism           January 22, 2012   16 / 19
Conclusion




    Parameters exist for mutualistic coexistence




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




    Parameters exist for mutualistic coexistence
    Resultant dynamics are oscillatory




   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




   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




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




    Empirical data
    More species
    Realistic network structure




   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 Andr´ Kraenkel
                                            e
                                 Paulo In´cio Prado
                                         a
                               Ayana de Brito Martins
                             Gabriel Andreguetto Maciel
                              Renato Mendes Coutinho
                         Funded by FAPESP and ICTP-SAIFR




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

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Intraguild mutualism

  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 4 / 19
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 5 / 19
  • 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) Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 6 / 19
  • 12. Simulation Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 7 / 19
  • 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) Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 8 / 19
  • 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 )] Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 9 / 19
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 9 / 19
  • 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 )] Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 11 / 19
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 11 / 19
  • 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 ) Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 12 / 19
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 12 / 19
  • 22. 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
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 14 / 19
  • 24. Simulations Resource Consumer 1 Consumer 2 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 15 / 19
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 16 / 19
  • 26. Conclusion Parameters exist for mutualistic coexistence Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 17 / 19
  • 27. Conclusion Parameters exist for mutualistic coexistence Resultant dynamics are oscillatory Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 17 / 19
  • 28. Conclusion Parameters exist for mutualistic coexistence Resultant dynamics are oscillatory Works for any mutualistic functional form Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 17 / 19
  • 29. 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
  • 30. Directions Empirical data More species Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 18 / 19
  • 31. Directions Empirical data More species Realistic network structure Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 18 / 19
  • 32. Directions Empirical data More species Realistic network structure Stochasticity Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 18 / 19
  • 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 Group 3 (ICTP-SAIFR) Intraguild mutualism January 22, 2012 19 / 19